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

TL;DR

The paper develops a Krylov–Veretennikov decomposition for functionals of measure-valued processes driven by SDEs with interaction on compact Riemannian manifolds. It combines Nash-embedding techniques to lift the problem to Euclidean space with smooth vector-field enlargements, an intrinsic Wasserstein calculus for measure derivatives, and Malliavin calculus to obtain an explicit Itô–Wiener expansion for functionals on the Wasserstein space $\mathcal{P}_2(M)$. The main contributions include existence and uniqueness of strong solutions on $M$, infinite differentiability with respect to the measure variable, and a concrete Krylov–Veretennikov expansion $f(\mu_t)=T_t f(\mu)+\sum_{i,y}...$ that decouples the stochastic and deterministic parts of the dynamics. This framework enables tractable analysis of long-term behavior and sensitivity of interacting particle systems on manifolds in terms of kernels on the Wasserstein space.

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 Itô 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.

Theorems & Definitions (38)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof : Proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof : Proof
• Definition 3.1: ren2021derivative