Title\thanksref{label1}
Title\thanksref{label1}
\selectlanguage{english} We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant $\epsilon$ goes to zero. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis. %{To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 340 (2005).} \vskip 0.5\baselineskip \selectlanguage{francais} % Text of abstract in French \noindent{R\'esum\'e} \vskip 0.5\baselineskip \noindent {Concentration de la premi\`ere fonction propre pour un op\'erateur elliptique du second ordre} %Your resume in French here. Nous \'etudions sur une vari\'et\'e Riemannienne compacte, les limites semiclassiques de la premi\`ere fonction propre associ\'ee \`a un op�rateur positif du second ordre positif divers quand la constante de diffusion $\epsilon$ tend vers z\'ero. Nous supposons que le terme d'ordre un est un champ de vecteur $b$, dont les ensembles r\'ecurrents sont des points hyperboliques ou des cycles ou des tores � deux dimensions. Les limites de la fonction propre normalis\'ee sont concentr\'ees sur les ensembles r\'ecurrents de dimension maximale o\`u la pression topologique est atteinte. Sur le cycles et les tores, les mesures limites sont absolument continues par rapport \`a la mesure de probabilit\`e invariante par $b$. Nous avons d\'etermin\'e ces limites en utilisant une analyse de type blow-up. %{Pour citer cet article~: D. Holcman, I. Kupka, C. R. Acad. Sci. %Paris, Ser. I 340 (2005).}
math-ph0506059.tex
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\selectlanguage{english}
\title{Concentration of the first eigenfunction for a second order elliptic operator}

\author{David Holcman \thanks{
Weizmann Institute of Science, department of Mathematics, Rehovot 76100,
Israel. D. H. is incumbent of the Madeleine Haas Russel Career Development
Chair.Visiting  Departement de Biologie, Ecole Normale Sup\'erieure, 
46, rue d'Ulm, 75005 Paris, France. } \and Ivan Kupka\thanks{
Universit\'e Paris VI, department of Mathematics, 175 rue du Chevaleret
75013 Paris, France. I.K. thanks the Weizmann Institute of Science for its
hospitality during the preparation of this paper.}}
\date{}
\maketitle

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%{\small Received *****; accepted after revision +++++\\
%Presented by Thierry Aubin}
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\begin{abstract}
\selectlanguage{english}
We study the semi-classical limits of the first eigenfunction of a positive
second order operator on a compact Riemannian manifold when the diffusion constant
 $\epsilon$ goes to zero. We assume that the first order term is given by a vector
 field $b$, whose recurrent components  are either   hyperbolic points  or cycles 
or two dimensional torii. The limits of the normalized  eigenfunctions
concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} 
is attained. On the cycles and torii, the limit measures are absolutely continuous 
with respect to the invariant probability measure on these sets. We have determined these limit 
measures, using a  blow-up analysis.

%{\it To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 340 (2005).}

\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
{\bf Concentration de la premi\`ere fonction propre pour un op\'erateur elliptique du second ordre}
%Your resume in French here.

Nous \'etudions sur une vari\'et\'e Riemannienne compacte, les limites semiclassiques de la premi\`ere fonction propre associ\'ee \`a un op�rateur positif du second ordre positif divers quand la
constante de diffusion $\epsilon$ tend vers z\'ero. Nous supposons que le terme d'ordre un est  un champ 
de vecteur $b$, dont les ensembles r\'ecurrents sont des points hyperboliques ou des cycles ou des tores � 
deux dimensions. Les limites de la fonction propre normalis\'ee sont concentr\'ees sur les ensembles
r\'ecurrents de dimension maximale o\`u la pression topologique est atteinte. Sur le cycles et les tores, 
les mesures limites sont absolument continues par rapport \`a la mesure de probabilit\`e invariante par $b$. Nous avons d\'etermin\'e ces limites en utilisant une analyse de type blow-up. 

