Existence of flows for linear Fokker\u2013Planck\u2013Kolmogorov equations and its connection to well-posedness

Marco Rehmeier

doi:10.1007/s00028-020-00569-y

Abstract

Let the coefficients a_{ij} and b_i, i,j le d, of the linear Fokker–Planck–Kolmogorov equation (FPK-eq.) ∂tμt=∂i∂j(aijμt)-∂i(biμt)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t\\mu _t = \\partial _i\\partial _j(a_{ij}\\mu _t)-\\partial _i(b_i\\mu _t) \\end{aligned}$$\\end{document}be Borel measurable, bounded and continuous in space. Assume that for every s in [0,T] and every Borel probability measure nu on mathbb {R}^d there is at least one solution mu = (mu _t)_{t in [s,T]} to the FPK-eq. such that mu _s = nu and t mapsto mu _t is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution mu ^{s,nu } for each pair (s,nu ) such that this family of solutions fulfills μts,ν=μtr,μrs,νfor all0≤s≤r≤t≤T,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mu ^{s,\\nu }_t = \\mu ^{r,\\mu ^{s,\\nu }_r}_t \\text { for all }0 \\le s \\le r \\le t \\le T, \\end{aligned}$$\\end{document}which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed.

Highlights

We are concerned with linear Fokker–Planck–Kolmogorov equations of the form
The aim of this paper is to prove similar results for the Fokker–Planck–Kolmogorov equation (1)
Assuming that the Cauchy problem for Eq (1) has at least one weakly continuous probability solution for every initial condition (s, ν), we prove the existence of a family of solutionss,ν such that

Summary

Introduction

We are concerned with linear Fokker–Planck–Kolmogorov equations of the form. In this work the authors prove the following: Given continuous and bounded coefficients ai j and bi , assume there exists at least one continuous solution to the martingale problem with start in x ∈ Rd at time s ≥ 0 for every pair (s, x). One can select a solution Ps,x to the martingale problem for every initial condition (s, x) such that the family (Ps,x )(s,x)∈R+×Rd is a strong Markov process on the space of continuous functions C(R+, Rd ) Such a consideration goes back to an earlier work of Krylov from 1973 In Theorem 3.16 we show: There exists exactly one such flow if and only if the Fokker–Planck–Kolmogorov equation is well-posed among weakly continuous probability solutions. This could be a direction of further research on this topic

Notation

Basic setting

The main results