Derivative over Wasserstein spaces along curves of densities

Abstract

In this paper, given any random variable $\hat{I}¾$ defined over a probability space $(\hat{I}\text{c◯},\mathcal{F},Q)$, we focus on the study of the derivative of functions of the form $L\mapsto F_Q(L):=f((LQ{)}_{\hat{I}}¾),$ defined over the convex cone of densities $L\in {\mathcal{L}}^Q:=\{L\in L^1(\hat{I}\text{c◯},\mathcal{F},Q;{\mathbb{R}}_{+}):E^Q[L]=1\}$ in $L^1(\hat{I}\text{c◯},\mathcal{F},Q).$ Here $f$ is a function over the space $\mathcal{P}({\mathbb{R}}^d)$ of probability laws over ${\mathbb{R}}^d$ endowed with its Borel $\ddot{I}ƒ$-field $\mathcal{B}({\mathbb{R}}^d)$. The problem of the differentiability of functions $F_Q$ of the above form has its origin in the study of mean-field control problems for which the controlled dynamics admit only weak solutions. Inspired by P.-L. Lions' results [18] we show that, if for given $L\in {\mathcal{L}}^Q$, $L^{\prime }\mapsto F_{LQ}(L^{\prime }):{\mathcal{L}}^{LQ}\to \mathbb{R}$ is differentiable at $L^{\prime }=1$, the derivative is of the form $g(\hat{I}¾)$, where $g:{\mathbb{R}}^d\to \mathbb{R}$ is a Borel function which depends on $(Q,L,\hat{I}¾)$ only through the law $(LQ{)}_{\hat{I}}¾$. Denoting this derivative by ${\partial }_{1}F((LQ{)}_{\hat{I}}¾,x):=g(x),x\in {\mathbb{R}}^d$, we study its properties, and we relate it to partial derivatives, recently investigated in [6], and, moreover, in the case when $f$ restricted to the 2-Wasserstein space ${\mathcal{P}}_2({\mathbb{R}}^d)$ is differentiable in P.-L. Lions' sense and $(LQ{)}_{\hat{I}}¾\in {\mathcal{P}}_2({\mathbb{R}}^d)$, we investigate the relation between the derivative with respect to the density of $F_Q(L)=f((LQ{)}_{\hat{I}}¾)$ and the derivative of $f$ with respect to the probability measure. Our main result here shows that ${\partial }_x{\partial }_{1}F((LQ{)}_{\hat{I}}¾,x)={\partial }_{\hat{I}}¼f((LQ{)}_{\hat{I}}¾,x),x\in {\mathbb{R}}^d,$ where ${\partial }_{\hat{I}}¼f((LQ{)}_{\hat{I}}¾,x)$ denotes the derivative of $f:{\mathcal{P}}_2({\mathbb{R}}^d)\to \mathbb{R}$ at $(LQ{)}_{\hat{I}}¾$.

55 pages