\section{Introduction}

This paper investigates the existence of traveling wave solutions for the heterogeneous reaction-diffusion equation
\begin{equation}\label{eqprinc}
 \partial_t u -\Delta u = f(t,u),\quad x\in\R^N,\ t\in\R,
\end{equation}
where $f=f(t,u)$ vanishes when $u=0$ or $u=1$,
is strictly positive for $u\in(0,1)$ and is of KPP type, that is,
$f(t,u)\leq f_u'(t,0)u$ for $(t,u)\in \R\times [0,1]$.

When $f$ is constant with respect to $t$, we recover the classical Fisher-KPP equation
\begin{equation}\label{eq-hom}
 \partial_t u -\Delta u = f(u), \quad x\in\R^N,\ t\in\R.
\end{equation}
It is well-known (see \cite{AronsonWeinberger, KPP}) that for all
$c\geq c^*=2\sqrt{f'(0)}$, there exists a {\em planar traveling wave}
of speed
$c$ in any direction  $e\in\mathbb{S}^{N-1}$, that is, a solution $u$ of
(\ref{eq-hom}) which can be written as
$u(x,t) = \phi(x\cdot e -ct)$, with $\phi\geq 0$, $\phi(-\infty)=1$ and
$\phi(+\infty)=0$. In this case,  the profile $\phi$ of the planar traveling
wave satisfies the
ordinary differential equation
$$-\phi''-c\phi' = f(\phi) \quad\hbox{in } \R.$$

When $f$ is periodic with respect to $t$, planar traveling waves no longer
exist and the relevant notion is that of
{\em pulsating traveling wave}. Assume for instance
that exists $T>0$ such that $f(t+T,u)= f(t,u)$ for all $(t,u) \in \R\times
[0,1]$. Then a pulsating traveling wave of speed
$c$ in direction $e\in\mathbb{S}^{N-1}$
is a solution $u$ of (\ref{eqprinc}) which can be written as
$u(x,t)=\phi (x\cdot e -ct,t)$, where $\phi\geq 0$, $\phi =\phi (z,t) $ is
$T-$periodic with respect to
 $t$, $\phi (-\infty,t)=1$ and $\phi (+\infty,t)=0$
for all $t$.
The profile $\phi$ then satisfies the time-periodic parabolic equation
$$\partial_t \phi-\partial_{zz}\phi-c\partial_z\phi = f(t,\phi)
\quad\text{in } \R\times\R.$$
The existence of pulsating traveling waves has been proved by Nolen, Rudd and
Xin \cite{NolenRuddXin}
 and the first author \cite{Nadinptf} under various hypotheses
in the more general framework of space-time periodic reaction-diffusion
equations
(see also \cite{Alikakos}
for time periodic bistable equations).
These results yield that, in the particular case of temporally periodic
equations,
a pulsating traveling wave of speed $c$ exists if and only if $c\geq
c^*=2\sqrt{\langle \mu\rangle}$, where
$\langle \mu\rangle=\frac{1}{T}\int_0^T f_u'(t,0)dt$.

Note that the notion of pulsating traveling wave has first been introduced in
the framework of space periodic reaction-diffusion
equations by Shigesada, Kawasaki and Teramoto
\cite{Shigesada} and Xin \cite{Xinperiodic} in parallel ways. Xin \cite{Xinperiodic}, Berestycki and Hamel \cite{BerestyckiHamel}
and Berestycki,
Hamel and Roques \cite{BerestyckiHamelRoques2} proved the existence of such
waves
in space periodic media under various hypotheses. In this case the minimal speed
$c^*$ for the existence of pulsating traveling waves is
not determined using the mean average of $x\mapsto f_u'(x,0)$, but rather by
means of a family of periodic principal eigenvalues to characterize $c^*$.

The case of a time almost periodic and bistable reaction term $f$ has been
investigated by Shen \cite{Shenap1, Shenap2}. A
time heterogeneous nonlinearity is said to be {\em bistable} if
there exists a smooth function $\theta : \R\to (0,1)$ such that
\begin{equation}\label{f-bistable}
\left\{\begin{array}{l}
f(t,0)=f(t,1)=f(t,\theta (t))=0 \ \text{ for }t\in\R,\\
f(t,s)<0 \ \text{ for }t\in\R,\ s\in (0,\theta (t)),\\
f(t,s)>0 \ \text{ for }t\in\R,\  s\in (\theta (t),1).
       \end{array}
\right.
\end{equation}
Shen constructed examples where
there exists no solution $u$ of the form $u(x,t)= \phi (x\cdot e -ct,t)$ such
that $\phi (-\infty,t)=1$, $\phi (+\infty,t)=0$ uniformly with respect to
$t\in\R$ and
$\phi=\phi (z,t)$ is almost periodic with respect to $t$. She proved that the
appropriate notion of wave in almost periodic media incorporates a time
dependence
of the speed $c=c(t)$. Namely, she defined an {\em almost periodic traveling
wave} as a solution $u$ of (\ref{eqprinc}) which can be written as
$u(x,t)= \phi (x\cdot e -\int_0^t c(s)ds,t)$, where $c=c(t)$ and $\phi=\phi
(z,t)$ are smooth functions which are almost periodic with respect to $t$ and
$\phi (-\infty,t)=1$ and $\phi (+\infty,t)=0$ hold uniformly with respect
to $t\in\R$.
She proved that for
nonlinearities $f$ which satisfy  (\ref{f-bistable}), there exists
an almost periodic traveling
wave and that its profile and speed are uniquely determined up to translation of
the profile and to addition of the derivative of a time
almost periodic function to the speed.

\smallskip

In order to handle general heterogeneous equations such as (\ref{eqprinc}), we will use
the notion of (almost planar) generalized transition wave introduced by
Berestycki and Hamel \cite{BerestyckiHamelgtw} and
Shen \cite{Shenbistable}.
This definition is a natural extension of the earlier
notions. In particular, it enables
a dependence of the speed with respect to time, as in almost periodic media.

\begin{defi}\label{def-gtf}
An (almost planar) {\em generalized transition wave} in the direction
$e\in\mathbb{S}^{N-1}$ of equation (\ref{eqprinc}) is a time-global solution $u$
which can be written as
$u(x,t)=\phi (x\cdot e -\int_0^t c(s)ds,t)$, where
$c\in L^\infty(\R)$ is bounded and $\phi : \R\times \R \to [0,1]$
satisfies
\begin{equation}
\lim_{z\to-\infty}\phi(z,t)=1 \ \hbox{ and }\ \lim_{z\to+\infty}\phi(z,t)=0\ \hbox{ uniformly in } t\in\R.
\end{equation}
The functions $\phi$ and $c$
are respectively called the {\em profile} and the
{\em speed} of the generalized transition wave $u$.
\end{defi}

We will only consider almost planar waves in the present paper and thus we will
omit to mention the almost planarity in the sequel. Notice that the speed $c$ and the profile $\phi$ in Definition
\ref{def-gtf} are not univocally determined: they can be replaced by $c+\xi$
and $\phi (z+\int_0^t \xi(s)ds,t)$ respectively, for any $\xi\in L^\infty(\R)$
such
that $t\in \R\mapsto \int_0^t \xi (s)ds$ is bounded.
If $u$ is a generalized transition wave, then its profile $\phi$ satisfies
\begin{equation}\label{def-tf}
\left\{\begin{array}{l}
\partial_t \phi-\partial_{zz}\phi -c(t)\partial_z \phi = f(t,\phi),\quad
z\in\R,\ t\in\R,\\
\displaystyle\lim_{z\to-\infty}\phi(z,t)=1 \ \hbox{ and }\
\lim_{z\to+\infty}\phi(z,t)=0\ \hbox{ uniformly in } t\in\R.\\
\end{array} \right.
\end{equation}

The existence of generalized transition waves has been proved by Shen in the
framework of time heterogeneous bistable
reaction-diffusion equations \cite{Shenbistable}. Berestycki and Hamel \cite{BerestyckiHamelgtw2} also proved the
existence of generalized transition waves
for the monostable equation (\ref{eqprinc}) when $t\mapsto f(t,u)$ converges as
$t\to\pm\infty$ uniformly in $u$
(see Section \ref{sec:examples} below).
Lastly, we learned while we were ending this paper that Shen proved the
existence of generalized transition waves when the coefficients are
uniquely ergodic in a very recent paper \cite{Shenue}. We will describe more precisely the differences between Shen's approach and ours in
Section \ref{sec:thm} below.
These are the only types of temporal heterogeneities for which the existence of generalized transition waves has been proved.

In space heterogeneous media, generalized transition waves have been shown to
exist for ignition type nonlinearity in dimension $1$. Namely, assuming that
$f=g(x)f_0(u)$,
with $g$ smooth, uniformly positive and bounded, and with $f_0$ satisfying
$$
\exists\theta\in (0,1),\qquad
\forall s\in [0,\theta]\cup \{1\},\quad f_0(s)=0,\qquad
\forall s\in (\theta,1),\quad f_0(s)>0.
$$
the existence of generalized transition waves has been obtained
in parallel ways by Nolen and Ryzhik \cite{NolenRyzhik} and Mellet, Roquejoffre
and Sire
\cite{MelletRoquejoffreSire}. These generalized transition waves attract the solutions of the Cauchy problem associated with front-like initial data
\cite{MelletNolenRoquejoffreRyzhik}. Zlatos gave some extensions of these results to multi-dimensional media
and more general nonlinearities \cite{Zlatosdisordered}.
Lastly, Nolen, Roquejoffre, Ryzhik and Zlatos constructed some space
heterogeneous KPP nonlinearities such that
equation (\ref{eqprinc}) does not admit any generalized transition wave
\cite{NRRZ}.
A different generalization of the notion of traveling wave has been introduced
in the framework of bistable reaction-diffusion equations with a small
perturbation in space of the homogeneous equation by Vakulenko and Volpert
\cite{VakulenkoVolpert}.

When the nonlinearity is homogeneous and of ignition or bistable type, we know
that there exists a unique speed associated with a planar traveling wave. The analogue
of this
property in heterogeneous media is that, for bistable or ignition type nonlinearities, the generalized transition wave is unique up to translation in time
(see \cite{BerestyckiHamelgtw2, MelletNolenRoquejoffreRyzhik, Shenbistable}). This means that the speed $t\mapsto c(t)$ of
the generalized transition wave is unique in some sense.

For monostable homogeneous nonlinearities, we know that there exists a unique
speed $c^*$ such that planar traveling waves of speed $c$ exist if and only if
$c\geq c^*$.
Moreover, it is possible to construct solutions which behave as planar traveling
waves with different speeds when $t\to \pm\infty$ (see \cite{HamelNadirashvili})
and one can remark that these solutions are generalized transition waves.
Hence,
we expect a wide range of speeds $t\mapsto c(t)$ to be associated with
generalized transition waves in heterogeneous media.
The aims of the present paper are the following.

\begin{itemize}
\item Prove the existence of generalized transition waves for time heterogeneous
monostable equations.

 \item Identify a set of speeds $t\mapsto c(t)$ associated with generalized transition waves.

\item Apply our results to particular nonlinearities such as random stationary ergodic.
\end{itemize}

\section{Statement of the results}

\subsection{Hypotheses}\label{sec:hyp}

In this paper we just assume the nonlinear
term $f(t,u)$
to be bounded and measurable with respect to $t$. The notion of solution
considered is that of
strong solution: a subsolution (resp.~supersolution) $u$ of
(\ref{eqprinc}) is a function in $W^{2,1}_{N+1,loc}(\R^N\times\R)$
\footnote{ $W^{2,1}_{p}(\mc{Q})$, $\mc{Q}\subset\R^N\times\R$, stands for the
space of functions $u$ such that
$u,\,\partial_{x_{i}}u,\,\partial_{x_i x_j}u,\,\partial_t u\in L^p(\mc{Q})$.}
satisfying
$$\partial_t u -\Delta u\leq f(t,u)
\quad\text{(resp.~$\geq f(t,u)$)},\quad \text{for a.e.~}x\in\R^N,\ t\in\R,$$
and a solution is a function $u$ which is both a sub and a supersolution.
It then follows from the standard parabolic theory that solutions belong to
$W^{2,1}_{p,loc}(\R^N\times\R)$, for all $p<\infty$, and then
they are uniformly continuous by the embedding theorem.

The fundamental hypothesis we make on $f$ is that it is of KPP
type.
Namely, $f(t,u)\leq\mu(t)u$, with $\mu(t):=f_u'(t,0)$, which means that
$f(t,\.)$ lies below its tangent at $0$ for all $t$. Hence,
we expect
the linearization at $u=0$ to play a crucial role in
the dynamics of the equation.

A typical $f$ we want to handle is
\begin{equation} \label{examplef} f(t,u)=\mu (t) u(1-u), \end{equation}
with
$\mu\in L^\infty (\R)$ and $\inf_{t\in\R}\mu(t)>0$.
This function fulfills the following set of hypotheses, which are the ones
we will require in the statements of our results.

\begin{hyp}\label{hyp:KPP}
The function $f=f(t,u)$
satisfies $f(\.,u)\in L^\infty(\R)$, for all
$u\in[0,1]$, and, as a function
of $u$, is Lipschitz continuous in $[0,1]$ and of class $C^1$ in a neighborhood
of
$0$, uniformly with respect to $t\in\R$. Moreover,
setting $\mu(t):=
f_u'(t,0)$, the following properties hold:
\begin{equation}\label{hyp-pos} \text{for a.e.~} t\in\R,\quad
f(t,0)=f(t,1)=0,\qquad
\forall u\in(0,1),\quad \essinf_{t\in\R}f(t,u)>0,
\end{equation}
\begin{equation} \label{hyp-KPP}
\text{for a.e.~}(t,u)\in\R\times [0,1],\quad f(t,u)\leq \mu (t) u,
\end{equation}
\begin{equation} \label{hyp-lower}
\exists C>0,\ \gamma,\delta\in(0,1],\quad \text{for a.e.~}(t,u)\in\R\times
[0,\delta],\quad
f(t,u)\geq \mu (t) u-Cu^{1+\gamma},
\end{equation}
\end{hyp}

Notice that (\ref{hyp-KPP}), (\ref{hyp-lower}) are fulfilled
if
$f(t,0)=0$ and $f(t,\.)$
is respectively concave and in
$C^{1+\gamma}([0,\delta])$, uniformly with respect to $t$.

\subsection{Existence of generalized transition waves for general nonlinearities} \label{sec:thm}

Since the range of speeds associated with planar traveling
waves in homogeneous media is given by $[2\sqrt{f'(0)},+\infty)$, we expect
to find similar constraints on the
speeds of generalized transition waves in heterogeneous media. The
constraint we will exhibit depends on the {\em least mean} of the speed.

\begin{defi}\label{def-mean}
For any given function $g\in L^\infty(\R)$, we define
$$\underline{g}:=\sup_{T>0}\,\inf_{t\in\R}
\frac{1}{T}\int_t^{t+T}g(s)\,ds$$
and we call this quantity the {\em least mean} of the function $g$
(over $\R$).
\end{defi}

The definition of least mean does not change if one
replaces the $\sup_{T>0}$ with $\lim_{T\to+\infty}$ (see Proposition
\ref{reformulation} below). Hence, if $g$ admits a {\em mean value}
$\langle
g\rangle$, i.e., if there
exists
\Fi{mv}
\langle g\rangle:=\lim_{T\to+\infty}
\frac{1}{T}\int_t^{t+T}g(s)\,ds,\quad\text{uniformly with respect to }t\in\R,
\Ff
then $\ul g=\langle g\rangle$.