%{\it Pour citer cet article~: D. Holcman, I. Kupka, C. R. Acad. Sci.
%Paris, Ser. I 340 (2005).}

\end{abstract}


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\selectlanguage{english}
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\section{Statement of the  problem }\label{Introduction}

On a compact Riemannian manifold $(M,g)$, we study the semi-classical limits 
of the first eigenfunction of a positive second order operator 
\begin{eqnarray}
{L}_{\epsilon}=\varepsilon \Delta _{g} +\theta (b)+c, 
\end{eqnarray}
when the diffusion constant $\epsilon$ goes to zero. $\Delta _{g}$ is 
the Laplace-Beltrami operator and $\theta (b)$ is the Lie derivative in the 
direction $b \in TM$. The function $c$ is chosen so that $%
L_{\epsilon }$ is positive. Note that $L_{\epsilon }$  cannot be conjugated in general to a
self-adjoint operator by scalar multiplication. The Krein-Rutman theorem implies that the first
 eigenvalue $\lambda _{\epsilon }$ is real positive, simple and the associated eigenfunction
 $u_{\epsilon }$ is of a fixed sign (see \cite{Kifer90,Pinsky}).
The  $L_2$ normalized  positive eigenfunction $u_{\epsilon }$,($\int_{V_{n}}u_{\epsilon }^{2}dV_{g} =1$), 
 is solution of 
\begin{eqnarray}  \label{edpfdt}
\epsilon \Delta _{g}u_{\epsilon }+(b,\nabla u_{\epsilon })+cu_{\epsilon }
=\lambda _{\epsilon }u_{\epsilon } .  \nonumber
\end{eqnarray}
 Let us introduce the following notation, 
\begin{eqnarray}
\Omega= b-\nabla \mathcal{L}\, ,v_{\epsilon }=e^{-\frac{\mathcal{L}}{%
2\epsilon }}u_{\epsilon }  \label{gauge} \,\, ,  c_{\epsilon } =\epsilon (c+\frac{\Delta _{g}\mathcal{L}}{2})+\Psi _{ \mathcal{L}},  \,\, , 
\Psi _{\mathcal{L}} =\frac{1}{4}(||{\nabla \mathcal{L}||}^{2}+2(\nabla 
\mathcal{L},\Omega )),  \nonumber
\end{eqnarray}%
where the function $\mathcal{L} M:\rightarrow \rR$ belongs to a special class of Lyapunov functions, associated to the vector field $ b$, as described in \cite{HK1}.
Equation (\ref{edpfdt}) is transformed into            
%\begin{eqnarray} \label{vrai}
$L_{\epsilon }(v_{\epsilon })=\epsilon ^{2}\Delta _{g}v_{\epsilon }+\epsilon
(\Omega ,\nabla v_{\epsilon })+c_{\epsilon }v_{\epsilon }=\epsilon \lambda
_{\epsilon }v_{\epsilon },$
and we impose the normalization condition $\int_{V_{n}}v_{\epsilon }^{2}dV_{g}=1.$

In this situation, a fundamental problem is to study 1) the limits of $\lambda_{\epsilon }$ 
2) the limits of the measures $v_{\epsilon }^2dV_g$, as $\epsilon$ goes to zero.

\noindent  \textbf{History of the problem} 

The study of $\lambda _{\epsilon }$ has its origin in the work of Kolmogorov about the mean first passage time (MFPT) of a random particle to the boundary of a bounded domain. For domains in $\rR^n$, very interesting contributions to the first problem were made, in particular by Friedlin-Wentzell \cite{FW} and Devinatz-Friedman \cite{Devi1,Fried}. In the global situation (compact Riemannian manifold),  Kifer  in \cite{Kifer90} made a fundamental contribution to the study of the limit of $\lambda _{\epsilon }$: When the recurrent set of the drift $b$ is a finite union of hyperbolic components \cite{Robinson}, $\lambda _{\epsilon }$ has a unique  limit, which can be explicitly determined in terms of the topological pressure of the components.

