Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs

摘要

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: $$partial_tmu(t)=L_{t,mu_t}^*mu_t, tge 0,$$ where $mu(t)$ is the image of $mu_t$under the projection $mathcal Ciximapsto xi(0)inmathbb R^d$, and$$L_{t,mu}(xi):= rac 1 2sum_{i,j=1}^d a_{ij}(t,xi,mu)rac{partial^2}{partial_{xi(0)_i} partial_{xi(0)_j}} +sum_{i=1}^d b_i(t,xi,mu)rac{partial}{partial_{xi(0)_i}}, tge 0,xiin mathcal C, muin mathcal P^{mathcal C}.$$ 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.