Notice that (\ref{hyp-pos}) and (\ref{hyp-KPP}) yield $\ul \mu>0$.
We are now in position to state our main result.

\begin{thm}\label{thm-existence}
Assume that $f$ satisfies Hypothesis \ref{hyp:KPP}
and let $e\in\mathbb{S}^{N-1}$.

1) For all $\gamma>2\sqrt{\ul \mu}$,
there exists a generalized transition wave $u$ in direction $e$ with a speed $c$
such that $\ul c= \gamma$
and a profile $\phi$ which is decreasing with respect to $z$.

2) There exists no generalized transition wave $u$ in direction $e$
with a speed $c$ such that
$\ul c< 2\sqrt{\ul\mu}$.
\end{thm}

The speeds we construct have the particular form
\begin{equation} \label{formc}
 c(t)=\frac{\mu(t)}{\kappa}+\kappa, \qquad \kappa\in (0, \sqrt{\ul \mu}).
\end{equation}
Hence, they keep some properties of the
function $\mu$.
In particular, if $\mu$ admits a mean value $\langle \mu \rangle$ then
our result implies that a generalized transition wave with a
speed $c$ such that $\langle c \rangle=\gamma$ exists if $\gamma>2\sqrt{\langle \mu
\rangle}$ and does not exist if $\gamma<2\sqrt{\langle \mu \rangle}$.
Of course, this construction is not exhaustive:
there might exist generalized transition waves with a speed $c$ which cannot be
written in the form (\ref{formc}), as exhibited in Example \ref{E:2speeds}
below.
More generally, trying to characterize the set of
speeds associated with generalized transition waves is a very hard task, since
this notion covers many types of speeds
(see \cite{BerestyckiHamelgtw, HamelNadirashvili}). Indeed, it is still an open
problem even in the case of homogeneous equations.

We did not manage to prove the existence of generalized transition waves with a
speed with least mean $2\sqrt{\ul \mu}$. We leave this extension as an open
problem, that we will discuss in Examples \ref{E:speeds} and \ref{E:minimal} of
Section \ref{sec:examples} below.

Theorem \ref{thm-existence} shows that the range of least means of the
speeds associated with generalized transition waves is a half-line, with
infimum $2\sqrt{\ul\mu}$.
If instead one considers other notions of mean then
the picture is far from being complete: our existence result implies that, for
every notion of mean $\mc{M}$
satisfying
$$\fa g\in L^\infty(\R),\quad
\mc{M}(g)\geq\ul g,\qquad
\fa\alpha,\beta>0,\quad\mc{M}(\alpha g+\beta)=\alpha\mc{M}(g)+\beta,$$
and every $\kappa\in(0,\sqrt{\ul
\mu})$, there exists a wave with speed $c$ satisfying
$\mc{M}(c)=\frac{\mc{M}(\mu)}\kappa+\kappa$, whereas there are no waves with
$\mc{M}(c)<2\sqrt{\ul\mu}$. But if $\mc{M}(\mu)>\ul\mu$ then
$2\sqrt{\ul\mu}<\frac{\mc{M}(\mu)}{\sqrt{\ul\mu}}+\sqrt{\ul\mu}$,
whence there is a gap between the thresholds of the existence and non-existence
results.

\bigskip

In order to conclude this section, we briefly comment the differences between
Shen's approach in \cite{Shenue} and the present one. First, Shen only
considered
uniquely ergodic coefficients. We refer to \cite{Shenue} for a precise definition but we would like to point out that if $\mu$ is uniquely ergodic,
then $\langle \mu \rangle=\lim_{T\to +\infty} \frac{1}{T}\int_{t}^{t+T}\mu (s)ds$ exists uniformly with respect to $t\in\R$.
This hypothesis is quite restrictive since it excludes, for example, general random stationary ergodic coefficients. Under this hypothesis, Shen proves
that a
generalized transition wave with a uniquely ergodic speed $c=c(t)$ satisfying
$\langle c \rangle = \gamma$ exists if and only if
$\gamma\geq 2\sqrt{\langle \mu\rangle}$. This is a slightly stronger result than
ours since it provides existence in the critical case $\gamma= 2\sqrt{\langle
\mu \rangle}$. Lastly,
Shen's approach is different since she uses a dynamical systems setting, while
we use a PDE approach inspired by \cite{BerestyckiHamel}.

\subsection{Application to random stationary ergodic equations}

We consider the reaction-diffusion equation with random nonlinear term
\begin{equation}\label{eq-random}
 \partial_t u -\Delta u = f(t,\omega,u),\quad
x\in\R^N,\ t\in\R,\ \omega\in\Omega.
\end{equation}
The function $f:\R\times \O\times [0,1]\to \R$ is a random function defined on
a probability space $(\O,\P,\mathcal{F})$. We assume that $(t,u)\mapsto
f(t,\omega,u)$
satisfies Hypothesis \ref{hyp:KPP} for almost every
$\omega\in\O$, and in addition that
\begin{equation} \label{hyp-vois1}
\fa t\in\R,\quad u\mapsto f(t,\omega,u)/u \hbox{ is nonincreasing in }
[1-\delta,1],
\end{equation}
where the constants $\gamma$ and $\delta$ depend on $\omega$.
Notice that (\ref{hyp-vois1})
is satisfied in particular when $u\mapsto f(t,\omega,u)$
is nonincreasing in $(1-\delta,1)$, since $f$ is positive there.
We further suppose that $f(t,\omega,u)$ is a stationary ergodic random
function with respect to $t$. Namely, there exists a group $(\pi_t)_{t\in\R}$ of
measure-preserving transformations of $\O$
such that
$$\forall (t,s,\omega,u)\in\R\times
\R\times \O\times [0,1],\quad
f(t+s,\omega,u)=f(t,\pi_s \omega,u),$$
and for all $A\in\mathcal{F}$, if $\pi_t A=A$ for all $t\in\mathbb{R}$, then
$\P(A)=0$ or $1$.

A generalization of the notion of traveling waves for equation (\ref{eq-random})
has been given by Shen in \cite{Shenrandom}.

\begin{defi}\label{def-randomtw} {\bf (see \cite{Shenrandom}, Def. 2.3)}
A {\em random transition wave} in the direction $e\in\mathbb{S}^{N-1}$ of
equation (\ref{eq-random}) is a
function $u:\R^N\times \R \times \Omega \to [0,1]$
which satisfies:
\begin{itemize}
 \item There exist two
bounded
measurable functions $\tilde{c}:\Omega \to \R$ and
$\tilde{\phi}:\R \times\Omega\to[0,1]$ such that $u$ can be written as
$$u(x,t,\omega) = \tilde{\phi}(x\cdot e -\int_0^t \tilde{c}(\pi_s
\omega)ds,\pi_t \omega) \hbox{ for all } (x,t,\omega)\in \R^N\times
\R\times\Omega.$$
\item For almost every $\omega\in\Omega$, $(x,t)\mapsto u(x,t,\omega)$ is
a solution of (\ref{eq-random}).
\item For almost every
$\omega\in\Omega$, $\displaystyle\lim_{z\to-\infty}\tilde{\phi}(z,\omega)=1$ and
$\displaystyle\lim_{z\to+\infty}\tilde{\phi}(z,\omega)=0$.
\end{itemize}
The functions $\tilde{\phi}$ and $\tilde{c}$ are respectively called the
{\em random profile} and the {\em random speed} of the random transition
wave $u$.
\end{defi}

Notice that if (\ref{eq-random}) admits a generalized transition wave for
a.e.~$\omega\in\O$, and the associated profiles $\phi(z,t,\omega)$ and speeds
$c(t,\omega)$ are stationary ergodic with respect to $t$, then the functions
$\t\phi(z,\omega):=\phi(z,0,\omega)$ and $\t c(\omega):=c(0,\omega)$ are the
profile and the speed of a random transition wave.

The existence of random transition waves has been proved in the framework of
space-time random stationary ergodic bistable nonlinearities by Shen
\cite{Shenrandom} and in the
framework of space random stationary ergodic ignition type nonlinearities by
Nolen and Ryzhik \cite{NolenRyzhik} (see also \cite{Zlatosdisordered} for some
extensions).

Starting from Theorem \ref{thm-existence}, we are able to characterize the
existence of random transition waves in terms of the least mean of their speed.
For a stationary ergodic function $g:\R\times\O\to\R$, the least mean of
$t\mapsto g(t,\omega)$ is independent of $\omega$, for every $\omega$ in a set
of probability $1$ (see Proposition \ref{pro:lmergodic} below). We call this
quantity the least mean of the random function $g$, and we denote it by $\ul g$.

\begin{thm}\label{thm-existencerandom}
Let $e\in\mathbb{S}^{N-1}$. Under the previous hypotheses,
for all $\gamma>2\sqrt{\ul\mu}$,
there exists a random transition wave $u$ in direction $e$ with random speed $\t
c$ such that $c(t,\omega):=\tilde{c}(\pi_t\omega)$ has least mean $\gamma$,
and a random profile $\t\phi$ which is decreasing with respect to $z$.
\end{thm}

This result is not an immediate corollary of Theorem \ref{thm-existence}.
In
fact, for given $\kappa\in (0,\sqrt{\ul \mu})$ and almost every $\omega\in\O$,
Theorem \ref{thm-existence} provides a generalized transition wave
with speed $c(\cdot,\omega)$ satisfying
$c=\frac{\mu}{\kappa}+\kappa$.
Hence, $c$ is stationary ergodic in $t$, but it is far from being obvious that
the same is true for the profile $\phi$.
Actually, to prove this we require the additional hypothesis
(\ref{hyp-vois1}). Instead, the non-existence of random transition waves with
speeds $c$ satisfying $\ul c<2\sqrt{\ul\mu}$ follows directly from Theorem
\ref{thm-existence}.

\subsection{Spreading properties}\label{sec:spreading}

When the nonlinearity $f$ does not depend on $t$, Aronson and Weinberger
\cite{AronsonWeinberger} proved that
if $u$ is the solution of the associated Cauchy problem, with an initial datum
which is ``front-like'' in direction $e$, then
for all $\sigma>0$,
$$\lim_{t\to +\infty}\,\inf_{x\leq
(2\sqrt{f'(0)}-\sigma)t}u(x,t)=1,\qquad
\lim_{t\to +\infty}\, \inf_{x\geq
(2\sqrt{f'(0)}+\sigma)t}u(x,t)=0.$$
This result is called a {\em spreading property} and means that the level-lines
of $u(t,\cdot)$ behave like $2\sqrt{f'(0)}t$ as $t\to +\infty$.
The aim of this section is to extend this property to the Cauchy problem
associated with (\ref{eqprinc}), namely,
\Fi{Cauchy}
\left\{\begin{array}{ll}
      \partial_t u -\Delta u = f(t,u),& x\in\R^N,\ t>0,\\
      u(x,0)=u_0(x), & x\in\R^N.
      \end{array}\right.
\Ff
Our result will involve once again the least mean of $\mu$, but this time over $\R_+$, because the
equation is defined only for $t>0$. For a given function $g\in L^\infty
((0,+\infty))$, we set
$$
\ul g_+:=\sup_{T>0}\,\inf_{t>0}
\frac{1}{T}\int_t^{t+T}g(s)\,ds.
$$
We similarly define the upper mean $\ol g^+$:
$$
\ol g^+:=\inf_{T>0}\,\sup_{t>0}
\frac{1}{T}\int_t^{t+T}g(s)\,ds.
$$
In \cite{BHN} Berestycki, Hamel and the first author partially extended the
result of \cite{AronsonWeinberger} to
general space-time heterogeneous equations. They showed in particular
that the level-lines
of $u(t,\cdot)$ do not grow linearly and can oscillate. They obtained some
estimates on the location of these level-lines, which are
optimal
when $t\mapsto f(t,u)$
converges as $t\to +\infty$ locally in $u$, but not when $f$ is periodic for
example.
These properties have been improved by Berestycki and the first author in
\cite{BN}, by using the notion of {\em generalized principal eigenvalues} in
order to estimate
more precisely the maximal and the minimal linear growths of the location of the
level-lines of $u(t,\cdot)$.
When $f$ only depends on $t$, as in the present paper, they proved that if
$u_0\in C^0(\R^N)$ is such that $0\leq u_0\leq 1$, $u_0\not\equiv 0$ and it is
compactly supported, then the solution $u$ of \eq{Cauchy} satisfies
\Fi{01}
\fa e\in\mathbb{S}^{N-1},\quad\lim_{t\to+\infty}u(x+\gamma te,t)=
\left\{\begin{array}{lll}
1 &\hbox{ if }& 0\leq\gamma <2\sqrt{\ul\mu_+}\\
0 &\hbox{ if }& \gamma>2\sqrt{\ol\mu^+}\\
\end{array}\right.
\quad\hbox{locally in } x.
\Ff
In \cite{BN}, this result follows from a more general statement, proved in the
framework of space-time heterogeneous equations using homogenization
techniques.
Here we improve \eq{01} by decreasing the threshold for the convergence to 0.
Our proof is based on direct arguments.

\begin{prop}\label{prop-spreading} Assume that $f$ satisfies Hypothesis
\ref{hyp:KPP} and let $u_0\in C^0 (\R^N)$ be such that $0\leq u_0 \leq
1$, $u_0\not\equiv0$. Then the solution $u$ of \eq{Cauchy} satisfies
\begin{equation}\label{eq:spreadinginf}
\fa\gamma<2\sqrt{\ul\mu_+},\quad
\lim_{t\to +\infty}\inf_{|x|\leq \gamma t} u(x,t)=1.
\end{equation}
If in addition $u_0$ is compactly supported then
\begin{equation}\label{eq:spreadingsup}
\fa\sigma>0,\quad
\lim_{t\to+\infty}\sup_{|x|\geq 2\sqrt{t\int_0^t \mu (s)ds} +\sigma t} u(x,t)
=0.
\end{equation}
\end{prop}

If $u_0$ is ``front-like'' in the direction $e$, then
$|x|\geq 2\sqrt{t\int_0^t \mu (s)ds} +\sigma t$
can be replaced by
$x\cdot e\geq 2\sqrt{t\int_0^t \mu (s)ds} +\sigma t$ in \eq{spreadingsup}.

\begin{rmq} Proposition \ref{prop-spreading} still
holds if (\ref{hyp-lower}),
(\ref{hyp-vois1}) are not satisfied and, in case of \eq{spreadinginf},
the KPP condition (\ref{hyp-KPP}) can also be dropped.
\end{rmq}

If $\frac1t\int_0^t \mu (s)ds\to\ul\mu_+$ as $t$ goes to $+\infty$ then the result of Proposition \ref{prop-spreading} is optimal.
Otherwise it does not describe in an exhaustive way the large time behavior of $u$.

\begin{opb} Assume that the hypotheses of Proposition \ref{prop-spreading}
hold and that
$$\ul\mu_+<\gamma<\liminf_{t\to+\infty}\frac1t\int_0^t \mu (s)ds.$$
What can we say about $\lim_{t\to+\infty}u(\gamma t e,t)$.
\end{opb}

\subsection{Examples.} \label{sec:examples}

We now present some examples in order to illustrate the notion of generalized
transition waves and to discuss the optimality of our results.

\begin{ex}\label{E:means}
{\em Functions without uniform mean.}\\
Set $t_1:=2$ and, for $n\in\N$,
$$\sigma_n:=t_n+n,\qquad \tau_n:=\sigma_n+n,\qquad t_{n+1}:=\tau_n+2^n.$$
The function $\mu$ defined by
$$\mu(t):=\left\{\begin{array}{ll}
                  3 & \text{if }t_n<t<\sigma_n,\ n\in\N,\\
		  1 & \text{if }\sigma_n<t<\tau_n,\ n\in\N,\\
                  2 & \text{otherwise}
                 \end{array}
\right.$$
satisfies
$$\ul\mu_+=1<\lim_{t\to+\infty}\frac1t\int_0^t \mu (s)ds=2<\ol\mu_+=3.$$
Therefore, $\mu$ does not admit a uniform mean $\langle\mu\rangle$
(over $\R_+$).
\end{ex}

\begin{ex}\label{E:speeds}
{\em Generalized transition waves with various choices of speeds.}\\
Fix $\alpha>1$ and consider the homogeneous reaction-diffusion equation
$$
\partial_t u -\partial_{xx}u = u(1-u)(u+\alpha).
$$
A straightforward computation yields that the function
$$U (z) = \frac{1}{1+e^{z/\sqrt2}}$$
is the profile of a planar traveling wave of speed $c_0 =
\sqrt{2} \alpha +\frac{1}{\sqrt{2}}$. Note that $c_0$ is strictly larger than the minimal speed $2\sqrt{\alpha}$
for the existence of traveling waves.

