Linearization of Nonlinear Fokker-Planck Equations and Applications

Abstract

Let P be the space of probability measures on R d . We associate a coupled nonlinear Fokker-Planck equation on R d , i.e. with solution paths in P, to a linear Fokker-Planck equation for probability measures on the product space R d ×P, i.e. with solution paths in P(R d × P). We explicitly determine the corresponding linear Kolmogorov operator L t using the natural tangent bundle over P with corresponding gradient operator ∇ P . Then it is proved that the diffusion process generated by L t on R d × P is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by L t in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. As applications, we prove the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on R d × P 2 , through the Feynman-Kac formula for the corresponding diffusion processes.

Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

“…We refer the reader to the appendix in [16], where for the space of probability measures P the tangent spaces…”

“…Here, the main result is Theorem 3.7. We use this result to prove an open conjecture of [16] (cf. Proposition 3.12) and present several consequences.…”

“…Hence, it seems natural to linearize (NLFPK) once more in order to obtain a linear equation, which is related to (DDSDE) and (NLFPK) in a natural way. By the results of [16], this linear equation is of type (SP-CE). Similar considerations prevail in the stochastic case, where one considers equations of type (DDSDE) with an additional source of randomness (we shall not pursue this direction in this work).…”

“…For a thorough introduction to the field, we refer to [5] and the references therein. As shown in [16], the deterministic nonlinear equation (NLFPK) is naturally associated to a first-order linear continuity equation on P(SP), the space of Borel probability measures on SP, of type…”

“…On the one hand, the superposition principles of Theorem 3.7 and Theorem 4.8 provide new structural results for nonlinear FPK-equations and its corresponding linearized equations on the space of probability measures over SP, involving a geometric interpretation of the latter. On the other hand, it is our future plan to further study the geometry of SP as initiated in [16] and this work to develop an analysis on such infinite-dimensional manifold-like spaces, which allows to solve linear equations of type (SP-CE) and (SP-FPK) on such spaces. By means of the results of this work, one can then lift such solutions to solutions to the nonlinear equations for measures (NLFPK) and (SNLFPK), thereby obtaining new existence results for these nonlinear equations for measures.…”