Path-Distribution Dependent SDEs: Well-Posedness and Asymptotic Log-Harnack Inequality

TL;DR

This paper studies path-distribution dependent stochastic differential equations, proving well-posedness and Lipschitz continuity under certain conditions, and establishing an asymptotic log-Harnack inequality extending previous results.

Contribution

It introduces new well-posedness results for path-distribution dependent SDEs and extends the asymptotic log-Harnack inequality to this setting.

Findings

Well-posedness under local integrability condition

Lipschitz continuity in initial values

Extension of asymptotic log-Harnack inequality

Abstract

We consider stochastic differential equations on $\mathbb R^d$ with coefficients depending on the path and distribution for the whole history. Under a local integrability condition on the time-spatial singular drift, the well-posedness and Lipschitz continuity in initial values are proved, which is new even in the distribution independent case. Moreover, under a monotone condition, the asymptotic log-Harnack inequality is established, which extends the corresponding result of [5] derived in the distribution independent case.