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 $(A_0,\dots ,A_k)$ of ${\mathbb{R}}^d$ then $(P_t(A_0),\dots ,P_t(A_k))$ is Dirichlet distributed with parameters $(t\alpha (A_0)\dots ,t\alpha (A_k))\mathrm{.}$ If ${\int }_{{\mathbb{R}}^d}\log (1+\mathrm{\parallel }x\mathrm{\parallel })\alpha (dx)<\mathrm{\infty }$ the random variable ${\int }_{{\mathbb{R}}^d}x{P}_t(dx)$ 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 $t\mapsto \mu (t\alpha )\mathrm{.}$ 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 $0\le s<t$, can we claim that $\mu$ is Cauchy distributed in ${\mathbb{R}}^d?$