The Annals of Probability

Integral identity and measure estimates for stationary Fokker–Planck equations

Wen Huang, Min Ji, Zhenxin Liu, and Yingfei Yi

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We consider a Fokker–Planck equation in a general domain in $\mathbb{R}^{n}$ with $L^{p}_{\mathrm{loc}}$ drift term and $W^{1,p}_{\mathrm{loc}}$ diffusion term for any $p>n$. By deriving an integral identity, we give several measure estimates of regular stationary measures in an exterior domain with respect to diffusion and Lyapunov-like or anti-Lyapunov-like functions. These estimates will be useful to problems such as the existence and nonexistence of stationary measures in a general domain as well as the concentration and limit behaviors of stationary measures as diffusion vanishes.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1712-1730.

Received: December 2013
Revised: January 2014
First available in Project Euclid: 3 June 2015

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Zentralblatt MATH identifier

Primary: 35Q84: Fokker-Planck equations 60J60: Diffusion processes [See also 58J65]
Secondary: 37B25: Lyapunov functions and stability; attractors, repellers

Fokker–Planck equation stationary measures measure estimates integral identity level set method


Huang, Wen; Ji, Min; Liu, Zhenxin; Yi, Yingfei. Integral identity and measure estimates for stationary Fokker–Planck equations. Ann. Probab. 43 (2015), no. 4, 1712--1730. doi:10.1214/14-AOP917.

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