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By investigating path-distribution dependent stochastic differentialequations, the following type of nonlinear Fokker--Planck equations forprobability measures $(\mu_t)_{t \geq 0}$ on the path space $\mathcalC:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t\mu(t)=L_{t,\mu_t}^*\mu_t,\ \ t\ge 0,$$ where $\mu(t)$ is the image of $\mu_t$under the projection $\mathcal C\ni\xi\mapsto \xi(0)\in\mathbb R^d$, and$$L_{t,\mu}(\xi):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,\xi,\mu)\frac{\partial^2}{\partial_{\xi(0)_i} \partial_{\xi(0)_j}}  +\sum_{i=1}^d b_i(t,\xi,\mu)\frac{\partial}{\partial_{\xi(0)_i}},\ \ t\ge 0,\xi\in \mathcal C, \mu\in  \mathcal P^{\mathcal C}.$$ Under reasonable conditions on the coefficients$a_{ij}$ and $b_i$, the existence, uniqueness, Lipschitz continuity inWasserstein distance, total variational norm and entropy, as well as derivativeestimates are derived for the martingale solutions.

Nonlinear Fokker-Planck equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs