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Path-Distribution Dependent SDEs: Well-Posedness and Asymptotic Log-Harnack Inequality
Feng-Yu Wang, Chenggui Yuan, Xiao-Yu Zhao · 2025-02-19 · via math.PR updates on arXiv.org

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.

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