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

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.