We now perturb this equation by adding some time heterogeneous bounded
function $\xi\in C^0(\R)$ in the nonlinearity:
\begin{equation}\label{example-2}
\partial_t v -\partial_{xx}v= v(1-v)(v+\alpha-\frac{1}{\sqrt{2}} \xi (t)),
\end{equation}
with $\|\xi\|_\infty < \sqrt{2}(\alpha-1)$ so that this equation is still of KPP
type.
Let $$v(x,t):=U(x-c_0 t +\int_0^t \xi (s)ds).$$
Since $U' = -\frac{U}{\sqrt{2}}
(1-U)$, one readily checks that $v$ is a generalized transition wave of
equation (\ref{example-2}), with
speed
$c(t)= c_0 +\xi (t)$. If $\langle\xi \rangle =0$,
then this generalized transition wave has a global mean speed $c_0$. But as
$\xi$
is arbitrary, the fluctuations around this global mean speed can be large.
This example shows that the speeds associated with generalized transition waves
can be very general, depending on the structure of the heterogeneous equation.
\end{ex}

\begin{ex}\label{E:minimal}
{\em Generalized transition waves with speeds $c$ satisfying $\ul c =
2\sqrt{\ul \mu}$.}
We can generalize the method used in the previous example to obtain generalized
transition waves with a speed with minimal least mean. Consider any homogeneous
function
$f: [0,1]\to [0,\infty)$ such that $f$ is Lipschitz-continuous, $f(0)=f(1)=0$,
$f(s)>0$ if $s\in (0,1)$ and $s\mapsto f(s)/s$ is nonincreasing.
Then we know from \cite{AronsonWeinberger} that for all $c\geq 2\sqrt{f'(0)}$,
there exists a decreasing function $U_c\in C^2(\R)$ such that $U_c(-\infty)=1$,
$U_c (+\infty)=0$ and $-U_c''-cU_c'=f(U_c)$ in $\R$.
It is well-known that $-U_c''(x)/U_c'(x)\to \lambda_c$ as $x\to
+\infty$, where $\lambda_c>0$ is the smallest root of
$\lambda\mapsto-\lambda^2+c\lambda_c-f'(0)$.
Moreover, writing the equation satisfied by $U_c'/U_c$, one can prove that
$U_c'\geq -\lambda_c U_c$ in $\R$.
It follows that the function $P_c
:  [0,1] \to[0,+\infty)$ defined by $P_c(u):=-U_c'(U_c^{-1}(u))$ for $u\in(0,1)$
and $P_c (0)=P_c (1)=0$ is Lipschitz-continuous.
Furthermore, it is of KPP type, because
$$P_c'(0)=-\lim_{x\to+\infty}\frac{U_c''(x)}{U_c'(x)}=\lambda_c,
\qquad \frac{P_c(u)}u=-\frac{U_c'(U_c^{-1}(u))}{U_c(U_c^{-1}(u))}
\geq\lambda_c.$$
We now consider a given function $\xi \in C^\alpha(\R)$ with least mean $\ul
\xi=0$. The function $v$ defined by $v(x,t):= U_c (x-ct -\int_0^t \xi (s)ds)$
satisfies
$$\partial_t v -\partial_{xx} v = f(v) +\xi (t) P_c(v)=: g(t,v)\quad \hbox{in }
\R.$$
It is clearly a generalized transition wave of this equation, with speed $c_1
(t)= c+\xi (t)$.
Let now see what Theorem \ref{thm-existence} gives. Here, $\mu (t)=
g_u'(t,0)=f'(0)+\lambda_c \xi (t)$ and thus $\ul \mu = f'(0)$ since $\ul
\xi=0$.

If $c>2\sqrt{f'(0)}$, then we know that there exists
a generalized transition wave $w$ with a speed $c_2(t)=\frac{\mu
(t)}{\kappa}+\kappa$, for some $0<\kappa<\sqrt{f'(0)}$ such that $\ul c_2=
c$. These two
conditions impose
$\kappa=\lambda_c$ and thus $c_2 (t)=\frac{f'(0)}{\lambda_c}+\xi (t)+\lambda_c=
c_1 (t)$. Hence, $c_1\equiv c_2$, which means that the speed obtained through
Theorem
\ref{thm-existence} is the speed of the generalized transition wave $v$.

The case $c=2\sqrt{f'(0)}$ is not covered by Theorem \ref{thm-existence}.
In this case, the speed $c_1$ of the generalized transition wave $v$ satisfies
$c_1(t)=2\sqrt{f'(0)}+\xi(t)=\frac{\mu (t)}{\lambda_c} +\lambda_c$.
Thus, in this example, it is possible to improve Theorem \ref{thm-existence}
part 1) to the case $\ul c= 2\sqrt{\ul \mu}$.
\end{ex}

\begin{ex}\label{E:2speeds}
{\em Speeds which cannot be written as
$c(t)=\frac{\mu(t)}{\kappa}+\kappa$.}\\
Consider a smooth positive function $\mu=
\mu (t)$ such that $\mu (t)\to \mu_1$ as $t\to -\infty$ and $\mu (t) \to \mu_2$
as $t\to +\infty$, with
$\mu_1>0$ and $\mu_2>0$. Let $f(t,u)= \mu (t) u (1-u)$. Then it has been proved
by Berestycki and Hamel (in a more general framework, see
\cite{BerestyckiHamelgtw2})
that if $\mu_1<\mu_2$, then for all $c_1\in [2\sqrt{\mu_1},+\infty)$, there
exists a generalized transition wave of equation (\ref{eqprinc}) with speed $c$
such that $c(t)\to c_1$ as $t\to -\infty$ and
$\frac{1}{t}\int_0^t c(s)ds\to c_2$ as $t\to +\infty$, where $c_2
=\frac{\mu_2}{\kappa_1}+\kappa_1$ and $\kappa_1$ is the smallest root of
$\kappa_1^2-\kappa_1 c_1 +\mu_1$. This result can be deduced from Theorem
\ref{thm-existence} when $c_1>2\sqrt{\mu_1}$, which even gives a stronger result
since we get $c(t)= \frac{\mu (t)}{\kappa_1}+\kappa_1$ for all $t\in\R$.

When $\mu_1>\mu_2$, then Berestycki and Hamel obtained a different result. Namely,
they prove that for all $c_1\in [2\sqrt{\mu_1},+\infty)$,
if $\kappa_1\geq \sqrt{\mu_2}$ (which is true in particular when
$c_1=2\sqrt{\mu_1}$),
there exists
a generalized transition wave of equation (\ref{eqprinc}) with speed $c$ such
that $c(t)\to c_1$ as $t\to -\infty$ and
$\frac{1}{t}\int_0^t c(s)ds\to 2\sqrt{\mu_2}$ as $t\to +\infty$.
In this case the speed $c$ cannot be put in the form $c(t)= \frac{\mu
(t)}{\kappa}+\kappa$ for some $\kappa>0$. Hence, the class of speeds we
construct in Theorem \ref{thm-existence}
is not exhaustive. Moreover, in this example, this class of speeds misses the
most important generalized transition waves: the one which travels with speed
$2\sqrt{\mu_1}$ when $t\to -\infty$ and $2\sqrt{\mu_2}$ when $t\to +\infty$. As
these two speeds are minimal near $t=\pm\infty$, one can expect this generalized
transition wave to
be attractive in some sense, as in homogeneous media (see \cite{KPP}).
\end{ex}

\section{Proof of the results}

As we said at the beginning of Section \ref{sec:hyp}, in this paper the terms
(strong)
sub and supersolution refer to functions in $W^{2,1}_{N+1,loc}$, satisfying the
differential inequalities a.e.
We say that a function
is a {\em generalized subsolution} (resp.~{\em supersolution}) if it is the
supremum (resp.~infimum) of a finite number of subsolutions
(resp.~supersolutions).

\subsection{Properties of the least mean}

We first give an equivalent formulation of the least mean
(see Definition \ref{def-mean}).

\begin{prop} \label{reformulation}
If $g\in L^\infty(\R)$ then its least mean $\ul g$ satisfies $$\ul
g=\lim_{T\to+\infty}\essinf_{t\in\R}\frac{1}{T}\int_t^{t+T}g(s)\,ds.$$
In particular, if $g$ admits a mean value
$\langle g\rangle$, defined by
\eq{mv}, then $\ul g=\langle g\rangle$.
\end{prop}

\begin{proof}
For $T>0$, define the following function:
$$F(T):=\inf_{t\in\R}\int_t^{t+T}g(s)\,ds.$$
We have that
$$\ul g=\sup_{T>0}\frac{F(T)}{T}\geq\limsup_{T\to+\infty}\frac{F(T)}{T}.$$
Therefore, to prove the statement we only need to show that
 $\liminf_{T\to+\infty}F(T)/T\geq\ul g$.
 For any $\e>0$, let $T_\e>0$ be such that $F(T_\e)/T_\e\geq\ul g-\e$.
 We use the notation $\lfloor x \rfloor$ to indicate the floor of the real
 number $x$ (that is, the greatest integer $n\leq x$) and we compute
 $$\fa T>0,\quad F(T)=\inf_{t\in\R}\left(\int_t^{t+\left\lfloor\frac{T}{T_\e}
 \right\rfloor T_\e} g(s)\,ds+\int_{t+\left\lfloor\frac{T}{T_\e}
 \right\rfloor T_\e}^{t+T} g(s)\,ds\right)\geq
 \left\lfloor\frac{T}{T_\e}\right\rfloor F(T_\e)-\|g\|_{L^\infty(\R)}T_\e.$$
 As a consequence,
 $$\liminf_{T\to+\infty}\frac{F(T)}{T}\geq
 \lim_{T\to+\infty}\left\lfloor\frac{T}{T_\e}\right\rfloor
 \frac{T_\e}{T}\frac{F(T_\e)}{T_\e}=\frac{F(T_\e)}{T_\e}\geq\ul g-\e.$$
 The proof is thereby achieved due to the arbitrariness of $\e$.
 \end{proof}

We now derive another characterization of the least mean.
This is the property underlying the fact that the existence of
generalized transition waves is expressed
in terms of the least mean of their
speeds.

\begin{lem}\label{lem:lm>0}
Let $B\in L^\infty(\R)$. Then
$$\ul B=\sup_{A\in W^{1,\infty}(\R)}\essinf_{t\in\R}(A'+B)(t).$$
\end{lem}

\begin{proof} If $B$ is a periodic function, then
$g(t):= \underline{B}-
B(t)$ is periodic with zero mean. Thus $A(t):=\int_0^t g(s)ds$ is bounded and
satisfies $A'+B\equiv\underline{B}$. This shows that
\Fi{lm<}
\ul B\leq\sup_{A\in W^{1,\infty}(\R)}\essinf_{t\in\R}(A'+B)(t)
\Ff
in this simple case. We will now
generalize this construction in order to handle general functions B.

Fix $m<\ul B$. By definition, there exists $T>0$ such that
$$\inf_{t\in\R}\frac1T\int_t^{t+T}B(s)\, ds>m.$$
We define
$$ \forall k\in\Z,\ \text{for a.e.~} t\in
[(k-1)T,kT),\quad
g(t):=-B(t)+\beta_k,\qquad\text{where }\
\beta_k:=\frac{1}{T}\int_{(k-1)T}^{kT}
B(s)\, ds.$$
Then we set $A(t):=\int_0^t g(s)ds$.
It follows that $A'+B\geq m$ and, since $\int_{(k-1)T}^{kT}
g(s)\, ds=0$, that
$$\|A\|_{L^\infty(\R)}\leq\|g\|_{L^\infty(\R)} T\leq2T\|B\|_{L^\infty(\R)}.$$
Therefore \eq{lm<} holds due to the arbitrariness of $m<\ul B$.

Consider now a function $A\in W^{1,\infty}(\R)$.
Owing to Proposition \ref{reformulation} we derive
\[\begin{split}
\ul B &=\lim_{T\to+\infty}\inf_{t\in\R}\frac{1}{T}\int_t^{t+T}B(s)\,ds
\geq\essinf_\R(A'+B)+
\lim_{T\to+\infty}\inf_{t\in\R}\frac{1}{T}\int_t^{t+T}(-A'(s))\,ds\\
&=\essinf_\R(A'+B)+\lim_{T\to+\infty}\inf_{t\in\R}\frac{1}{T}(A(t)-A(t+T))=
\essinf_\R(A'+B).
\end{split}\]
This concludes the proof.
\end{proof}

\begin{rmq}\label{rmq:lm>0}
 In the proof of Lemma \ref{lem:lm>0} we have shown the following fact: if
 $\eta\in\N\cup\{+\infty\}$, $T>0$ are such that
 $$m:=\inf_{\su{k\in\N}{k\leq\eta}}\frac1T\int_{(k-1)T}^{kT}
 B(s)\, ds>0,$$
 then there exists $A\in W^{1,\infty}((0,\eta T))$ satisfying
 $$\essinf_{[0,\eta T)}(A'+B)=m,\qquad
 \|A\|_{L^\infty((0,\eta T))}\leq2T
 \|B\|_{L^\infty((0,\eta T))}.$$
\end{rmq}

\subsection{Construction of the generalized transition waves when $\ul c>2\sqrt{\ul \mu}$.}

In order to construct generalized transition waves, we will use appropriate sub
and supersolutions.
The particular form of the speeds (\ref{formc})
will naturally emerge from constraints on the exponential supersolution.

\begin{prop}\label{pro:subsuper}
Under the assumptions of Theorem \ref{thm-existence}, for all $\gamma>
2\sqrt{\ul\mu}$, there exists a function $c\in L^\infty(\R)$, with
$\ul c=\gamma$, such that (\ref{def-tf}) admits some uniformly continuous
generalized sub and supersolutions
$\ul\phi(z,t)$, $\ol\phi(z)$ satisfying

$$0\leq\ul\phi<\ol\phi\leq1,\qquad
\ol\phi(+\infty)=0,\qquad\ol\phi(-\infty,t)=1\  \hbox{
uniformly in } t\in\R,$$
$$\exists\xi\in\R,\ \ \inf_{t\in\R}\ul\phi(\xi,t)>0,\qquad
\forall z\in\R,\ \ \inf_{t\in\R}(\ol\phi-\ul\phi)(z,t)>0.$$
$$\ol\phi\ \text{ is nonincreasing in }\R,\qquad
\fa\tau>0,\ \
\lim_{z\to+\infty}\frac{\ol\phi(z+\tau)}{\ul\phi(z,t)}<1
\  \hbox{ uniformly in } t\in\R.$$
\end{prop}

\begin{proof}
Fix $\gamma>2\sqrt{\ul{\mu}}$. We choose $c$ in such a way that the
linearized equation
around $0$ associated with (\ref{def-tf}) admits an exponential solution of the
type $\psi(z)=e^{-\kappa z}$, for some $\kappa>0$. Namely,
$$0=\partial_t \psi-\partial_{zz}\psi -c(t)\partial_z
\psi-\mu(t)\psi=
[-\kappa^2+c(t)\kappa-\mu(t)]\psi,\quad \text{for a.e.~}t\in\R.$$
It follows that $c\equiv\kappa+\kappa^{-1}\mu$. Imposing $\ul c=\gamma$
yields
$$\gamma=\lim_{T\to +\infty}\inf_{t\in\R}
\frac{1}{T}\int_t^{t+T}
[\kappa+\kappa^{-1}\mu(s)]ds=\kappa+\kappa^{-1}\ul\mu.$$
Since $\gamma>2\sqrt{\ul{\mu}}$, the equation
$\kappa^2-\gamma\kappa+\ul\mu=0$
has two positive solutions. We take the smallest one,
that is,
$$\kappa=\frac{\gamma-\sqrt{\gamma^2-4\ul\mu}}2.$$
Extending $f(t,\.)$ linearly outside $[0,1]$, we can assume that $\psi$ is a
global supersolution. We then set $\ol\phi(z):=\min(\psi(z),1)$.

