Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation (original) (raw)
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The fast diffusion equation with measure data
We show the existence of solutions to the fast diusion equation with a general nite and positive Borel measure as the right hand side source term. u t − ∆u m = µ, where µ is a general nite, positive Borel measure. Our main result, Theorem 3.6, states that a solution to (1.1) exists, and provides a sharp estimate in a parabolic Sobolev space. The power m is allowed to take values in the supercritical range: m c < m < 1, where m c = (n − 2) + /n. The fundamental diculty in existence results of this kind is that general measures do not belong to the dual of the natural function space for weak solutions. An explicit illustration of this is provided by the celebrated Barenblatt solution B m , given by (3.7) below. The Barenblatt solution satises (1.1) with the right hand side given by Dirac's delta. However, the weak gradient of B m m is not an L 2 function, so the usual Sobolev class of weak solutions is not large enough to admit solutions with general right hand side measures.
Fokker-Planck equations with terminal condition and related McKean probabilistic representation
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Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.
Nonlinear Fokker-Planck equations with time-dependent coefficients
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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) − ∆(a(t, x, u(t, x))u(t, x)) + div(b(t, x, u(t, x))u(t, x)) = 0 in (0, ∞) × R d , u(0, x) = u 0 (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. Is is proved also that, if u 0 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.
From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE
The Annals of Probability, 2020
We construct weak solutions to the McKean-Vlasov SDE dX(t) = b X(t), dL X(t) dx (X(t)) dt + σ X(t), dL X(t) dt (X(t)) dW (t) on R d for possibly degenerate diffusion matrices σ with X(0) having a given law, which has a density with respect to Lebesgue measure, dx. Here L X(t) denotes the law of X(t). Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well known superposition principle to obtain weak solutions of the above SDE.
Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation
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We prove the existence of an invariant measure µ for the transition semigroup P t associated with the fast diffusion porous media equation in a bounded domain O ⊂ R d , perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator N of P t in L 2 (H −1 (O), µ) is characterized as the closure of a secondorder elliptic operator in H −1 (O). Moreover, we construct the Sobolev space W 1,2 (H −1 (O), µ) and prove that D(N ) ⊂ W 1,2 (H −1 (O), µ).
The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations omega\omegaomega, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.
Projecting the Fokker-Planck Equation onto a finite dimensional exponential family
1996
In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the orthogonal projection of the evolution of the density of a diffusion process onto a finite dimensional exponential manifold. It has been shown by D. Brigo (1996) that the projected evolution can always be interpreted as the evolution of the density of a different diffusion process. We give also a compactness result when the dimension of the exponential family increases, as a first step towards a convergence result to be investigated in the future. The infinite dimensional exponential manifold structure introduced by G. Pistone and C. Sempi is used and some examples are given.