% and $v_{\epsilon }$ as $\epsilon$ goes to zero is
% unknown in general. However,  addtional assumptions are given on b, this problem 
%has been extensively studied: when $c=0$, on compact set $B$ of $\rR^n$, $L_{\epsilon }$ is the 
%Fokker-Planck operator and $\lambda _{\epsilon }$ is exactly the mean first passage time(MFPT) it takes for a trajectory following the stochastic equation: 
%\begin{eqnarray}
%dX_{\epsilon }(t)=b(X_{\epsilon }(t))dt+\sigma (X_{\epsilon }(t))d\mathbf{w}
%\end{eqnarray}
%where $\mathbf{w}$ is the N-dimensional classical Brownian process, to escape the domain B for the first time. This problem orginated in the work of Kolmogorov.
% When no recurrent point of b are located in B (Levinson's conditions),  $\lambda _{\epsilon }\sim_{\epsilon \sim 0} \frac{C}{\epsilon}$. If B contains a single attractor point, then $\lambda _{\epsilon }\sim_{\epsilon \sim 0} e^{\frac{-C}{\epsilon}}$, where $C>0$. This result was obtained by Wencell, in a more general context, Friedman \cite{Fried1,Fried,Devi1}, see also \cite{FW}. More explicit results are obtained in \cite{Devi2}. Using singular perturbation techniqued and blow up analysis, an explicit computation of the MFPT was obtained by Matkowsky and Schuss in \cite{Mat}, see also \cite{Mat2}.
% On a compact Riemannian manifold,  Kifer \cite{Kifer90} gave a complete description of $\lambda _{\epsilon }$,  when the drift satisfies the well known Axiom-A assumption \cite{Robinson}. In that case, $\lambda _{\epsilon }$ converges to the topological pressure, denoted TP.
%\begin{eqnarray*}
%TP &=& {\sup_{{1\leq {\ j}\leq N }} }TP(K_{j}),\hbox{ where } \\
%TP(K_{j}) &=&\sup_{\mu }\{h_{\mu }+\int_{V}(c-\frac{d\det D\phi _{t}^{u}}{dt}%
%)d\mu \mid \mu \in P(V),\hbox{ support }\mu \subset K_{j}\,,\ \mu \,\ \phi
%_{t}-invariant\}
%\end{eqnarray*}%
%where $h_{\mu }$ is the metric entropy \cite{Kifer90}, $\phi _{t}$ is
%the flow of the vector field $b$ and $D\phi _{t}^{u}$ is the tangent mapping
%of $\phi $ restricted to the unstable bundle $T_{K_{j}}^{u}V$ of $K_{j}$ and 
%$\det D\phi _{t}^{u}$ is the determinant of $\ D\phi _{t}^{u}$ with respect
%to the metric $g.$  There are  measure $\mu $, solution of the variational problem
%associated to topological pressure, which are $\phi _{t}-invariant$. $\mu _{j}\in P(V)$ with support in $K_{j}$ such that 
%\begin{eqnarray}
%TP(K_{j})=h_{\mu _{j}}+\int_{V}(c-\frac{d\det D\phi _{t}^{u}}{dt})d\mu _{j}
%\end{eqnarray}
%Up to relabeling the $% K_{j}$ we can assume that $TP(K_{j})=TP$ if 1$\leq j\leq l$, and $%
%TP(K_{j})=TP$ \ if $j>l.$ An extremal  measure can be written $\mu =\sum_{j=1}^{l}p_{j}\mu _{K_{j}}$ where $\mu _{K_{j}}$ is supported on the set $K_{j}$, $p_{j}\geq
%0,\sum_{j=1}^{l}p_{j}=1,$.

\noindent\textbf{Limits of the first eigenfunction}

Problem 2) was much less studied than problem 1). However in the local situation, there are some interesting contributions in the paper \cite{Devi2} by Devinatz-Friedman. Otherwise very little was known. Here, we study this problem 2) in the spirit of  Kifer.  We consider drift $b$ such that the recurrent set is the union of a finite number of hyperbolic components, which are  either points, cycles or 2-dimensional torii. 

Let us mention that problem 2) is related to questions  studied in the unique ergodic conjecture \cite{Sarnak},
 where the limits of eigenfunctions are studied.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Notation and assumptions}

We consider here vector fields $b$, related to the Morse-Smale
fields, which satisfy some strong compatibility conditions with respect to the
metric $g$. More specifically we shall assume that for each field $b$:

\textrm{I)}The recurrent set is a finite union of stationary points, limit
cycles and two dimensional torii.

\textrm{II)} The stationary points are hyperbolic and for each such $P$ the
stable and unstable manifolds belonging to $P$ are orthogonal at $P$ with
respect to the metric $g.$

\textrm{III)} Any limit cycle $S$ has a tubular neighborhood $\mathcal{T}_{S}$
provided with a covering map $\Phi :\mathbb{R}^{m-1}\times \mathbb{R}%
\longrightarrow \mathcal{T}_{S}$ having the following properties:

(a) for all $(x^{\prime },\theta )\in \mathbb{R}^{m-1}\times \mathbb{R},\Phi
^{-1}\circ \Phi (x^{\prime },\theta )=\{(x^{\prime },\theta +nT_{S})|n\in 
\mathbb{Z}\}$, $T_{S}$ minimal period of S. 

