NONLINEAR FOKKER-PLANCK EQUATIONS FOR PROBABILITY MEASURES ON PATH SPACE AND PATH-DISTRIBUTION DEPENDENT SDES

摘要

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker-Planck equations for probability measures (mu(t)) t = 0 on the path space l := C([-r(0), 0]; R-d), is analyzed:partial derivative(t)mu(t) = L-t*(,mu t)mu t, t = 0,where mu(t) is the image of mu(t) under the projection l (sic) xi bar right arrow xi(0) is an element of R-d, andL-t(,mu)(xi) :=1/2 Sigma(d)(i,j=1) aij(t, xi, mu) partial derivative(2)/partial derivative(xi(0)i)partial derivative(xi)(0)(j)+ Sigma(d)(i=1) bi(t, xi, mu)partial derivative/partial derivative(xi(0)i), t = 0, xi is an element of l, mu is an element of P-l.Under reasonable conditions on the coefficients a(ij) and b(i), the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.