Krylov-Veretennikov decomposition for measure-valued processes induced by SDEs with interaction on Riemannian manifolds

TL;DR

This paper develops a comprehensive framework for measure-valued solutions of SDEs with interaction on manifolds, establishing existence, uniqueness, and smooth dependence, along with an explicit Wiener decomposition.

Contribution

It introduces a novel approach to analyze measure-valued SDEs on manifolds, including explicit decompositions and differentiability results.

Findings

Existence and uniqueness of strong solutions under general conditions

Explicit Itô Wiener decomposition for measure-valued processes

Malliavin differentiability and smooth dependence on measures

Abstract

We introduce a framework for stochastic differential equations (SDEs) with interaction on compact, connected, $d$-dimensional manifolds. For SDEs whose drift and diffusion coefficients may depend on both the state variable and the empirical distribution, we establish existence and uniqueness of strong solutions under general regularity assumptions. We study the associated measure valued process on the Wasserstein space over the manifold, deriving an explicit It\^o Wiener decomposition. We prove Malliavin differentiability of the solution and, using directional derivatives in the Wasserstein space, establish smooth dependence of the solution on the measure component for a class of coefficients.