## The Annals of Probability

### Integral identity and measure estimates for stationary Fokker–Planck equations

#### Abstract

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

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

Dates
Revised: January 2014
First available in Project Euclid: 3 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1433341318

Digital Object Identifier
doi:10.1214/14-AOP917

Mathematical Reviews number (MathSciNet)
MR3353813

Zentralblatt MATH identifier
1319.35268

#### Citation

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. https://projecteuclid.org/euclid.aop/1433341318

#### References

• [1] Albeverio, S., Bogachev, V. and Röckner, M. (1999). On uniqueness of invariant measures for finite- and infinite-dimensional diffusions. Comm. Pure Appl. Math. 52 325–362.
• [2] Arapostathis, A., Borkar, V. S. and Ghosh, M. K. (2012). Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and Its Applications 143. Cambridge Univ. Press, Cambridge.
• [3] Bensoussan, A. (1988). Perturbation Methods in Optimal Control. Wiley, Chichester.
• [4] Bhattacharya, R. N. (1978). Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 541–553.
• [5] Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.
• [6] Bogachev, V. I., Da Prato, G. and Röckner, M. (2008). On parabolic equations for measures. Comm. Partial Differential Equations 33 397–418.
• [7] Bogachev, V. I., Kirillov, A. I. and Shaposhnikov, S. V. (2012). Integrable solutions of the stationary Kolmogorov equation. Dokl. Math. 85 309–314.
• [8] Bogachev, V. I., Krylov, N. V. and Rökner, M. (2009). Elliptic and parabolic equations for measures. Russian Math. Surveys 64 973–1078.
• [9] Bogachev, V. I., Krylov, N. V. and Röckner, M. (2001). On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Partial Differential Equations 26 2037–2080.
• [10] Bogachev, V. I. and Rökner, M. (2001). A generalization of Has’minskiĭ’s theorem on the existence of invariant measures for locally integrable drifts. Theory Probab. Appl. 45 363–378.
• [11] Bogachev, V. I., Rökner, M. and Shaposhnikov, S. V. (2012). On positive and probability solutions of the stationary Fokker–Planck–Kolmogorov equation. Dokl. Math. 85 350–354.
• [12] Bogachev, V. I., Rökner, M. and Shtannat, V. (2002). Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Mat. Sb. 193 3–36.
• [13] Bogachev, V. I. and Röckner, M. (2002). Invariant measures of diffusion processes: Regularity, existence, and uniqueness problems. In Stochastic Partial Differential Equations and Applications (Trento, 2002). Lecture Notes in Pure and Applied Mathematics 227 69–87. Dekker, New York.
• [14] Bogachev, V. I., Röckner, M. and Shaposhnikov, S. V. (2011). On uniqueness problems related to elliptic equations for measures. J. Math. Sci. (N. Y.) 176 759–773.
• [15] Bogachev, V. I., Röckner, M. and Stannat, W. (2000). Uniqueness of invariant measures and essential $m$-dissipativity of diffusion operators on $L^{1}$. In Infinite Dimensional Stochastic Analysis (Amsterdam, 1999). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52 39–54. R. Neth. Acad. Arts Sci., Amsterdam.
• [16] Guillemin, V. and Pollack, A. (1974). Differential Topology. Prentice-Hall, Englewood Cliffs, NJ.
• [17] Has’minskiĭ, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 179–196.
• [18] Has’minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis 7. Sijthoff & Noordhoff, Alphen aan den Rijn.
• [19] Huang, W., Ji, M., Liu, Z. and Yi, Y. (2013). Steady states of Fokker–Planck equations, Parts I–III. Submitted.
• [20] Huang, W., Ji, M., Liu, Z. and Yi, Y. (2013). Concentration and limit behaviors of stationary measures. Preprint.
• [21] Huang, W., Ji, M., Liu, Z. and Yi, Y. (2014). Convergence of Gibbs measures. Submitted.
• [22] Skorohod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. Amer. Math. Soc., Providence, RI.
• [23] Veretennikov, A. Y. (1987). Bounds for the mixing rate in the theory of stochastic equations. Theory Probab. Appl. 32 273–281.
• [24] Veretennikov, A. Y. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115–127.
• [25] Veretennikov, A. Y. (1999). On polynomial mixing and the rate of convergence for stochastic differential and difference equations. Theory Probab. Appl. 44 361–374.