Electronic Journal of Probability

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

Nadia Belaribi and Francesco Russo

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Abstract

The 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 $\partial_t u = \partial^2_{xx} u^m$ with $m\in]0,1[$. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition.

Article information

Source
Electron. J. Probab. Volume 17 (2012), paper no. 84, 28 pp.

Dates
Accepted: 2 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062406

Digital Object Identifier
doi:10.1214/EJP.v17-2349

Mathematical Reviews number (MathSciNet)
MR2988399

Zentralblatt MATH identifier
1268.82024

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter 60J60: Diffusion processes [See also 58J65] 60H07: Stochastic calculus of variations and the Malliavin calculus 35C99: None of the above, but in this section 35K10: Second-order parabolic equations 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 65C05: Monte Carlo methods 65C35: Stochastic particle methods [See also 82C80]

Keywords
Fokker-Planck fast diffusion probabilistic representation non-linear diffusion stochastic particle algorithm

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Belaribi, Nadia; Russo, Francesco. Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation. Electron. J. Probab. 17 (2012), paper no. 84, 28 pp. doi:10.1214/EJP.v17-2349. https://projecteuclid.org/euclid.ejp/1465062406


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