The Dirichlet curve of a probability in $\mathbb{R}^d$

If $α$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the Dirichlet random probability $P_t\sim\mathcal{D}(tα) ;$ 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 $(tα(A_0)\ldots,tα(A_k)).$ If $\int_{\mathbb{R}^d}\log(1+\|x\|)α(dx)<\infty$ the random variable $\int_{\mathbb{R}^d}xP_t(dx)$ of $\mathbb{R}^d$ does exist and we denote by $μ(tα)$ its distribution. The Dirichlet curve associated to the probability $α$ is the map $t\mapsto μ(tα).$ It has simple properties like $\lim_{t\searrow 0}μ(tα)=α$ and $\lim_{t\rightarrow \infty}μ(tα)=δ_m$ when $m=\int_{\mathbb{R}^d} xα(dx)$ exists. The present paper shows first that if $m$ exists and if $ψ$ is a convex function on $\mathbb{R}^d$ then $t\mapsto \int_{\mathbb{R}^d}ψ(x)μ(tα)(dx)$ is a decreasing function, which means that $t\mapsto μ(tα)$ 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 $μ(tα)=μ(sα)$ for some $0\leq s<t$, can we claim that $μ$ is Cauchy distributed in $\mathbb{R}^d?$

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