Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation

Abstract

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThe object of this paper is the uniqueness for a d-dimensional Fokker-Planck type equation with inhomogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so-called Barenblatt's solution of the fast diffusion equation which is the partial differential equation ∂tu = ∂ 2 xx u m with m ∈]0, 1[. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition.(1.5) where β : R → R is a monotone non-decreasing function such that β(0) = 0 and u 0 is a finite measure. When β(u) = u m , m ∈]0, 1[ and u 0 = δ 0 , two difficulties arise: first, the coefficient β is of singular type since it is not locally Lipschitz, second, the initial condition is a measure. Another type of singular coefficient is βis a Heaviside function and u c > 0 is some critical value, see e.g. [2]. Problem (1.2) with m ∈]0, 1[ was studied by several authors. For a bounded integrable function as initial condition, the equation in (1.2) is well-stated in the sense of distributions, as a by product of the classical papers [10, 6] on (1.5) with general monotonous coefficient β. Electron. J. Probab. 0 (2012), no. 0, 1-28. ejp.ejpecp.org We now state the main result of the paper which concerns uniqueness for the Fokker-Planck type equation with measurable, time-dependent, (possibly degenerated and unbounded) coefficients. It generalizes [7, Theorem 3.8] where the coefficients were bounded and one-dimensional.The theorem below holds with two classes of hypotheses: (B1), operating in the multidimensional case, and (B2), more specifically in the one-dimensional case. Theorem 3.1. Let a be a Borel nonnegative function on [0, T ] × R d . Let z i : [0, T ] → M(R d ), i = 1, 2, be continuous with respect to the weak topology on finite measures on M(R d ). Let z 0 be an element of M(R d ). Suppose that both z 1 and z 2 solve the problem ∂ t z = ∆(az) in the sense of distributions with initial condition z(0, ·) = z 0 .Then z := (z 1 −z 2 )(t, ·) is identically zero for every t under the following requirement.

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

“…The theorem below plays the analogous role as Theorem 3.8 in [5] or Theorem 3.1 in [4]. We recall that our Fokker-Planck SPDE has possibly degenerate measurable coefficients.…”

“…[6] and references therein. More particularly, concerning uniqueness, in addition we draw the attention to Proposition 3.4 [5] and Theorem 3.1 of [4]. As far as we know this is the first time that a Fokker-Planck equation as (1.1) is considered in the literature, in particular for uniqueness, except for the unpublished work by the same authors [2].…”

Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

“…If β(u) = u.|u| m−1 , m ∈]0, 1[, the partial differential equation in (1.1) is in fact the so-called fast diffusion equation. In the case when d = 1, [16] provides a probabilistic representation for the Barenblatt type solutions of (1.1).…”

“…When d = 1 and a is bounded this was the object of [19,Theorem 3.8]. Extensions where considered in [16,Theorem 3.1] when a is possibly degenerate and is allowed to be unbounded under some technical conditions. In fact [16,Theorem 3.1] also deals with the multidimensional case.…”

“…Extensions where considered in [16,Theorem 3.1] when a is possibly degenerate and is allowed to be unbounded under some technical conditions. In fact [16,Theorem 3.1] also deals with the multidimensional case. When a is bounded and given two measure-valued solutions z 1 and z 2 , the Fokker-Planck uniqueness theorem applies saying that z 1 = z 2 whenever (z 1 − z 2 )(t, ·) has a density z(t, ·) for almost all t and z belongs to…”