Let $C$, $\gamma$ be the constants in (\ref{hyp-lower}).
Our aim is to find a function $A\in
W^{1,\infty}(\R)$ and a constant $h>\kappa$ such that the
function $\vp$ defined by $\vp(z):=\psi(z)-e^{A(t)-hz}$
satisfies
\begin{equation}\label{ulphi}
\partial_t\vp-\partial_{zz}\vp-c(t)\partial_z\vp
\leq\mu(t)\vp-C\vp^{1+\gamma},\quad\text{for a.e.~} z>0,\ t\in\R.
\end{equation}
By direct computation we see that
$$\partial_t \vp-\partial_{zz}\vp-c(t)\partial_z\vp-\mu(t)\vp
=[-A'(t)+h^2-c(t)h+\mu(t)]e^{A(t)-hz}.$$
Hence (\ref{ulphi}) holds if and only if
$$\text{for a.e.~} z>0,\ t\in\R,\quad
A'(t)+B(t)\geq C\vp^{1+\gamma}e^{hz-A(t)},\qquad\text{where }
B(t):=-h^2+c(t)h-\mu(t).$$
Let $\kappa< h<(1+\gamma)\kappa$. Since
$$\forall z>0,\ t\in\R,\quad
\vp^{1+\gamma}e^{hz-A(t)}\leq
e^{[h-(1+\gamma)\kappa]z-A(t)}
\leq e^{-A(t)},$$
if $\essinf_\R(A'+B)>0$ then the desired inequality follows by adding
a large constant to $A$.
Owing to Lemma \ref{lem:lm>0}, this condition is fulfilled by a suitable
function $A\in W^{1,\infty}(\R)$ as soon as $\ul B>0$.
Let us compute
\[\begin{split}
\ul B &=
\lim_{T\to +\infty}\inf_{t\in\R}
\frac{1}{T}\int_t^{t+T}h\left[
\kappa-h+\mu(s)\left(\frac1\kappa-\frac1h\right)\right ] ds \\
& =h\left[
\kappa-h+\ul\mu\left(\frac1\kappa-\frac1h\right)\right ]\\
&= -h^2+\gamma h-\ul\mu.
\end{split}\]
Since $\kappa$ is the smallest root of $-x^2+\gamma x-\ul\mu=0$, we can choose
$h\in(\kappa,(1+\gamma)\kappa)$ in such a way that
$\ul B>0$. Therefore, there exists $A\in
W^{1,\infty}(\R)$ such that (\ref{ulphi}) holds. Up to adding a suitable
constant $\alpha<0$ to $A$, it is
not
restrictive to assume that $\vp$ is less than the constant $\delta$ in
(\ref{hyp-lower}). Hence, $\vp$ is a subsolution of (\ref{def-tf}) in
$(0,+\infty)\times\R$.
Since $\vp\geq0$ if and only if $z\geq(h-\kappa)^{-1}A(t)$,
$\alpha$ can be chosen in such a way that $\vp\leq0$ for $z\leq0$.
Whence, due to the arbitrariness of the extension of $f(t,u)$ for $u<0$, it
follows that $\ul\phi(z,t):=\max(\vp(z,t),0)$ is a generalized
subsolution.
Finally, for $\tau>0$, it holds that
$$\lim_{z\to+\infty}\frac{\ol\phi(z+\tau)}{\ul\phi(z,t)}=
\lim_{z\to+\infty}\frac{e^{-\kappa\tau}}{1-e^{A(t)-(h-\kappa)z}}=e^{-\kappa\tau}
.$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm-existence} part 1)]
Let $c,\ \ul\phi,\ \ol\phi$ be the functions given by Proposition
\ref{pro:subsuper}, with $\ul c=\gamma$. For $n\in\N$, consider the solution
$\phi_n$ of the problem
\Fi{-n}
\left\{\begin{array}{ll}
\partial_t \phi-\partial_{zz}\phi -c(t)\partial_z \phi = f(t,\phi),&
z\in\R,\ t>-n, \\
\phi(z,-n)=\ol\phi(z), & z\in\R.
\end{array} \right.
\Ff
The comparison principle implies that $\ul\phi\leq\phi_n\leq\ol\phi$ and, since
$\ol\phi$ is nonincreasing, that $\phi_n(\.,t)$ is nonincreasing too.
Owing to the parabolic estimates and the embedding theorem, using
a diagonal extraction method we can
find a subsequence of $(\phi_n)_{n\in\N}$ converging  weakly in $W^{2,1}_p(K)$
and
strongly in $L^\infty(K)$, for any compact $K\subset\R\times\R$ and any
$p<\infty$, to a solution $\phi$ of
$$\partial_t \phi-\partial_{zz}\phi -c(t)\partial_z \phi = f(t,\phi),\quad
z\in\R,\ t\in\R.$$
The function $\phi$ is nonincreasing in $z$ and
satisfies $\ul\phi\leq\phi\leq\ol\phi$. Applying the parabolic strong maximum
principle to $\phi(z-z_0,t)-\phi(z,t)$, for every $z_0>0$, we find that $\phi$
is decreasing in~$z$.

It remains to prove that $\phi(-\infty,t)=1$ uniformly with respect to $t\in\R$.
Set
$$\theta:=\lim_{z\to-\infty}\inf_{t\in\R}\phi(z,t).$$
Our aim is to show that $\theta=1$.
Let $(t_n)_{n\in\N}$ be such that $\lim_{n\to\infty}\phi(-n,t_n)=\theta$.
We would like to pass to the limit in
the sequence of equations satisfied by the $\phi(\.-n,\.+t_n)$, but this is not
possible due to the presence of the drift term. To
overcome this difficulty we come back to the fixed coordinate system by
considering
the functions $(v_n)_{n\in\N}$ defined by
$$v_n(z,t):=\phi (z-n-\int_{t_n}^{t_n+t}c(s)ds,t+t_n).$$
These functions are solutions of
$$\partial_t v_n-\partial_{zz}v_n= f(t+t_n,v_n),\quad
z\in\R,\ t\in\R,$$
and satisfy $\lim_{n\to\infty}v_n(0,0)=\theta$ and
$\liminf_{n\to\infty}v_n(z,t)\geq\theta$ locally uniformly in
$(z,t)\in\R\times\R$.
The same diagonal extraction method as before
shows that $(v_n)_{n\in\N}$ converges (up to subsequences) weakly in
$W^{2,1}_{p,loc}$ and strongly in $L^\infty_{loc}$ to some function $v$
satisfying
$$\partial_t v-\partial_{zz}v=g(z,t)\geq0,\quad\text{for a.e.~}z\in\R,\
t\in\R,$$
where $g(z,t)$ is the weak limit in $L^p_{loc}(\R\times\R)$ of (a subsequence
of) $f(t+t_n,v_n)$.
Moreover, $v$ attains its minimum value $\theta$ at $(0,0)$.
As a consequence, the strong maximum principle yields
$v=\theta$ in $\R\times(-\infty,0]$. In particular, $g=0$ a.e.~in
$\R\times(-\infty,0)$.
Using the Lipschitz continuity of $f(t,\.)$,
we then derive
$$\text{for a.e.~}(z,t)\in\R\times(-\infty,0),\quad
0=g(z,t)\geq\essinf_{s\in\R}f(s,\theta).$$
Therefore, hypothesis \ref{hyp-pos} yields
$\theta=0$ or $1$.
The proof is then concluded by noticing that
$$\theta=\lim_{z\to-\infty}\inf_{t\in\R}\phi(z,t)
\geq\inf_{t\in\R}\phi(\xi,
t)\geq\inf_{t\in\R}
\ul\phi(\xi,t)>0$$
($\xi$ being the constant in Proposition \ref{pro:subsuper}).
\end{proof}

\subsection{Non-existence of generalized transition waves when $\ul c <
2\sqrt{\ul \mu}$.}\label{sec:nonE}

This section is dedicated to the proof of the lower bound for the least mean of
admissible speeds - Theorem \ref{thm-existence} part 2.
This is achieved by comparing the generalized transition waves with some
subsolutions whose level-sets propagate at speeds less than $2\sqrt{\ul
\mu}$. The construction of the
subsolution is based on an auxiliary result which
is quoted from \cite{BHRossi} and reclaimed in the Appendix here.

\begin{lem}\label{lem:0nT}
 Let $g\in L^\infty(\R)$, $\eta\in\N\cup\{+\infty\}$ and $T>0$ be such that
 $$\gamma_*:=\inf_{\su{k\in\N}{k\leq\eta}}\,
 2\sqrt{\frac1T\,\int_{(k-1)T}^{kT}g(s)\,
 ds}>0.$$
 Then for all $0<\gamma<\gamma_*$, there exists a uniformly continuous
 subsolution $\ul v$ of
 \Fi{0etaT}
 \partial_t v-\Delta v=g(t)v, \quad x\in\R^N,\ t\in(0,\eta T),
 \Ff
 such that
 $$0\leq\ul v\leq1,\qquad \ul v(x,0)=0 \text{ for }|x|\geq R,\qquad
 \inf_{\su{0\leq t<\eta T}{|x|\leq\gamma t}}\ul v(x,t)\geq C,$$
 where $R$, $C$ only depend on $T$, $\gamma_*-\gamma$, $N$,
 $\|g\|_{L^\infty((0,\eta T))}$ and not on $n$.
\end{lem}

\begin{proof}
Fix $\gamma\in(0,\gamma^*)$. By Lemma \ref{lem:h} in the Appendix,
there exist $h\in C^{2,\alpha}(\R)$ and $r>0$, both
depending on $\gamma$ and $\gamma_*-\gamma$, satisfying
$$h=0\ \text{in }(-\infty,0],\qquad
h'>0\ \text{in }(0,r),\qquad
h=1\text{ in }[r,+\infty),$$
$$Q\leq \gamma+1,\quad
4C-Q^2\geq\frac12(\gamma_*-\gamma)^2
\qquad\Rightarrow\qquad-h''+Qh'-Ch<0\quad\text{in }(0,r).$$
Note that $r$, $h$ actually depend on
$\gamma_*-\gamma$ and $\|g\|_\infty$, because $\gamma<\gamma_*\leq\|g\|_\infty$.
We set
$$\ul v(x,t):=e^{A(t)}h(R-|x|+\gamma t),$$
where $R>r$ and $A:\R\to\R$ will be chosen later. This function solves
$$\partial_t \ul v-\Delta\ul v-g(t)\ul v
=\left[-h''(\rho)+\left(\gamma+\frac{N-1}{|x|}\right)h'(\rho)-C(t)h(\rho)\right]
e^{A(t)},\quad x\in\R^N,\ t\in(0,\eta T).$$
with $\rho=R-|x|+\gamma t$ and $C(t)=g(t)-A'(t)$.
Since $h'$ is nonnegative and vanishes in $[r,+\infty)$, it
follows that
$$\partial_t \ul v-\Delta\ul v-g(t)\ul v
\leq[-h''(\rho)+Qh'(\rho)-C(t)h(\rho)]e^{A(t)},\quad \text{for a.e.~} x\in\R^N,\
t\in(0,\eta T),$$
with $Q=\gamma+\frac{N-1}{R-r}$. We write
$$4C(t)-Q^2=B(t)-4A'(t),\quad\text{where }
B(t)=4 g(t)-\left(\gamma+\frac{N-1}{R-r}\right)^2.$$
and we compute
$$m:=\inf_{\su{k\in\N}{k\leq\eta}}\frac1T
\int_{(k-1)T}^{kT}B(s)\,
ds=\gamma_*^2-\left(\gamma+\frac{N-1}{R-r}\right)^2.$$
Hence, since $\gamma_*^2-\gamma^2\geq(\gamma_*-\gamma)^2$, it is possible to
choose $R$, depending on $N$, $r$, $\|g\|_\infty$ and $\gamma_*-\gamma$, such
that $m\geq\frac12(\gamma_*-\gamma)^2$.
By Remark \ref{rmq:lm>0}
there exists a function $A\in W^{1,\infty}(\R)$ such that
$$\min_{[0,\eta T)}(B-4A')=m,\qquad \|A\|_{L^\infty((0,\eta T))}\leq\frac T2
\|B\|_{L^\infty((0,\eta T))}\leq4T\|g\|_{L^\infty((0,\eta T))}.$$
Consequently, $4C-Q^2\geq\frac12(\gamma_*-\gamma)^2$ a.e.~in $(0,\eta T)$ and,
up to increasing $R$ if need be, $Q\leq\gamma+1$.
Therefore, $\ul v$ is a subsolution of \eq{0etaT}. This concludes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm-existence} part 2)]
Let $u$ be a generalized transition wave with speed $c$.
Definition \ref{def-gtf} yields
\Fi{Le}
\lim_{L\to+\infty}\,\inf_{\su{x\.e<-L}{t\in\R}}u(x+e\int_0^tc(s)ds,t)=1,
\qquad
\lim_{L\to+\infty}\,\sup_{\su{x\.e>L}{t\in\R}}u(x+e\int_0^tc(s)ds,t)=0.
\Ff
By the definition of least mean, for all $\e>0$, there exists $T>0$ such
that
$$\fa T'\geq T,\quad
\frac1{T'}\inf_{t\in\R}\int_t^{t+T'}c(s)\,ds<\ul c+\e,\qquad
\fa t\in\R,\quad\frac1T\int_t^{t+T}\mu(s)\,ds>\ul\mu-\e.$$
For $n\in\N$, let $t_n$ be such that
$$\frac1{nT}\int_{t_n}^{t_n+nT}c(s)\,ds<\ul c+2\e.$$
Taking $\e$ small enough in such a way that $2\sqrt{\ul\mu-2\e}>\e$, we find
that
$$\gamma_*^n:=\inf_{k\in\{1,\dots,n\}}2\sqrt{\frac1T\,\int_{(k-1)T}^{kT}
(\mu(s+t_n)-\e)\,
ds}>2\sqrt{\ul\mu-2\e}>\e.$$
Let $(\ul v_n)_{n\in\N}$ be the functions obtained applying Lemma
\ref{lem:0nT} with $g(t)=\mu(t+t_n)-\e$
and $\gamma=\gamma^n:=\gamma_*^n-\e$, and
let $R$, $C$ be the associated constants, which are independent of $n$.
By the regularity hypothesis on $f$, there exists $\sigma\in(0,1)$ such that
$f(t,w)\geq(\mu(t)-\e)w$ for $w\in[0,\sigma]$. As a consequence, the functions
$\ul u_n(x,t):=\sigma \ul v_n(x,t-t_n)$ satisfy
$$\partial_t \ul u_n-\Delta \ul u_n\leq f(t,\ul u_n), \quad \text{for a.e.~}
x\in\R^N,\
t\in(t_n,t_n+nT).$$
By \eq{Le}, for $L$ large enough we have that
$$\fa t\in\R,\quad\inf_{|x|<R}u(x-Le+e\int_0^tc(s)ds,t)\geq\sigma.$$
Thus, up to replacing $u(x,t)$ with $u(x-Le,t)$,
it is not restrictive to assume that
$$\fa n\in\N,\ x\in\R^N,\quad u(x+e\int_0^{t_n}c(s)ds,t_n)\geq \ul u_n(x,t_n).$$
The comparison principle then yields
$$\fa n\in\N,\ x\in\R^N,\ t\in(t_n,t_n+nT),\quad
u(x+e\int_0^{t_n}c(s)ds,t)\geq \ul u_n(x,t).$$
Therefore,
$$\liminf_{n\to\infty}u(\gamma^n nT e+e\int_0^{t_n}c(s)ds,t_n+nT)\geq
\liminf_{n\to\infty}\ul u_n(\gamma^n nT,t_n+nT)\geq\sigma C,$$
whence, owing to \eq{Le}, we deduce that
\[\begin{split}
+\infty &>\limsup_{n\to\infty}\left(\gamma^n nT +\int_0^{t_n}c(s)ds
-\int_0^{t_n+nT}c(s)ds\right)\\
&=\limsup_{n\to\infty}\left(\gamma^n nT -\int_{t_n}^{t_n+nT}c(s)ds\right)\\
&\geq \limsup_{n\to\infty}\left(2\sqrt{\ul\mu-2\e}-\e-\ul c-2\e\right)nT.
\end{split}\]
That is, $\ul c\geq2\sqrt{\ul\mu-2\e}-3\e$.
Since $\e$ can be chosen arbitrarily small, we eventually infer that $\ul
c\geq2\sqrt{\ul\mu}$.
\end{proof}

\begin{rmq}
The same arguments as in the above proof yield the non-existence of fronts
$(\phi,c)$ such that $\ul c_{\pm}<2\sqrt{\ul\mu_\pm}$, where,
for a given function $g\in L^\infty(\R)$,
$$\underline{g}_+:=\sup_{T>0}\,\inf_{t>0}
\frac{1}{T}\int_t^{t+T}g(s)\,ds,\qquad
\underline{g}_-:=\sup_{T>0}\,\inf_{t<0}
\frac{1}{T}\int_{t-T}^T g(s)\,ds$$
\end{rmq}

\section{The random stationary ergodic case}

To start with, we show that the temporal least mean of a stationary ergodic
function
is almost surely independent of $\omega$.

\begin{prop}\label{pro:lmergodic}
For a given bounded
measurable function $\tilde{g}:\O\to\R$, the mapping
$$\omega\,\mapsto\,\sup_{T>0}\,\inf_{t\in\R}
\frac{1}{T}\int_t^{t+T}\tilde{g}(\pi_s\omega)\,ds$$
is constant in a set of probability measure $1$.
We call this constant value the {\em least mean} of the random stationary ergodic function $g$ defined by $g(t,\omega):= \tilde{g}(\pi_t \omega)$
and we denote it by $\ul g$.
\end{prop}

\begin{proof}
The result follows immediately from the ergodicity of the process
$(\pi_t)_{t\in\R}$. Indeed, setting
$$G(\omega):=\sup_{T>0}\,\inf_{t\in\R}\frac{1}{T}\int_t^{t+T}\tilde{g}(\pi_s\omega)\,ds,$$
for given $\e>0$, there exists a set $\mc{A}_\e\in\mc{F}$, with
$\P(\mc{A}_\e)>0$, such that $G(\omega)<\essinf_\O G+\e$ for
$\omega\in\mc{A}_\e$.
It is easily seen that $\mc{A}_\e$ is invariant under the action of
$(\pi_t)_{t\in\R}$, and then $\P(\mc{A}_\e)=1$. Owing to the arbitrariness of
$\e$, we infer that $G$ is almost surely equal to $\essinf_\O G$.
\end{proof}

The proof of Theorem \ref{thm-existencerandom} relies on a general uniqueness
result for the profile of  generalized transition waves that share the
same behavior at infinity. This, in turn, is derived using the following strong
maximum principle-type property.

\begin{lem}\label{lem:I}
Let $c\in L^\infty(\R)$ and assume that $f$ satisfies the regularity
conditions
in Hypothesis \ref{hyp:KPP}.
Let $I$ be an open interval and $\vp,\psi$ be respectively a generalized sub
and supersolution of
$$\partial_t\phi-\partial_{zz}\phi-c(t)\partial_z\phi
=f(t,\phi),\quad z\in I,\ t\in\R,$$
which are uniformly continuous and
satisfy $0\leq\vp\leq\psi\leq1$ in $I\times\R$. Then, the set
$$\{z\in I\ : \ \inf_{t\in\R}(\psi-\vp)(z,t)=0\}$$
is either empty or coincides with $I$.
\end{lem}

\begin{proof}
Clearly, it is sufficient to prove the result for strong sub and
supersolutions. We achieve this by showing that the set
$$J:=\{z\in I\ : \ \inf_{t\in\R}(\psi-\vp)(z,t)=0\}$$
is open and closed in the topology of $I$.
That it is closed follows immediately from the uniform continuity of $\vp$ and
$\psi$.

Let us show that it is open. Suppose that there exists $z_0\in J$. There
is a sequence $(t_n)_{n\in\N}$ such that $(\psi-\vp)(z_0,t_n)$ tends to $0$ as
$n$ goes to infinity. For $n\in\N$, define $\Phi_n(z,t):=(\psi-\vp)(z,t+t_n)$.
These functions satisfy
$$\partial_t\Phi_n-\partial_{zz}\Phi_n-c(t+t_n)\partial_z\Phi_n
-\zeta(z,t+t_n)\Phi_n\geq0\quad \text{for a.e.~}z\in I,\ t\in\R,$$
where
$$\zeta(z,t):=\frac{f(t,\psi)-f(t,\vp)}{\psi-\vp}$$
belongs to $L^\infty(I\times\R)$
due to the Lipschitz-continuity of $f$. Let $\delta>0$ be such that
$[z_0-\delta,z_0+\delta]\subset I$. We now make use of the parabolic weak
Harnack
inequality (see e.g. Theorem 7.37 in \cite{Lie}). It provides two constants
$p,C>0$
such that
$$\fa n\in\N,\quad
\|\Phi_n\|_{L^p((z_0-\delta,z_0+\delta)\times(-2,-1))}
\leq C\inf_{(z_0-\delta,z_0+\delta)\times(-1,0)}\Phi_n
\leq C\Phi_n(z_0,0).$$
Whence $(\Phi_n)_{n\in\N}$ converges to $0$ in
$L^p((z_0-\delta,z_0+\delta)\times(-2,-1))$. By the Arzela-Ascoli
theorem we then infer that, up to subsequences, $(\Phi_n)_{n\in\N}$ converges to
$0$ uniformly in $(z_0-\delta,z_0+\delta)\times(-2,-1)$. This means that
$(z_0-\delta,z_0+\delta)\subset J$.
\end{proof}

\begin{prop}\label{pro:!}
Assume that $c\in L^\infty(\R)$ and that $f$ satisfies (\ref{hyp-vois1}). Let
$\vp,\ \psi$ be a subsolution and a positive supersolution of (\ref{def-tf})
which are uniformly continuous and satisfy
$$0\leq\vp,\psi\leq1,\qquad
\psi(\.,t) \text{ is nonincreasing},\qquad\fa\tau>0,\quad
\lim_{R\to+\infty}\sup_{\su {z>R}{t\in\R}}\frac{\vp(z,t)}{\psi(z-\tau,t)}<1.$$
Then $\vp\leq\psi$ in $\R\times\R$.
\end{prop}

\begin{proof}
Since $\vp,\ \psi$ are uniformly continuous and satisfy $\vp(+\infty,t)=0$,
$\psi(-\infty,t)=1$ uniformly in $t$, applying Lemma \ref{lem:I}, first with
$\psi\equiv1$ and then with $\vp\equiv0$, we derive
\Fi{supinf}
\fa r\in\R,\quad\sup_{(r,+\infty)\times\R}\vp<1,\quad
\inf_{(-\infty,r)\times\R}\psi>0.
\Ff
Let $\delta\in(0,1)$ be the constant in (\ref{hyp-vois1}).
By hypothesis, there exists $\rho\in\R$ such that $\psi>1-\delta$ in
$(-\infty,\rho)\times\R$. Let $\chi:\R\to[0,1]$ be a smooth function satisfying
$\chi=1$ in $(-\infty,\rho]$, $\chi=0$ in $[\rho+1,+\infty)$. Define the family
of functions $(\psi^{\e,\tau})_{\e,\tau\geq0}$ by setting
$$\psi^{\e,\tau}(z,t):=[1+\e\chi(z)]\psi(z-\tau,t).$$
Since $\lim_{\e,\tau\to0^+}\psi^{\e,\tau}\equiv\psi$, the statement is proved
if we show that $\psi^{\e,\tau}\geq\vp$ for all $\e,\tau>0$.
The $\psi^{\e,\tau}$ are nondecreasing with respect to both $\e$ and $\tau$.
Moreover, there
exists $z_0\in\R$ such that $\psi^{0,1}>\vp$ in $(z_0,+\infty)\times\R$. On the
other hand, for all $\e>0$, there exists $\tau\geq0$ such that
$\psi^{\e,\tau}>1\geq\vp$ in $(-\infty,z_0]\times\R$. Consequently, for all
$\e>0$, we have that $\psi^{\e,\tau}\geq\vp$ for $\tau$ large enough. Define
$$\fa\e>0,\quad\tau(\e):=\min\{\tau\geq0\ : \ \psi^{\e,\tau}\geq\vp\}.$$
The function $\e\mapsto\tau(\e)$ is nonincreasing and it holds that
$\psi^{\e,\tau(\e)}\geq\vp$. We argue by contradiction, assuming that there
exists $\t\e>0$ such that $\tau(\t\e)>0$.
By hypothesis, we have that
\Fi{psi>vp}
\exists\, h>1,\ R\in\R,\quad
\psi^{0,\tau(\t\e)/2}\geq h\vp\quad\text{in }(R,+\infty)\times\R.
\Ff
Fix $\e\in(0,\t\e]$. We know that
$\tau(\e)\geq\tau(\t\e)>0$ and, for $\tau\in(0,\tau(\e))$,
$\inf_{\R\times\R}(\psi^{\e,\tau}-\vp)<0$.
Hence, from \eq{psi>vp} it follows that, for $\tau\in[\tau(\t\e)/2,\tau(\e))$,
$\inf_{(-\infty,R]\times\R}(\psi^{\e,\tau}-\vp)<0$. Thus, by the uniform
continuity of $\psi$ we get
$\inf_{(-\infty,R]\times\R}(\psi^{\e,\tau(\e)}-\vp)=0$.
We now use the assumption (\ref{hyp-vois1}). Since
$\psi^{0,\tau(\e)}\geq\psi>1-\delta$ in $(-\infty,\rho)\times\R$,
for a.e.~$z\in(-\infty,\rho)$, $t\in\R$ we have
$$\partial_t \psi^{\e,\tau(\e)}-\partial_{zz}\psi^{\e,\tau(\e)}
-c(t)\partial_z
\psi^{\e,\tau(\e)} =(1+\e)f(t,\psi^{0,\tau(\e)})\geq
f(t,\psi^{\e,\tau(\e)})$$
(where we have extended $f$ by $0$ in $\R\times(1,+\infty)$).
By hypothesis, we can find $R_\e<\rho$ such that
$\inf_{(-\infty,R_\e]\times\R}\psi^{\e,\tau(\e)}>1\geq\sup\vp$.
Consequently, Lemma \ref{lem:I} yields
$\inf_{(-\infty,\rho-1]\times\R}(\psi^{\e,\tau(\e)}-\vp)>0$.
It follows that $R>\rho-1$ and that
\Fi{psi=vp}
\inf_{(\rho-1,R]\times\R}(\psi^{\e,\tau(\e)}-\vp)=0.
\Ff
In order to pass to the limit $\e\to0^+$ in the above expression, we notice
that, by \eq{supinf}, there exists $\tau_0>0$ such that
$$\inf_{(-\infty,R]\times\R}\psi^{0,\tau_0}>\sup_{[\rho-1,R]\times\R}\vp.$$
As a consequence, \eq{psi=vp} implies that the nonincreasing function $\tau$
is bounded form above by $\tau_0$, and then there exists
$\tau^*:=\lim_{\e\to0^+}\tau(\e)$.
Letting $\e\to0^+$ in the inequality
$\psi^{\e,\tau(\e)}\geq\vp$ and in \eq{psi=vp} yields
$$\psi^{0,\tau^*}\geq\vp\ \text{
in }\R\times\R,\qquad\inf_{[\rho-1,R]\times\R}(\psi^{0,\tau^*}
-\vp)=0.$$
Thus, using once again Lemma \ref{lem:I} and then the inequality \eq{psi>vp}
we derive
$$0=\inf_{t\in\R}(\psi^{0,\tau^*}-\vp)(R,t)\geq\inf_{t\in\R}(\psi^{0,
\tau(\t\e)/2} -\vp)(R , t)
\geq(1-h^{-1})\inf_{t\in\R}\psi^{0,\tau(\t\e)/2}(R,t).$$
This contradicts \eq{supinf}.
We have shown that $\tau(\e)=0$ for all $\e>0$. That is, $\psi^{\e,\tau}\geq\vp$
for all $\e,\tau>0$.
\end{proof}

We are now in position to prove Theorem \ref{thm-existencerandom}.

