Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dirichlet curve of a probability in $\mathbb{R}^d$   If $\alpha$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the
Dirichlet random probability $P_t\sim\mathcal{D}(t\alpha) ;$ it is such that
for any measurable partition $(A_0,\ldots,A_k)$ of $\mathbb{R}^d$ then
$(P_t(A_0),\ldots,P_t(A_k))$ is Dirichlet distributed with parameters
$(t\alpha(A_0)\ldots,t\alpha(A_k)).$ If
$\int_{\mathbb{R}^d}\log(1+\|x\|)\alpha(dx)<\infty$ the random variable
$\int_{\mathbb{R}^d}xP_t(dx)$ of $\mathbb{R}^d$ does exist and we denote by
$\mu(t\alpha)$ its distribution. The Dirichlet curve associated to the
probability $\alpha$ is the map $t\mapsto \mu(t\alpha).$ It has simple
properties like $\lim_{t\searrow 0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow
\infty}\mu(t\alpha)=\delta_m$ when $m=\int_{\mathbb{R}^d} x\alpha(dx)$ exists.
The present paper shows first that if $m$ exists and if $\psi$ is a convex
function on $\mathbb{R}^d$ then $t\mapsto
\int_{\mathbb{R}^d}\psi(x)\mu(t\alpha)(dx)$ is a decreasing function, which
means that $t\mapsto \mu(t\alpha)$ is decreasing according to the Strassen
convex order of probabilities. The second aim of the paper is to prove a group
of results around the following question: if $\mu(t\alpha)=\mu(s\alpha)$ for
some $0\leq s<t$, can we claim that $\mu$ is Cauchy distributed in
$\mathbb{R}^d?$

32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc