Nonlinear Fokker-Planck equations with time-dependent coefficients

2021/10/24 by Barbu, Viorel, Rockner, Michael
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2110.12460

Abstract

An operatorial based approach is used here to prove the existence and uniqueness of a strong solution u to the time-varying nonlinear Fokker--Planck equation u_t(t,x)-Δ(a(t,x,u(t,x))u(t,x))+\rm div(b(t,x,u(t,x))u(t,x))=0 in (0,∞)× ℝ u(0,x)=u₀(x), x∈ℝ^d in the Sobolev space H^-1(ℝ^d), under appropriate conditions on the a:[0,T]×ℝ^d×ℝ→ℝ and b:[0,T]×ℝ^d×ℝ→ℝ^d. It is proved also that, if u₀ is a density of a probability measure, so is u(t,⋅) for all t≥0. Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that u(t) is the density of its time marginal law. MSC: 60H15, 47H05, 47J05. Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential equation, Sobolev space, periodic solution.

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