\begin{proof}[Proof of Theorem \ref{thm-existencerandom}]
First, we fix
$\omega\in\Omega$ such that $\mu(\cdot,\omega)$ admits
$\underline{\mu}$ as a least mean.
By Theorem \ref{thm-existence} there exists a generalized transition wave in
direction $e$ with a speed $c(t,\omega)$ such that $\ul c(\.,\omega)= \gamma$
and a profile $\phi(z,t,\omega)$ which is decreasing with respect to $z$.
Moreover, $c(t,\omega)=\kappa+\kappa^{-1}\mu(t,\omega)$, where $\kappa$ is the
unique solution in $(0,\sqrt{\underline{\mu}})$ of $\kappa+\kappa^{-1}
\underline{\mu} =\gamma$.
For $s\in\R$, we set $\phi^s(z,t,\omega):=\phi^s(z,t-s,\pi_s\omega)$. As $f$ and
$c$ are random stationary, the functions $\phi$ and $\phi^s$ satisfy the
same equation
$$\partial_t \phi-\partial_{zz}\phi
-c(t,\omega)\partial_z\phi = f(t,\omega,\phi),\quad z\in\R, t\in\R.$$
We further know that $\ul\phi\leq\phi\leq\ol\phi$, where
$\underline{\phi}=\underline{\phi}(z,t,\omega)$ and
$\overline{\phi}=\overline{\phi}(z)$ are given by Proposition
\ref{pro:subsuper}. We point out that $\ol\phi(z)\equiv\min(e^{-\kappa z},1)$
does not depend on $\omega$.
For $\tau>0$, we get
$$\lim_{R\to+\infty}\sup_{\su
{z>R}{t\in\R}}\frac{\phi^s(z,t,\omega)}{\phi(z-\tau,t,\omega)}\leq
\lim_{R\to+\infty}\sup_{\su {z>R}{t\in\R}}\frac{\ol\phi(z)}
{\ul\phi(z-\tau,t,\omega)}<1.$$
Hence,
Proposition \ref{pro:!} gives $\phi^s(\cdot,\cdot,\omega)\leq\phi
(\cdot,\cdot,\omega)$. Exchanging the roles of $\phi^s$, $\phi$ we derive
$\phi^s(\cdot,\cdot,\omega)\equiv\phi(\cdot,\cdot,\omega)$ for almost every $\omega$, that is,
$$\fa (z,t,s,\omega)\in\R\times\R\times \R\times \Omega,\quad  \phi
(z,t+s,\omega)=\phi (z,t,\pi_s\omega).$$
Set $\tilde{\phi} (z,\omega):= \phi (z,0,\omega)$ and $\t
c(\omega):=c(0,\omega)$. We see that the function $u$ defined by
$u(x,t,\omega):=
\t\phi (x\cdot e -\int_0^t \tilde{c}(\pi_s\omega)ds,\pi_t\omega)$ is a random
transition wave with random speed $\tilde{c}$ and random
profile $\tilde{\phi}$.
\end{proof}

\section{Proof of the spreading properties}

This Section is dedicated to the proof of Proposition \ref{prop-spreading}.
We prove separately properties \eq{spreadinginf} and \eq{spreadingsup}.

\begin{proof}[Proof of \eq{spreadinginf}]
Let $\gamma<\gamma'<2\sqrt{\ul\mu}$.
Take $0<\e<\ul\mu$ in such a way that
$\gamma'<2\sqrt{\ul\mu-\e}$. By
definition of least mean over $\R_+$ there exists $T>0$ such that
$$2\sqrt{\inf_{t>0}\frac1T\int_t^{t+T}(\mu(s)-\e)ds}>\gamma'.$$
Hence
$$\gamma^*:=\inf_{k\in\N}\,2\sqrt{\frac1T\int_{(k-1)T}^{kT}(\mu(s+1)-\e)ds}
>\gamma'.$$
Let $\ul v$ be the subsolution given by Lemma \ref{lem:0nT} with
$g(t)=\mu(t+1)-\e$, $\eta=+\infty$ and $\gamma$ replaced by $\gamma'$.
Since $\ul v(\.,0)$ is compactly supported and $u(\.,1)$ is positive by the
strong maximum principle, it is possible to normalize $\ul v$ in such a way that
$\ul v(\.,0)\leq u(\.,1)$.
Moreover, by further decreasing $\ul v$ if need be, it is not restrictive
to assume that
$$\partial_t\ul  v-\Delta\ul  v=(\mu(t+1)-\e)\ul v\leq f(t+1,\ul v), \quad
\text{for a.e.~} x\in\R^N,\ t>0.$$
Therefore,
$u(x,t+1)\geq \ul v (x,t)$ for $x\in\R^N$, $t>0$ by the comparison principle.
Whence,
\Fi{m}
m:=\liminf_{t\to +\infty}\inf_{|x|\leq \gamma't} u(x,t+1)
\geq\liminf_{t\to +\infty}\inf_{|x|\leq \gamma' t}\ul v(x,t)>0.
\Ff
If $f$ was smooth with respect to $t$, then the conclusion would follow from Hypothesis \ref{hyp:KPP} (condition
(\ref{hyp-pos}) in particular) through classical arguments. But as we only
assume $f$ to be bounded and measurable in $t$, some more arguments are needed
here.

Set
$$\theta:=\liminf_{t\to +\infty}\inf_{|x|\leq \gamma t} u(x,t),$$
and let $(x_n)_{n\in\N}$ and $(t_n)_{n\in\N}$ be such that
$$\lim_{n\to\infty}t_n=+\infty,\qquad
\forall n\in\N,\ \ |x_n|\leq\gamma t_n,
\qquad\lim_{n\to\infty}u(x_n,t_n)=\theta.$$
Usual arguments show that, as $n\to\infty$, the functions
$u_n(x,t):=u(x+x_n,t+t_n)$ converge (up to subsequences) weakly in
$W^{2,1}_{p,loc}(\R^N\times\R)$, for any $p<\infty$, and strongly in
$L^\infty_{loc}(\R^N\times\R)$ to a solution $v$ of
$$\partial_t v-\Delta v= g(x,t),\quad x\in\R^N,\ t\in\R,$$
where $g$ is the weak limit in $L^p_{loc}(\R^N\times\R)$ of
$f(t+t_n,u_n)$. We further see that $v(0,0)=\theta$ and that $v$ is bounded
from below by the constant $m$ in \eq{m}. Let $(\xi_k,\tau_k)_{k\in\N}$ be a
minimizing sequence for $v$, that is,
$$\lim_{k\to\infty}v(\xi_k,\tau_k)=\eta:=\inf_{\R^N\times\R}v.$$
In particular, $0<m\leq\eta\leq\theta$.
The same arguments as before, together with the strong maximum principle, imply
that the $v_k(x,t):=v(x+\xi_k,t+t_k)$ converge (up to subsequences) to
$\eta$ in, say, $L^\infty(B_1\times(-1,0))$.
For $h\in\N\backslash \{0\}$, let $k_h, n_h\in\N$ be such that
$$\|v_{k_h}-\eta\|_{L^\infty(B_1\times(-1,0))}<\frac1h,\qquad
\|u_{n_h}-v\|_{L^\infty(B_1(\xi_{k_h})\times
(\tau_{k_h}-1,\tau_{k_h}))}<\frac1h.$$
Hence, the functions $\t
u_h(x,t):=u(x+x_{n_h}+\xi_{k_h},
t+t_{n_h}+\tau_{k_h})$ satisfy
$\|\t u_h-\eta\|_{L^\infty(B_1\times(-1,0))}<\frac2h$ and then $(\t
u_h)_{h\in\N}$ converges to $\eta$ uniformly in $B_1\times(-1,0)$. On the other
hand, it converges (up to subsequences) to a solution $\t v$ of
$$\partial_t\t v-\Delta\t  v=\t g(x,t),\quad x\in B_1,\ t\in(-1,0),$$
where $\t g$ is the weak limit in $L^p(B_1\times(-1,0))$ of
$f(t+t_{n_h}+\tau_{k_h},\t u_h)$. As a consequence,
$$\text{for a.e.~}x\in B_1,\ t\in(-1,0),\quad 0=\t g(x,t)\geq\essinf_{s\in\R}
f(s,\eta).$$
Hypothesis \ref{hyp-pos} then yields $1=\eta\leq\theta$.
\end{proof}

\begin{proof}[Proof of \eq{spreadingsup}]
Let $R>0$ be such that $\supp u_0\subset B_R$. For all $ \kappa>0$ and
$e\in\mathbb{S}^{N-1}$, we define
$$v_{\kappa,e} (x,t):= e^{-\kappa (x\cdot e-R-\kappa t)+\int_0^t \mu (s)ds}. $$
Direct computation shows that the functions $v_{\kappa,e}$ satisfy
$$
\partial_t v_{\kappa,e} -\Delta v_{\kappa,e} -\mu (t)
v_{\kappa,e} =0,\quad x\in\R^N,\ t>0,
\qquad\fa x\in B_R,\quad v_{\kappa,e}(x)>1.$$
Hence, by (\ref{hyp-KPP}), they are supersolutions of \eq{Cauchy} and then
they are greater than $u$ due to the comparison principle.
Let $\sigma>0$, $x\in\R^N$ and $t>0$ be such that
$|x|\geq2\sqrt{t\int_0^t\mu(s)ds}+\sigma t$. Applying the inequality $u(x,t)\leq
v_{\kappa,e}(x,t)$ with $e=\frac x{|x|}$ and $\kappa=\frac{|x|-R}{2t}$ yields
$$u(x,t)\leq\exp\left(-\frac{(|x|-R)^2}{4t}
+\int_0^t\mu(s)ds\right).$$
If in addition $t>R/\sigma$ then $|x|-R\geq2\sqrt{t\int_0^t\mu(s)ds}+\sigma
t-R>0$, whence
$$u(x,t)\leq\exp\left(-\frac{\sigma^2t}{4}-\frac{R^2}{4t}-\sigma\sqrt{t\int_0^t\mu(s)ds}
+R\sqrt{\frac1t\int_0^t\mu(s)ds}+\frac{R\sigma}2\right).$$
Since the right hand side tends to $0$ as $t\to+\infty$, \eq{spreadingsup}
follows.
\end{proof}


\section{Introduction} \label{sec:intro}

The Whirlpool Galaxy (NGC~5194, M51) and its companion (NGC~5195) are a nearby interacting galaxy pair at a distance of $8.58\pm0.1\text{ Mpc}$ \citep{McQuinn2016} in the constellation Canes Venatici. NGC 5194 is a face-on grand design spiral galaxy that lends itself well to studies of its spiral arms, globular clusters, and X-ray binaries XRBs. There have been a large number of studies of the X-ray sources in M51 going back decades. \cite{Terashima2004} studied the X-ray point source population observed in the two of the earliest M51 {\it Chandra X-Ray Observatory} (\chandra) observations (ObsId 354 \& 1622). In a follow-up, \cite{Terashima2006} investigated the candidate optical counterparts to those X-ray point sources  using \hst\ (with the additional \chandra\ observation ObsId 3932). More recently, \cite{Kuntz2016} used most of the available (at the time) \chandra\ data.

X-ray binaries (XRBs) are gravitationally bound systems containing a compact object (black hole or neutron star) accreting matter from a main sequence or massive star companion. XRBs fall generally into two main classes: low-mass (LMXBs) and high mass (HMXBs), distinguished by the mass of the companion star. A LMXB has X-ray emission originating in an accretion disk supplied by Roche-lobe overflow of a low-mass ($M_\text{donor}\le 1.0\,M_\odot$) stellar companion. A HMXB has X-ray emission originating in the accretion of the stellar wind of a high-mass ($M_\text{donor}>8.0\,M_\odot$) stellar companion. HMXBs are divided into two sub-classes: those with O-type companions and those with Be companions. Some excellent reviews on XRBs are found in \cite{Shapiro1983}, \cite{Tauris2006}, and \cite{Remillard2006}.

Images of nearby spiral galaxies taken with \chandra\ reveal bright X-ray sources, many of which are believed to be HMXBs. \citet{Fabbiano1989,Fabbiano2006} present an extensive summary of the X-ray source populations in nearby spiral galaxies. X-ray properties such as X-ray luminosity, hardness ratios, and variability can be utilized to identify and study X-ray binaries \citep{Kaaret2001,Prestwich2003,Luan2018,Jin2019,Sell2019}.

\hst\ provides another avenue to investigate XRBs. While the optical magnitudes of LMXBs will be too small to detect, the massive donor stars of HMXBs can be detected with \hst\ at the distance to M51. For example, Supergiant donors are bright, with $M_{V}$ brighter than $\approx-6.5$ \citep{CI1998}, while Be donors tend to be fainter, with typical $M_{V}$ ranging from $-2$ to $-5$ \citep{McBride2008}. Deep photometry with \hst\ can therefore be used to distinguish between the two main classes of HMXBs.

In this paper we present \chandra\ X-ray and \hst\ optical data analysis on the X-ray sources and their stellar counterpart candidates in the M51 system. In \S\ref{sec:observations}, we describe the X-ray and optical observations used in this study, discuss the results in \S\ref{sec:results}, in \S\ref{sec:conclusions}, we describe our conclusions. Due to the large amount of information, here (Paper I) we will primarily show the methodology used to compile our results. We will present a more in-depth analysis of the X-ray source population, as well as analysis of individual sources in a follow up study (Paper II).

\section{Observations and Data Reduction} \label{sec:observations}

\subsection{Chandra X-ray Observatory}\label{subsec:Chandra}

The Whirlpool Galaxy was the focus of many \chandra\ programs since 2000. For this work, we select data with exposure time $t_{\text{exp}} \ge 10.0\text{ ks}$, resulting in 13 \chandra\ observations, the longest of which is 189\,ks. Information about the X-ray data is listed in Table~\ref{tbl-1}. The data were taken with the Advanced CCD Imaging Spectrometer (ACIS) instrument onboard \chandra. The data were analyzed with the Chandra Interactive Analysis of Observations (CIAO) software version $4.10$ and Chandra Calibration Data Base (CALDB) version 4.7.9\footnote{\href{http://cxc.harvard.edu/ciao/}{http://cxc.harvard.edu/ciao/}}

We aligned all datasets with \emph{US Naval Observatory Robotic Astrometric Camera} (USNO URAT1\footnote{\href{https://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/urat}{https://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/urat}}) Catalog using the CIAO scripts \texttt{wcs\_match} and \texttt{wcs\_update}. Taking into account the new aspect ratio solution and bad pixel files, the observation event files were merged into one event file using \texttt{merge\_obs}. The CIAO script \texttt{mkpsfmap} was run on the full merged event file, taking the minimum PSF map size at each pixel location.

We used the CIAO's Mexican-hat wavelet source detection routine \texttt{wavdetect} \citep{Freeman2002} on the merged data to create source lists. Wavelets of 1, 2, 4, 6, 8, 12, 16, 24, and 32 pixels and a detection threshold of $10^{-6}$ were used, which typically results in one spurious detection per million pixels.