(b) at any point ($0$,$\theta )\in $ $\mathbb{R}^{m-1}\times \mathbb{R}$,\ $%
\left( \Phi \right) ^{\ast }g_{(0,\theta )}=\sum_{n=1}^{m-1}dx_{n}^{2}
+g^{m m }(\theta )d\theta ^{2}$,($\theta=x^m$).

(c) at any point ($0$,$\theta )\in \mathbb{R}^{m-1}\times \mathbb{R}$,
$
\left( \Phi \right) _{\ast }b=\frac{\partial }{\partial \theta }%
+\sum_{i,j=1}^{m-1}B_{ij}x_{j}\frac{\partial }{\partial x_{i}},$
up to term of order two in $x^{\prime }=(x_{1},..,x_{m-1}),$ canonical
coordinates on $\mathbb{R}^{m-1}.$

(d) the $\left( m-1\right) \times (m-1)$ matrix $B=$\{$B_{ij}|1\leq i,j\leq
m-1$\} is hyperbolic and its stable and instable spaces are orthogonal with
respect to the Euclidean metric $\sum_{n=1}^{m-1}x_{n}^{2}$ on $\mathbb{R}%
^{m-1}$. 

\textrm{IV)} Any 2-dimensional torus $R$ has a tubular neighborhood $\mathcal{T}_{R}$
provided with a diffeomorphism $\Phi :\mathbb{R}^{m-2}\times \mathbb{R}%
^{2}\longrightarrow T_{R}$ having the following properties:

(a) at any point ($0$,$\theta )\in $ $\mathbb{R}^{m-2}\times \mathbb{R}^{2}$, 
$\theta =(\theta _{1},\theta _{2})$, $\theta _{1},\theta _{2}$ cyclic
coordinates,\ 

$\, \, \, \, \left( \Phi ^{-1}\right) ^{\ast }g_{(0,\theta
)}=\sum_{n=1}^{m-2}dx_{n}^{2}+a(\theta )d\theta _{1}^{2}+2b(\theta )d\theta
_{1}d\theta _{2}+c(\theta )d\theta _{2}^{2}$

(b) at any point ($0$,$\theta )\in \mathbb{R}^{m-2}\times \mathbb{R}^{2}$ ,
$
\, \, \, \, \left( \Phi \right) _{\ast }\left( b\right) (0,\theta )=\ \ k_{1}%
\frac{\partial }{\partial \theta ^{1}}+k_{2} \frac{\partial }{\partial
\theta ^{2}}+\sum_{i,j=1}^{m-2}B_{ij}x_{j}\frac{\partial }{\partial x_{i}} \hbox{ where: }
$

(i) the $\left( m-2\right) \times (m-2)$ matrix $B=$\{$B_{ij}|1\leq i,j\leq
m-2$\} is hyperbolic and its stable and instable spaces are orthogonal with
respect to the Euclidean metric $\sum_{n=1}^{m-2}x_{n}^{2}$ on $\mathbb{R}%
^{m-2}$,

(ii) $k_1,k_2 \in \hbox{\bb R}$ and $\frac{k_1}{k_2} \in \hbox{\bb R}-%
\hbox{\bb Q}$. 
% the vector field $\ b_{1}(\theta )\frac{\partial }{\partial \theta _{1}}%
%+b_{2}(\theta )\frac{\partial }{\partial \theta _{2}}$ on $\mathbb{T}$ can
%be $C^{\infty }$ conjugated to a constant field $k_{1}\frac{\partial }{%
%\partial \theta _{1}}+k_{2}\frac{\partial }{\partial \theta _{2}}$ where the
%ratio $\frac{k_{1}}{k_{2}}\in \mathbb{R}-\mathbb{Q}$. In the remainder of
%the paper, $(\theta _{1},\theta _{2})$ will denote the cyclic coordinates on 
%$\mathbb{T}$ in which $b$can be written  $k_{1}\frac{\partial }{\partial
%\theta _{1}}+k_{2}\frac{\partial }{\partial \theta _{2}}.$
A torus with such a flow will be called an irrational torus.

