The Trotter product formula for nonlinear Fokker–Planck flows

One proves herein that the flow S(t), generated by the nonlinear Fokker–Planck equation ρt−Δβ(ρ)+div(a(ρ)ρ)=0 in (0,∞)×Rd, is expressed by the Trotter product formulaS(t)ρ0=limn→∞⁡(SA1(tn)SA2(tn))nρ0 inL1(Rd), where SA1(t) is the flow (continuous semigroup) generated in L1(Rd) by the nonlinear diffusion operator A1(ρ)=−Δβ(ρ), while SA2(t) is that generated in L1(Rd) by the conservation law operator A2(ρ)=div(a(ρ)ρ) defined in the entropy sense. As an application, one obtains a split-product formula for the McKean–Vlasov stochastic differential equation associated with the Fokker–Planck equation.