We followed standard CIAO procedures\footnote{\url{http://cxc.harvard.edu/ciao/threads/wavdetect_merged/}}, using an exposure-time-weighted average PSF map in the calculation of the merged PSF. We detected a total of 497 X-ray sources in the merged dataset. In this paper we focus on the sources that are also withing the \hst\ field-of-view, of which there are left 334 (Figure~\ref{fig:m51}). The \texttt{srcflux} CIAO tool was then run individually on each observation (using the coordinates found by \texttt{wavdetect}). The data have been restricted to the energy range between 0.5 and 7.0\,keV and filtered in three energy bands, 0.5--1.2\,keV (soft), 1.2--2.0\,keV (medium), and 2.0--7.0\,keV (hard). We corrected our source catalog to the effects of neutral hydrogen absorption along the line of sight using the Galactic Neutral Hydrogen Density Calculator (COLDEN\footnote{\href{https://cxc.harvard.edu/toolkit/colden.jsp}{https://cxc.harvard.edu/toolkit/colden.jsp}}) tool, finding a mean neutral hydrogen absorption along the line of sight to each source of $n_{\rm {H}} = (1.53\pm0.03)\times10^{20}\text{ cm}^{-2}$. Our fluxes are consistent with the Chandra Source Catalog v2 (CSC\footnote{\url{https://cxc.harvard.edu/csc/}}).

\begin{deluxetable}{cccccc}
\tablecaption{\chandra\ Observations\label{tbl-1}}
\tablewidth{0pt}
\tablehead{
\colhead{ObsId} & \colhead{Date} & \colhead{Detector} & \colhead{Mode\tablenotemark{a}} & \colhead{PI} &
\colhead{Exp\tablenotemark{b}}}
\startdata
354 & 2000-06-20 & ACIS-S & F & Wilson & 15 \\
1622 & 2001-06-23 & ACIS-S & VF & Wilson & 29 \\
3932 & 2003-08-07 & ACIS-S & VF & Terashima & 50 \\
12562 & 2011-06-12 & ACIS-S & VF & Pooley & 10 \\
12668 & 2011-07-03 & ACIS-S & VF & Soderberg & 10 \\
13813 & 2012-09-09 & ACIS-S & F & Kuntz & 180 \\
13812 & 2012-09-12 & ACIS-S & F & Kuntz & 180 \\
15496 & 2012-09-19 & ACIS-S & F & Kuntz & 40 \\
13814 & 2012-09-20 & ACIS-S & F & Kuntz & 190 \\
13815 & 2012-09-23 & ACIS-S & F & Kuntz & 68 \\
13816 & 2012-09-26 & ACIS-S & F & Kuntz & 74 \\
15553 & 2012-10-10 & ACIS-S & F & Kuntz & 38 \\
19522 & 2017-03-17 & ACIS-I & F & Brightman & 40 \\
\enddata
\tablenotetext{a}{F = ``Faint'', VF = ``Very Faint''}
\tablenotetext{b}{Proposed exposure in ks.}
\end{deluxetable}

\subsection{Hubble Space Telescope} \label{subsec:hst}

A six-image mosaic image of M51 with the {\it Hubble Space Telescope} (\hst) Advanced Camera for Surveys was obtained by the Hubble Heritage Team\footnote{\href{https://archive.stsci.edu/prepds/m51/index.html}{https://archive.stsci.edu/prepds/m51/index.html}} (PI: Beckwith, program GO~10452) in January 2005 (see \citealt{Mutchler2005}). The pixel scale of these observations is $0.05''$\,pix$^{-1}$, corresponding to 2.1\,pc\,pix$^{-1}$ at the observed distance of M51. The full mosaic consists of four bands $I$, $V$, $B$, and $H\alpha$ with exposure times of $1360$, $1360$, $2720$, and $2720$ seconds, respectively. The total exposure time is thus $t_{exp} = 8160$\,s over 96 separate exposures. We identified sources in each of the four \hst\ images to align with the URAT1 Catalog and improve the absolute astrometry of the images (similar to \chandra). The common sources totaled 43, distributed across the M51 system. In IRAF, the command \texttt{ccmap} was run on all four of the \hst\ images. The \texttt{ccmap} command finds a six-parameter linear coordinate transformation (plate solution) that takes the $(X,Y)$ centroids and maps them to the more accurate astrometric positions (URAT1 Catalog). In the four bands ($I,V,H\alpha,B$) the mean $(\text{RA},\text{Dec})$ offsets were $(0.142'',0.119'')$, $(0.141'',0.124'')$, $(0.143'',0.124'')$, and $(0.144'',0.117'')$, respectively. We identified candidate \hst\ point sources that fell within $0.5''\,(10\,\text{px})$ of the 334 \chandra\ X-ray point source centroids in our X-ray catalog. We chose $0.5''$ to limit the total number of sources in the catalog while making sure all candidate optical counterparts were identified.

We used the AstroPy package \texttt{photutils}\footnote{\href{https://photutils.readthedocs.io/en/stable/}{https://photutils.readthedocs.io/en/stable/}} to perform photometry calculations on the candidate \hst\ sources. Within \texttt{photutils} we created a circular aperture of radius $r=3.0$ px around each source. The background counts were summed within an annulus centered on each \hst\ point source with inner radius $r_{in}=8.0$\,px and outer radius $r_{out}=11.0$\,px. We corrected for the encircled energy fraction (EEF) using the most recent ACS encircled energy values\footnote{\href{https://www.stsci.edu/hst/instrumentation/acs/data-analysis/aperture-corrections}{https://www.stsci.edu/hst/instrumentation/acs/data-analysis/aperture-corrections}}. The output of \texttt{photutils} on the \hst\ data includes the corrected $(I,V,H\alpha,B)$ magnitudes in the VegaMag system\footnote{\href{https://www.stsci.edu/hst/instrumentation/acs/data-analysis/zeropoints}{https://www.stsci.edu/hst/instrumentation/acs/data-analysis/zeropoints}} for each candidate point source.

\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{m51_final.pdf}
\caption{\label{fig:m51}Combined \hst\ (r: F658N, g: F555W, b: F435W) image with candidate X-ray point source centroids overlaid as white crosses. North is to the right.}
\end{center}
\end{figure*}

\begin{figure*}
\begin{center}
\includegraphics[width=\columnwidth]{fb0-9.pdf}
\includegraphics[width=\columnwidth]{fb10-19.pdf}
\caption{\label{fig:fb09} Broad X-ray flux light curves for the top twenty (by net counts) X-ray sources during ObsId 13813, 13812, 15496, 13814, 13815, 13816, and 15553. {\bf Left}: Broad X-ray flux light curves of the brightest 1-5 (black curves) and next brightest 6-10 (copper curves) X-ray sources. {\bf Right}: Broad X-ray flux light curves of the next brightest 10-15 (black curves) and next brightest 16-20 (copper curves) X-ray sources. The black triangles indicate upper limits, while the dashed black and copper line indicates one particular source ($x_{id}=199$: RA: 13:30:06.0397, DEC: +47:15:42.477) was outside the FOV for ObsIDs 13814, 13815, and 13816 and thus had no measured X-ray counts in those observations, except for a 90\% upper limit on the counts during ObsID 13814. In both panels, in the 30-day window of these observations, some sources vary in flux by approximately two orders of magnitude. {\bf Note}: Line thickness increases with decreasing net counts within each group of five curves. Also note the different $y$-axis scales. The time-averaged mean broad flux error of these twenty sources over the 30-day window of observations in this figure is $\langle\delta F_{X,b}\rangle\simeq\,5.7\times10^{-15}\,{\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$, and for most of the data points the error is within the line thickness.}
\end{center}
\end{figure*}

\subsection{Optical Counterparts}\label{subsec:counterparts}

Candidate point source optical counterparts were found by identifying the brightest \hst\ point source within the \chandra\ positional uncertainty of each X-ray source. We used a 90\% confidence level positional uncertainty of $0.5''$ typical for a $5'$ off-axis X-ray source with 50 counts (see Eq. 12 in \citealt{Kim2007}). This positional uncertainty corresponds to $20.8$\,pc at the distance of M51. In total, there are 173 such candidate optical counterparts. The closest \hst\ source to the X-ray centroid is not always the brightest and often is not visible in all four \hst\ bands, so we justify the optical counterpart candidate identification process in this way. It is possible that the true physical counterparts are invisible in the four \hst\ bands and we identify the incorrect physical counterpart using this method, but it seems to capture the majority of the sources sufficiently. We used the $I$-band images to select the brightest candidate optical counterpart within the \chandra\ $1\sigma$ uncertainty. If we select the closest candidate optical counterpart in the same way, we pick up $\sim 65\%$ (113/173) of the same sources; that is, 60 of the candidate \hst\ counterparts are the brightest, but not closest sources to the X-ray centroids.

\section{Results \& Discussion}\label{sec:results}

\subsection{X-ray Variability}\label{subsec:xrayvariability}

We look for short term X-ray variability using seven \chandra\ observations from 2012 (ObsId 13813, 13812, 15496, 13814, 13815, 13816, and 15553). These observations span over a one month period (see Table~\ref{tbl-1}). In Figure~\ref{fig:fb09}, we plot the broad X-ray flux light curves of the brightest twenty (by net counts) X-ray sources. The brightest ten are shown in the left panel, while the next brightest ten are shown in the right panel. In the 30-day window of these observations, some sources vary in flux by approximately two orders of magnitude.

We calculate a reduced chi-square statistic, $\chi^2_\nu$, for each broad flux light curve in the 30-day window as a measure of variability. We assume the null hypothesis that the underlying broad X-ray flux light curve is described by a uniform function whose value is the weighted mean of the flux across the seven observations in the 30-day window. The chi-square statistic $\chi^2$ used here is defined as
\begin{align}
    \chi^2 &\equiv \sum\limits_{i=1}^\nu \frac{(F_i-\mu_i)^2}{\sigma_i^2},
\end{align}
where $F_i$ is the X-ray flux of the $i$th source, $\mu_i$ is the weighted mean error of the associated $i$th flux measurement, and $\sigma_i^2$ is the variance of the $i$th flux measurement. The reduced chi-square statistic is simply $\chi^2_\nu\equiv\chi^2/\nu$. The mean of the chi-square distribution is $\nu$, so that $\chi^2_\nu = 1$ is a natural value with which to compare results. The value of $\chi^2_\nu$ should be approximately unity if the null hypothesis is to be accepted. Large values of $\chi^2_\nu$ indicate that the null hypothesis should be rejected. Thus, the sources with $\chi^2_\nu \ge 1$ are sources whose flux varies greatly in the 30-day window and we label them ``variable'' sources. Sources with $\chi^2_\nu \lesssim 1$ are sources that have a light curve in the 30-day window that is consistent with the null hypothesis (uniform flux).

Approximately $80\%$ ($266/334$) of the sources are considered variable by our $\chi^2_\nu \ge 1$ criterion. Approximately $69\%$ (120/173) of the sources with at least one detected candidate stellar counterpart and no cluster counterparts are variable (see our upcoming follow-up Paper II for a discussion of cluster counterparts), while about $77\%$ (124/161) of the sources without a stellar or cluster counterpart are variable. In addition, about $76\%$ (22/29) of the X-ray sources that have both an associated candidate stellar source and candidate cluster are considered variable. There is a strong positive correlation between the variability and flux of the X-ray sources. Our findings are consistent with the inter-observation variability reported in the CSC (for sources that overlap, which is the majority of sources), even though we have limited our variability study to data within this 30-day window. We speculate that the observed strong correlation is due to the small uncertainty associated with very bright sources (see Figure~\ref{fig:fb09}), i.e. the time-averaged mean broad flux error over the 30-day window of observations is $\langle\delta F_{X,b}\rangle\simeq\,5.7\times10^{-15}\,{\rm erg}\,{\rm cm}^{-2}\,{\rm s}^{-1}$.

\subsection{X-ray Hardness Ratios}\label{subsec:hardness}

We calculate two X-ray hardness ratios (HRs), ``soft'' and ``hard'' ($HR_1$ and $HR_2$, respectively), for all X-ray sources using the same seven observations as follows:
\begin{align}
    HR_1 &\equiv \frac{M-S}{M+S}\quad\text{and}\\
    HR_2 &\equiv \frac{H-M}{H+M},
\end{align}
\noindent where $S$, $M$, and $H$ are the X-ray counts in each of the \chandra\ bands (soft, medium, and hard) discussed in \S\S\ref{subsec:Chandra}. We also calculate the associated uncertainty in each of the hardness ratios.

In Figure~\ref{fig:hrchi2}, top panel, we plot the X-ray color-color diagram for all sources, colored by the logarithm of their reduced chi-square statistic calculated in the 30-day window discussed in \S\S\ref{subsec:xrayvariability}. The two X-ray colors are the measurements from the longest of the observations in the 30-day window, ObsId 13814.

Hardness ratio diagrams, such as our Figure~\ref{fig:hrchi2} and Figure~4 in \cite{Prestwich2003} (which uses a different definition\footnote{In \cite{Prestwich2003}, they define the hard and soft X-ray colors as $HR_1\equiv(M-S)/T$ and $HR_2\equiv(H-M)/T$, respectively, where $S$, $M$, $H$, and $T\equiv S+M+H$ are the soft, medium, hard, and total X-ray counts, respectively.} of the X-ray hardness ratios), have been used historically to assist with revealing the nature of the X-ray sources. The majority of the variable sources ($\chi^2_\nu \ge 1$) lie in the XRB (LMXB and HMXB) regions of the figure (see e.g., \citealt{Prestwich2003}), while most of the low variability sources lie in the region of the diagram that is generally occupied by thermal supernova remnants. However, it is well established that X-ray information alone is not enough to accurately identify the nature of unknown X-ray sources. Therefore, we use the X-ray colors together with optical information (see Section~\ref{subsec:opticalcounterparts}) to classify these sources.

In Figure~\ref{fig:hrchi2} we also plot the X-ray color-color diagrams of the brightest twenty (by net counts) X-ray sources; the top ten brightest in the middle panel and the next ten brightest in the bottom panel. Overlaid in all three plots are the hardness color evolution tracks of various accretion disk models. In blue are power law models with increasing photon index $\Gamma$ from $0.4-4$, in orange are absorbed power law models with increasing hydrogen column density, in green are disk blackbody models with temperature ranging from $0.02-2.0\,{\rm keV}$, and in red are absorbed bremsstrahlung models with temperature ranging from $0.1-10.0\,{\rm keV}$. These color-color diagrams contain X-ray colors from all available data in the 30-day window, with appropriate $1\sigma$ error bars. Each source has multiple (same plotted color) points in the diagram, and the color-color evolution is thus apparent. Typically, the color-color evolution is $\lesssim 0.5$ in either color over the entire data set. This suggests that while some spectral change may occur over the 30-day period, the accretion process for these sources does not change dramatically. There are a few sources that appear to significantly change their spectral properties as indicated by movement in the plane of the X-ray color-color diagram, for example the middle panel of Figure~\ref{fig:hrchi2}. It is possible, however, that the movement in the color-color diagram could arise due to a drop in flux, which would raise signficant uncertainties in the location of a source in the diagram due to low count statistics. The error bars are large enough for many of these sources that the spectral evolution cannot be confidently confirmed. Attempting to track the spectral evolution of fainter sources becomes meaningless due to the large uncertainties associated with the X-ray hardness ratio measurements. Detailed analysis of bright sources will be presented in Paper II.

\begin{figure}
\begin{center}
\includegraphics[trim= 5mm 5mm 5mm 5mm clip, width=\columnwidth]{Fig_3_top.pdf}
\includegraphics[trim= 5mm 5mm 5mm 3mm clip, width=\columnwidth]{Fig_3_middle.pdf}
\includegraphics[trim= 5mm 5mm 5mm 3mm, clip, width=\columnwidth]{Fig_3_bottom.pdf}
\caption{\label{fig:hrchi2} {\bf Top}: X-ray color-color diagram for X-ray sources, colored by the logarithm of their reduced chi-square statistic calculated in the 30-day window. The values of $HR_1$ and $HR_2$ in the figure are those of the longest observation (ObsId 13814) in the 30-day window. Overlaid in all three plots are the hardness color evolution tracks of various accretion disk models, see text for details.
{\bf Middle}: Color-color diagram of the brightest ten sources (by net counts), with $1\sigma$ error bar shown in black.
{\bf Bottom}: Color-color diagram of the next brightest ten sources (by net counts), with $1\sigma$ error bar shown in black.
}
\end{center}
\end{figure}

\subsection{X-ray Luminosity Function}\label{subsec:xrayluminosity}

The X-ray luminosity function (XLF) for all the 334 X-ray point source candidates in M51 can be approximated as a power law within a certain luminosity range. We use the differential luminosity function defined as
\begin{align}
    \frac{dN(>L_0)}{dL} &= A\biggl(\frac{L}{L_0}\biggr)^{-\alpha},\label{eq:dXLF}
\end{align}
\noindent where the luminosity $L_0$ is an arbitrary lower limit and $A$ is some normalization constant. Integrating Eq.~\ref{eq:dXLF} gives the luminosity function within a particular range:
\begin{align}
    N(>L_0) &= \frac{AL_0}{1-\alpha}\biggl(\frac{L}{L_0}\biggr)^{1-\alpha},
\end{align}
and the fractional luminosity function is given by
\begin{align}
    f(>L_0) &= \frac{AL_0}{N_{tot}(1-\alpha)}\biggl(\frac{L}{L_0}\biggr)^{1-\alpha},
\end{align}
where $f(>L_0)$ is the fraction of sources with $L>L_0$ and $N_{tot}$ is the total number of sources.