(iii) Small divisor condition: There exist constants $C>0,\alpha>0$ such that for all $m_{1},m_{2} \in \zZ $, $ |m_{1}k_{1}+m_{2}k_{2}| \geq C(m_{1}^{2}+m_{2}^{2})^{\alpha }  \label{sd}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Statement of the main result}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The spirit of our result can be summarized as follow:

\emph{\textquotedblleft\ On a Riemannian manifold, for any choice of a
special Lyapunov function $\mathcal{L}$, vanishing at order 2 on the
recurrent sets of the field, the limits as }$\varepsilon $ \emph{tends to 0
of the normalized measures $e^{-\frac{\mathcal{L}}{2\epsilon }}u_{\epsilon
}^{2}dV_{g}$ are concentrated on the components of the recurrent sets which
are of maximal dimension and where the topological pressure is achieved.}

The construction of the function  $\mathcal{L}$ is detailed in \cite{HK1}.
\begin{theorem}\label{th1} 
On a compact Riemannian manifold $(M,g)$, let $b$ be a 
Morse-Smale vector field and $\mathcal{L}$ be a special Lyapunov function
for $b$. Consider the normalized positive eigenfunction $u_{\epsilon }>0$ of
the operator $L_{\epsilon }=\epsilon \Delta +\theta (b)+c$, associated to
the first eigenvalue $\lambda _{\epsilon }$.
\begin{enumerate}
\item The recurrent set $S$ of $b$ is a union of a finite set of stationary points $S^s$, 
a finite set of periodic orbits $S^p$ and a finite set of two dimensional irrational torii $S^t$.
The limit set of a  normalized measure $\frac{ u_{\epsilon }^{2}e^{-\mathcal{L}/\epsilon }dV_{g}}
{\int_{V_{n}}u_{\epsilon}^{2}e^{-\mathcal{L}/\epsilon }dV_{g}}$, is contained in the set of probability measure $\mu$ of the form   
\begin{eqnarray*}
\mu =\sum_{P\in S^s_{tp}}c_{P}\delta _{P}+\sum_{\Gamma \in
S^p_{tp}}a_{\Gamma }\delta _{\Gamma }+\sum_{\hbox{\bbbis T}\in S^t_{tp}}b_{%
\hbox{\bbbis T}}\delta_{\hbox{\bbbis T} }
\end{eqnarray*}
where $S^s_{tp}$ (resp.$S^p_{tp},S^t_{tp}$) is the subset of $S^s$ (resp.$S^p,S^t$), where the topological pressure is attained. $\delta _{P}$ is the Dirac measure at P.

For $\Gamma \in S^p$ and h $\in C(V )$, 
$ \delta _{\Gamma }(h)= \int_{0}^{T_{\Gamma }} f_\Gamma(\theta) h(\Gamma
(\theta )) d\theta$, where $\theta \in \mathbb{R\longrightarrow }\Gamma (\theta )\in V$ is a
solution of $b$ representing $\Gamma$.%and $b_0(\theta) = d\theta(b(\Gamma(\theta)))$ 
(see the notations for the precise definition of $\theta$). The periodic function
 $f_\Gamma$ is given by 
\begin{eqnarray*}
f_\Gamma(\theta) = \exp \{  -{\int_0^\theta{ c(\Gamma(s)) } ds} + \frac{\theta}{T_\Gamma}\int_0^{T_\Gamma} c(\Gamma(s)) ds \}
\end{eqnarray*}
and $T_\Gamma$ is the minimal period of $\Gamma$.

$\hbox{ For {\bbbis T}} \in S^t \hbox{ and }\,  h \in C(V ), \delta_{\hbox{\bbbis T} }(h) =\int_{\hbox{\bbbis T}}h(\theta _{1},\theta
_{2}) f_{\hbox{\bbbis T}}(\theta _{1},\theta _{2})dS_{\hbox{\bbbis T}},$
where $dS_{\hbox{\bbbis T}}$ is the unique probability measure on $\hbox{\bbbis T}$ invariant under the action of the field b and $f_{\hbox{\bbbis T}}$ is the unique solution of maximum 1, of the equation 
\begin{eqnarray*}
k_1 \frac{\partial f}{\partial \theta_1}+k_2 \frac{\partial f}{\partial
\theta_2} +cf = \mu_2 f \hbox{ where } \mu_2=\int_{\hbox{\bbbis T}} c dS_{\hbox{\bbbis T}}.
\end{eqnarray*}
\item The coefficients $c_{P}, a_{\Gamma}, b_{\hbox{\bbbis T}}$ obey the following rule:
If at least one coefficient $b_{\hbox{\bbbis T}}>0$, then for all cycle $a_{\Gamma }=0$ and
 all points $c_{P}=0$. If all coefficients  $b_{\hbox{\bbbis T}}=0$, and at least one coefficient $a_{\Gamma }>0$ then all $c_{P}=0$.
\end{enumerate}
\end{theorem}

\section{Remark on the proofs and future perspectives }

The basic techniques are the Blow-up and stochastic analysis and the theory of Ornstein-Uhlenbeck operator, but they are too involved to be presented here. As a buy product, we estimate the decay of the first eigenfunction near the recurrent sets.

