Derivative over Wasserstein spaces along curves of densities

TL;DR

This paper studies the differentiability of functions over Wasserstein spaces along curves of densities, linking derivatives with respect to densities to derivatives of functions on probability measures, with applications to mean-field control problems.

Contribution

It establishes a connection between derivatives of functions over densities and derivatives on Wasserstein spaces, extending Lions' differentiability framework to new settings.

Findings

Derived explicit forms for derivatives of functions over Wasserstein spaces.

Linked density derivatives to measure derivatives in the Wasserstein space.

Provided conditions under which these derivatives coincide and relate to mean-field control.

Abstract

In this paper, given any random variable $\xi$ defined over a probability space $(\Omega,\mathcal{F},Q)$, we focus on the study of the derivative of functions of the form $L\mapsto F_Q(L):=f\big((LQ)_{\xi}\big),$ defined over the convex cone of densities $L\in\mathcal{L}^Q:=\{ L\in L^1(\Omega,\mathcal{F},Q;\mathbb{R}_+):\ E^Q[L]=1\}$ in $L^1(\Omega,\mathcal{F},Q).$ Here $f$ is a function over the space $\mathcal{P}(\mathbb{R}^d)$ of probability laws over $\mathbb{R}^d$ endowed with its Borel $\sigma$-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 $L\in\mathcal{L}^Q$, $L'\mapsto F_{LQ}(L'):\mathcal{L}^{LQ}\rightarrow\mathbb{R}$ is differentiable at $L'=1$, the derivative is of the form $g(\xi)$, where $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is a Borel function which depends on $(Q,L,\xi)$ only through the law $(LQ)_\xi$. Denoting this derivative by $\partial_1F((LQ)_\xi,x):=g(x),\, x\in\mathbb{R}^d$, 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 $(LQ)_{\xi}\in\mathcal{P}_2(\mathbb{R}^d)$, we investigate the relation between the derivative with respect to the density of $F_Q(L)=f\big((LQ)_{\xi}\big)$ and the derivative of $f$ with respect to the probability measure. Our main result here shows that $\partial_x\partial_1F((LQ)_\xi,x)=\partial_\mu f((LQ)_\xi,x),\ x\in \mathbb{R}^d,$ where $\partial_\mu f((LQ)_\xi,x)$ denotes the derivative of $f:\mathcal{P}_2(\mathbb{R}^d)\rightarrow \mathbb{R}$ at $(LQ)_\xi$.