Uniqueness for a class of stochastic Fokker–Planck and porous media equations

Abstract

Summary. The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate.We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.

Probabilistic Representation for Solutions to Nonlinear Fokker--Planck Equations

“…in the sense of (2.2) with u(t, x)dx dt replaced by v(t, x)dx dt. This was, however, achieved in certain cases (see [6], [7] and also [16]). As explained in the introduction of this paper, we look at generalized (= entropic) solutions for a special case of (2.1).…”

“…in the sense of (2.2) with u(t, x)dx dt replaced by v(t, x)dx dt. This was, however, achieved in certain cases (see [6], [7] and also [16]).…”

Doubly probabilistic representation for the stochastic porous media type equation

“…ψ : R → R is Lipschitz and that the functions belong to W 1,∞ . The proof of Theorem B.1 is a consequence of the result stated in Theorem B.1 of [24], see also [7].…”

“…The proof makes use of the similar arguments as in Theorem 3.8 of [14] or Theorem 3.1 in [10], in a randomized form. The full proof is given in [24] Theorem 4.2, see also [7].…”