We conjecture that under $L_p$ normalization, where p$>$1, the limit measures are similar to the one we obtained, except that the coefficients are different. To our knowledge, it is still unknown if the limit measure is unique or not. But this is a difficult problem as explained partially in \cite{Simon} and in more detail in \cite{HK1}.  
%When b is Hamiltonian in the KAM situation, we conjecture that the limit measures are concentrated on The study of the limit measures,

% etc, etc

% The Appendices part is started with the command \appendix;
% appendix sections are then done as normal sections
% \appendix

% \section{}
% \label{}

% The Acknowledgements are an un-numbered section
%\section*{Acknowledgements}


\begin{thebibliography}{00}
% please try to use the bibitem system -
% the references should be in alphabetical order of authors' names.
% Articles with a single author first, author will 1 co-author next,
% then author with several co-authors;


% \bibitem{label}
% Text of bibliographic item

\bibitem{Devi1}  Devinatz A. Ellis R. Friedman, A.,  The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II.  Indiana Univ. Math. J. 23  (1973--1974), 991--1011

\bibitem{Devi2} {Devinatz A.}-- Friedman A-- Asymptotic Behavior of
the Principal Eigenfunction for a Singularly Perturbed Dirichlet Problem 
\emph{Indiana Univ. Math. J. 27, No. 1, 1978} p143-157.

\bibitem{Fried}
 Friedman, A.,  The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives.  Indiana Univ. Math. J.  22 
 (1972/73), 1005--1015

%\bibitem{Fried1} Friedman, A. Stochastic differential equations and
%applications, Vol II, Academic Press, 1976.

\bibitem{FW} {Freidlin, M. I.-- Wentzell, A. D.} Random perturbations of
dynamical systems, 260. Springer-Verlag, New York, 1984.

\bibitem{HK1} Holcman D.--Kupka I. Singular perturbation for the first
eigenfunction and blow up analysis, to appear May 2006 in Forum Math and
 http://arxiv.org/pdf/math-ph/0412088.


\bibitem{Kifer90} Kifer Y., Principal eigenvalues, topological pressure and stochastic stability 
of equilibrium states. Israel J. Math. 70 (1990),
no. 1, 1--47.

\bibitem{Pinsky} Pinsky, R. Positive harmonic functions and diffusion.
Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press,
Cambridge, 1995.

%\bibitem{Mat}  Matkowsky, B. J.; Schuss, Z. On the problem of exit.  Bull. Amer. Math. Soc.  82  (1976), no. 2, 321--324.
%\bibitem{Mat2} Naeh, T., Klosek, M. M.; Matkowsky, B. J.; Schuss, Z. A direct
% approach to the exit problem.  SIAM J. Appl. Math.  50  (1990),  no. 2, 595--627. 

\bibitem{Robinson} Robinson C. Dynamical systems. Stability, symbolic dynamics,
and chaos. Second edition CRC Press, 1995.

\bibitem{Sarnak} P. Sarnak, Arithmetic quantum chaos, First annual R. A. Blygth Lectures, March15-19, 1993. U. of Toronto, Department of Mathematics.

%\bibitem{label}
\bibitem{Simon} Simon, B., Semi-classical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions.  Ann. Inst. H. Poincar\'e Sect. A (N.S.)  38  (1983),  no. 3, 295--308.
\end{thebibliography}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Sketch of a proof }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The basic techniques are the Blow-up analyzis of equation \ref{vrai} where the operator converges to Ornstein-Ulhembeck type operator.  More precisley: by using gauge transformation involving a special Lyapunov function $\mathcal{L}$, the first order term in the operator(\ref{fpo}) 
is transformed into one which becomes arbitrarily small $\epsilon $ goes to zero.

The second step is the blow-up analysis of the eigenfunction. In the case of a limit cycle, we study in an appropriate coordinate system, the limit of the sequence $ w_{\epsilon }(x, \theta)=\frac{v_{\epsilon }(\sqrt{\epsilon }x,\theta)}{sup v_{\epsilon }}$, which converge in $\rR^n$ to a function solution of 
\begin{equation}
-\Delta w+\sum_{i,j=1}^{m}\Omega _{ij}(P)x_{j}\frac{\partial w}{\partial
x_{i}}+[c(P)+\frac{\Delta _{E}\mathcal{L}}{2}(P)+\psi _{2}(x)]w=\lambda w,
\label{bua}
\end{equation}% 
where $\sum_{i,j=1}^{m}\Omega _{ij}(P)x_{j}\frac{\partial }{\partial x_{i}}$
is the linear part of the field $\Omega $ at P, $\psi _{2}(x)$ is a
quadratic form representing the terms of order two in the Taylor development
of $\Psi _{\mathcal{L}}$ at P and $\lambda \geq 0$ is equal to the
topological pressure. 

In (\cite{HK1}) it has been proved that the concentration occurs on the recurrent sets of
the field, and that the supports of the limit measures are contained in the
set where the function $\Psi _{\mathcal{L}}$ defined in (\ref{gauge})
vanishes. We show that if a component of the
recurrent set is a cycle or an irrational 2-dimensional torus, the
restriction of any limit measure to it is absolutely continuous with respect
to the natural measure (length, area) induced by the flow of b on the
component. An irrational torus is one such that the restriction of the flow
of the field to it diffeomorphic to the classical linear irrational flow.
Moreover we give explicit formulas for the densities of these limit measures.

In the third step, we compute the precise exponential decay estimates for the eigenfunction,
For that purpose, we use the Feynman-Kac formula for the eigenfunction and formulate the associated control problem for a specific distance.
\bigskip
Remark: If b=c=0 then the limit measures are uniform. 
 
math-ph0506059.extracted_bib
\begin{thebibliography}{00}
% please try to use the bibitem system -
% the references should be in alphabetical order of authors' names.
% Articles with a single author first, author will 1 co-author next,
% then author with several co-authors;


% \bibitem{label}
% Text of bibliographic item

\bibitem{Devi1}  Devinatz A. Ellis R. Friedman, A.,  The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II.  Indiana Univ. Math. J. 23  (1973--1974), 991--1011

\bibitem{Devi2} {Devinatz A.}-- Friedman A-- Asymptotic Behavior of
the Principal Eigenfunction for a Singularly Perturbed Dirichlet Problem 
\emph{Indiana Univ. Math. J. 27, No. 1, 1978} p143-157.

\bibitem{Fried}
 Friedman, A.,  The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives.  Indiana Univ. Math. J.  22 
 (1972/73), 1005--1015

%\bibitem{Fried1} Friedman, A. Stochastic differential equations and
%applications, Vol II, Academic Press, 1976.

\bibitem{FW} {Freidlin, M. I.-- Wentzell, A. D.} Random perturbations of
dynamical systems, 260. Springer-Verlag, New York, 1984.

\bibitem{HK1} Holcman D.--Kupka I. Singular perturbation for the first
eigenfunction and blow up analysis, to appear May 2006 in Forum Math and
 http://arxiv.org/pdf/math-ph/0412088.


\bibitem{Kifer90} Kifer Y., Principal eigenvalues, topological pressure and stochastic stability 
of equilibrium states. Israel J. Math. 70 (1990),
no. 1, 1--47.

\bibitem{Pinsky} Pinsky, R. Positive harmonic functions and diffusion.
Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press,
Cambridge, 1995.

%\bibitem{Mat}  Matkowsky, B. J.; Schuss, Z. On the problem of exit.  Bull. Amer. Math. Soc.  82  (1976), no. 2, 321--324.
%\bibitem{Mat2} Naeh, T., Klosek, M. M.; Matkowsky, B. J.; Schuss, Z. A direct
% approach to the exit problem.  SIAM J. Appl. Math.  50  (1990),  no. 2, 595--627. 

\bibitem{Robinson} Robinson C. Dynamical systems. Stability, symbolic dynamics,
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\bibitem{Sarnak} P. Sarnak, Arithmetic quantum chaos, First annual R. A. Blygth Lectures, March15-19, 1993. U. of Toronto, Department of Mathematics.

%\bibitem{label}
\bibitem{Simon} Simon, B., Semi-classical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions.  Ann. Inst. H. Poincar\'e Sect. A (N.S.)  38  (1983),  no. 3, 295--308.
\end{thebibliography}