An important luminosity is the Eddington luminosity of a $1.4\,M_\odot$ compact object (the typical mass of NSs) accreting at the Eddington rate:
\begin{align}
    \nonumber\dot{L}_E &\equiv \dot{M}_Ec^2 = \nonumber\frac{4\pi Gm_pc}{\sigma_T}M\\
&\simeq 1.76\times10^{38}\text{ erg$\,$s$^{-1}$}\,\biggl(\frac{M}{1.4\,M_\odot}\biggr),\label{eq:Eddi}
\end{align}
\noindent where $\dot{M}_E$ is the Eddington accretion rate, $M$ is the mass of the accretor, $G$ is Newton's gravitation constant, $m_p$ is the proton mass, $c$ is the speed of light, and $\sigma_T$ is the Thomson scattering cross section for electrons.

In Figure~\ref{fig:XLF}, we plot the combined XLF (total and fractional) on various cuts of the data. The purple curve is the full sample of the $86\%$ (288/334) of X-ray sources that have a measured X-ray luminosity in ObsId 13814 (the observation with the longest exposure time). The green curve is the $39\%$ (130/334) of X-ray sources that have a stellar counterpart in \hst\ (within 10\,px). Across few orders of magnitude of X-ray luminosity starting at $L_{X,b} \ge 2\times10^{36}\text{ erg$\,$s$^{-1}$}$ the curves follow a power law $N(>L_{X,b})\propto L_{X,b}^{-0.65}$, i.e. we fit a power-law to the differential luminosity function with $\alpha = 1.65\pm0.03$. This is consistent with XLFs for star forming galaxies dominated by HMXBs, for example \cite{Lehmer2019} who find $\alpha = 1.59\pm0.05$ for M51. The blue curve represents the X-ray sources that have no stellar or cluster counterparts within 10\,px, $40\%$ (133/334) and has the  same slope. The black vertical dashed line indicates the Eddington luminosity of a canonical NS (e.g., $1.4\,M_\odot$) accretor of $L_{Edd}\simeq 1.8\times10^{38}\text{ erg$\,$s$^{-1}$}$. Fewer than $10\%$ of the sources have an X-ray luminosity that is greater than $L_{Edd}$ for the typical NS accretor.

A major obstacle in studying the extragalactic XRB population is differentiating HMXBs from LMXBs, which cannot be done by their X-ray properties alone. One attempt to solve this problem was done by \cite{Mineo2012} who used galactocentric distance to distinguish between the two types of XRBs. However, many galaxies, including spirals such as M51, show a spatially mixed population of ``young'' and ``old'' XRBs. Our results show that combining \chandra\ and \hst\ data can break this degeneracy.

\begin{figure*}
\begin{center}
\includegraphics[width=\columnwidth]{XLF_10pix.pdf}
\includegraphics[width=\columnwidth]{XLF_frac_10pix.pdf}
\caption{\label{fig:XLF} {\bf Left}: Combined X-ray luminosity functions. The purple curve is the full sample of 288/334 X-ray sources that have a measured X-ray luminosity in ObsId 13814. For $L_{X,b}\ge 2\times10^{36}\text{ erg$\,$s$^{-1}$}$, the purple curve follows a power law $N(>L_{X,b})\propto L_{X,b}^{1-\alpha}$ where $\alpha = 1.65\pm0.03$; see text for details. The green curve is a cut of the full sample with 130/334 X-ray sources that have only a stellar \hst\ candidate counterpart within 10 px. The blue curve is a cut of the full sample with 133/334 X-ray sources that have no stellar or cluster counterparts within 10 px. The vertical dashed black line is the Eddington luminosity of a $1.4\,M_\odot$ accretor, i.e. $L_{Edd}\simeq 1.76\times10^{38}\text{ erg$\,$s$^{-1}$}$ (see Eq. \ref{eq:Eddi} in the text).
{\bf Right}: Combined fractional X-ray luminosity functions. Same as the left panel but normalized.
}
\end{center}
\end{figure*}

\subsection{Optical Counterparts to X-ray Sources}\label{subsec:opticalcounterparts}

Due to the distance to M51 there are issues with crowding and source confusion. Many X-rays sources ($51\%$; 173/334) have at least one \hst\ stellar counterpart within the $0\farcs5$ \chandra\ positional uncertainty, whereas ($75\%$; 252/334) have at least one \hst\ stellar counterpart within the 2$\sigma$ \chandra\ uncertainty. Just over half, $51\%$ ($88/173$), of sources that have at least one detected \hst\ stellar counterpart within 1$\sigma$ have at least two detected candidate stellar counterparts.

Selecting the counterpart candidate can be challenging in cases where there are more than two or more optical sources in the search radius. One method of choosing the donor star candidate is to select the closest optical source to the \chandra\ position. On the  other hand, large fraction of the XRBs in M51 are expected to be HMXBs with early-type stars as the donors. Therefore, an alternative method of selecting an optical counterpart is to select the brightest optical sources within the 1$\sigma$ radius ($10$ px).
In Figure~\ref{fig:b-v_v-i} we plot the $B-V$ and $V-I$ color-magnitude diagrams for the candidate \hst\ optical counterparts that are the brightest or closest within $10$ px of the X-ray point source centroids. If we select the closest candidate \hst\ optical counterpart within $10$ px, out of 173 total candidate optical counterparts, $\sim65.3\%$ (113/173) of the sources are the same. That is, 113 of the \hst\ counterparts are both the brightest and the closest source within $10$ px. As expected, selecting the closest optical counterpart to the X-ray sources is biased toward fainter (and older) stellar sources. However, we performed a two-sample Kolmogorov-Smirnov (K-S) test on the following data from Figure~\ref{fig:b-v_v-i}:
\begin{enumerate}
\item $M_{V,closest}$ vs. $M_{V,brightest}$ ($y$-axis of both panels)
\item $(m_B-m_V)_{closest}$ vs. $(m_B-m_V)_{brightest}$ ($x$-axis of left panel)
\item $(m_V-m_I)_{closest}$ vs. $(m_V-m_I)_{brightest}$ ($x$-axis of right panel),
\end{enumerate}
and found that in each case the null hypothesis $H_0$, namely that the two samples in 1, 2, and 3 above are drawn from the same unknown underlying continuous distribution, cannot be rejected. The two-sample KS test statistic $D$ and $p$-values for each of the three tests above are:
\begin{enumerate}
\item $D = 0.13194$ and $p = 0.15056$
\item $D = 0.07639$ and $p = 0.77899$
\item $D = 0.06250$ and $p = 0.93376$.
\end{enumerate}
At a level of significance $\alpha=0.05$, we cannot reject $H_0$ since in each case $p\ge\alpha$. Thus we cannot claim a statistically significant difference in choosing either the closest or the brightest sources as the candidate optical counterpart to our X-ray sources. The mean photometric error is approximately $0.1$ mag in $V$ and $I$. In Table~\ref{tbl-2}, we select the brightest source as the donor star candidate in case of multiple matches.

Also in Figure~\ref{fig:b-v_v-i} we plot four mass tracks: $5\,M_\odot$, $8\,M_\odot$, $20\,M_\odot$, and $40\,M_\odot$, respectively from bottom to top, taken from the MESA Isochrones \& Stellar Tracks\footnote{\href{http://waps.cfa.harvard.edu/MIST/index.html}{http://waps.cfa.harvard.edu/MIST/index.html}} (see \citealt{Dotter2016,Choi2016,Paxton2011,Paxton2013,Paxton2015,Paxton2018}). The initial protosolar bulk metallicity for the models used is $Z_i = 0.0147$, with extinction $A_V=0$ ($R_V=3.1$). It is clear from the color-magnitude diagram that most of the candidate \hst\ optical counterparts lie above the $8\,M_\odot$ mass track, indicating that most of our candidate sources are likely HMXBs. In classifying the candidate sources as HMXBs, there are no statistically significant differences in choosing either the brightest (black) or closest (light orange) sources (see the two-sample K-S test discussion above).

\begin{figure*
\begin{center}
\includegraphics[width=85mm]{b-v_final.pdf}
\includegraphics[width=85mm]{v-i_final.pdf}
\caption{\label{fig:b-v_v-i}Color-magnitude diagrams for the potential HST optical counterpart candidates that are the brightest (black) or closest (light orange) within 10 px of the X-ray source centroids. The dark orange indicates a candidate optical counterpart is both the brightest and closest to the X-ray centroid. There are 173 candidate optical counterparts within 10 px of the 334 \emph{Chandra} X-ray point sources. For both panels, the four mass tracks from bottom to top are from MESA Isochrones \& Stellar Tracks (MIST; see text) and have masses $5\,M_\odot$, $8\,M_\odot$, $20\,M_\odot$, and $40\,M_\odot$, respectively. {\bf Left}: $B-V$ color-magnitude diagram. {\bf Right}: $V-I$ color-magnitude diagram.}
\end{center}
\end{figure*}

\section{Conclusions}\label{sec:conclusions}

In this study we presented a catalog and statistical analysis of archival \chandra\ and \hst\ data of point sources in the interacting galaxy pair NGC 5194/5195 (M51).

\begin{itemize}

\item Using standard CIAO procedures, we detected 334 X-ray point sources in the merged thirteen \chandra\ observations. We corrected the data for neutral hydrogen absorption along the line of sight and improved the astrometry using the USNO URAT1 catalog.

\item We identified 173 candidate optical counterparts to the X-ray sources in our catalog by finding the brightest \hst\ point sources within 10 px of the X-ray source. We found no statistically different results by choosing the closest \hst\ point sources by performing a two-sample Kolmogorov-Smirnov test (see text for details). Similar to \chandra\ the astrometry of the data was corrected by using the USNO URAT1 catalog and applying a six-parameter plate transformation.

\item We calculated a reduced chi-square statistic, $\chi_\nu$, as a measurement of the broad flux variability in a 30-day window of the longest seven observations for the X-ray sources in our catalog and found that approximately $80\%$ of the sources are considered variable, i.e. $\chi_\nu\ge 1$.

\item Approximately $69\%$ of the sources with at least one detected candidate stellar counterpart (but no cluster counterpart) are considered variable and about $77\%$ of the sources without a stellar counterpart are variable (see our upcoming follow-up Paper II for a discussion of candidate cluster counterparts to our X-ray sources).

\item The majority of optical counterparts are above the 8\,M$_\odot$ line in  Figure~\ref{fig:b-v_v-i}, which is consistent with these sources being HMXB candidates.

\item There is a strong positive correlation between the broad X-ray flux and the variability of the X-ray sources in the 30-day window, consistent with the interobservation variability in the CSC catalog.

\item We calculated X-ray hardness ratios for all sources and found that the majority of the variable sources lie in the XRB region of the X-ray color-color diagram (e.g., hard or absorbed X-ray sources; see Figure~\ref{fig:hrchi2}).

\item The broad X-ray luminosity function above a few times $10^{36}\, {\rm erg\,s^{-1}}$ follows a power law  $N(>L_{X,b})\propto L_{X,b}^{1-\alpha}$ with $\alpha=1.65\pm0.03$, consistent with X-ray luminosity functions of star-forming galaxies dominated by HMXBs.

\item Most of the brightest 20 sources do not show any evidence of of flux variability.

\item Fewer than $10\%$ of the X-ray sources have a broad X-ray luminosity greater than the Eddington luminosity of a typical NS accretor.
\end{itemize}

As mentioned earlier, a detailed analysis of individual sources will be presented in a follow-up paper.

\software{CIAO \citep{Fruscione2006}, IRAF \citep{Tody1986,Tody1993}, SAOImageDS9 \citep{Joye2003}, NumPy \citep{Harris2020}, SciPy \citep{Virtanen2020}, Matplotlib \citep{Perez2007}, Astropy \citep{Astropy2013,Astropy2018}}

\section{Acknowledgments}\label{sec:acknowledgments}

We thank an anonymous referee for constructive comments.

\begin{deluxetable*}{cccccccccrr}
\tablecaption{Catalog of X-ray sources\label{tbl-2}}
\tablewidth{0pt}
\tablehead{
\colhead{$X_\text{ID}$} & \colhead{RA} & \colhead{Dec} & \colhead{$C_\text{net}/10^3$} & \colhead{$F_{X,b}/(10^{-15}\text{erg}\,\text{s}^{-1})$} &
\colhead{$I$} & \colhead{$V$} & \colhead{$H\alpha$} & \colhead{$B$} & \colhead{$HR_1$} & \colhead{$HR_2$}}
\startdata
$178$ & $202.504274$ & $47.228885$ &  $6.73$ & $441$ & $-8.26$ & $-7.58$ & $-8.08$ & $-6.62$ & $0.073$ & $0.012$\\
$62$ & $202.531533$ & $47.185057$ & $6.03$ & $263$ & $-6.70$ & $-5.10$ & $-7.27$ & $-4.69$ & $-0.083$ & $-0.37$\\
$166$ & $202.496180$ & $47.221801$ & $3.63$ & $240$ & $-6.18$ & $-3.83$ & $-4.91$ & $-2.62$ & $0.64$ & $0.069$\\
$190$ & $202.473939$ & $47.243281$ & $2.25$ & $153$ & $-$ & $-$ & $-$ & $-$ & $0.14$ & $0.027$\\
$154$ & $202.414552$ & $47.212141$ & $2.05$ & $109$ & $-6.35$ & $-6.58$ & $-6.34$ & $-6.67$ & $0.023$ & $-0.15$\\
$100$ & $202.470043$ & $47.194362$ & $1.90$ & $62.4$ & $-8.49$& $-7.47$ & $-8.05$ & $-6.49$ & $-0.64$ & $-0.79$\\
$94$ & $202.430548$ & $47.193022$ & $1.77$ & $70.7$ & $-6.97$ & $-$ & $-$ & $-$ & $-0.83$ & $-0.82$\\
$149$ & $202.416637$ & $47.210268$ & $1.65$ & $64.1$ & $-5.72$ & $-5.74$ & $-6.14$ & $-5.67$ & $-0.36$ & $-0.55$\\
$168$ & $202.517998$ & $47.222471$	& $1.38$ & $92.0$ & $-8.06$ & $-7.23$ & $-8.15$ & $-6.80$ & $0.22$ & $-0.033$\\
$54$ & $202.489953$ & $47.180183$ & $1.08$ & $56.7$ & $-7.11$ & $-5.65$ & $-5.86$ & $-5.42$ & $0.065$ & $-0.21$
\enddata
\end{deluxetable*}

\newpage