Oleksandr Misiats¹, Oleksandr Stanzhytskyi², Viktoriia Mogylova³ and Ihor Korol^4,5
1Virginia Commonwealth University, Richmond, VA, US
²Taras Shevchenko National University of Kyiv, Ukraine
³Igor Sirkorsky National Polytechnic University of Kyiv, Ukraine
⁴The John Paul II Catholic University of Lublin, Poland
⁵Uzhhorod National University, Ukraine

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

Chapter DOI: https://doi.org/10.52305/MVCD1139

Abstract

This work is devoted to investigation of large time asymptotic behavior of stochastic partial differential equations (SPDEs). This problem is closely related to the problem of existence and uniqueness of stationary solutions and invariant measures, which in certain cases possess attractive properties. We obtain the sufficient conditions for the existence and uniqueness of invariant measures for a large class of SPDEs of reaction-diffusion type.

Keywords: invariant measure, stationary solution, dissipation condition, monotonicity condition, stochastic homogenization, non-Lipschitz coefficients

References

[1] Volpert V., Elliptic Partial Differential Equations. Volume 2: Reaction-Diffusion
Equations, Springer, 2014.
[2] Da Prato G. and Zabczyk J., Stochastic equations in infinite dimensions, Encyclopedia
of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge,
1992.
[3] Maslowski B. and Seidler J., On sequentially weakly Feller solutions to SPDEs, Atti
Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., vol. 10, no. 2,
pp. 69–78, 1999.
[4] Goldys B. and Maslowski B., Uniform exponential ergodicity of stochastic dissipative
systems, Czechoslovak Mathematical Journal, vol. 51, no. 4, pp. 745–762, 2001.
[5] Brzezniak Z. and Gatarek D., Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces, Stochastic Processes and their Applications,
vol. 84, no. 2, pp.187–225, 1999.
[6] Hieber M., Misiats O., and Stanzhytskyi O., On the bidomain equations driven by
stochastic forces, Discrete and Continuous Dynamical Systems, vol. 40, no. 11, pp.
6159–6177, 2020.
[7] Misiats O., Stanzhytskyi O., and Yip N., Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, Journal of
Theoretical Probability, vol. 29, no. 3, pp. 996–1026, 2016.
[8] Misiats O., Stanzhytskyi O., and Yip N., Invariant measures for stochastic reactiondiffusion equations with weakly dissipative nonlinearities, An International Journal
of Probability and Stochastic Processes, vol. 92, no. 8, pp. 1197–1222, 2020.
[9] Misiats O., Stanzhytskyi O., and Yip N., Asymptotic analysis and homogenization of
invariant measures, Stochastic and Dynamics, vol. 19, no. 2, 1950015, 2019.
[10] Stanzhytskyi O., Investigation of the exponential dichotomy of It ˆo stochastic systems
by means of quadratic forms, Ukrainian Mathematical Journal, vol. 53, no. 11, pp.
1882–1894, 2001.
[11] Stanzhytskyi O., Investigation of invariant sets of Itˆo stochastic systems by means of
Lyapunov function, Ukrainian Mathematical Journal, vol. 53, no. 2, pp. 323–327,
2001.
[12] Stanzhytsky A., Misiats O., and Stanzhytskyi O., Invariant measure for neutral
stochastic functional differential equations with non-lipschitz coefficients, Evolution
Equations and Control Theory, doi:10.3934/eect.2022005
[13] Clark J., Misiats O., Mogylova V. and Stanzhytskyi O., Invariant measure for stochastic functional differential equations in hilbert spaces, arXiv:2011.07034.
[14] Maslowski B., Uniqueness and stability of invariant measure for stochastic differential
equations in Hilbert spaces, Stochastics, vol. 28, no. 2, pp. 85–114, 1989.
[15] Doob J., Asymptotic properties of markoff transition probabilities, Transactions of
the American Mathematical Society, vol. 63, pp. 393–421, 1948.
[16] Wang F., Harnack inequalities for Stochastic Partial Differential Equations, SpringerVerlag, 2013.
[17] Wang F.-Y. and Zhang T., Log-Harnack inequality for mild solutions of SPDEs with
multiplicative noise, Stochastic Processes and Their Applications, vol. 124, no. 3, pp.
1261–1274, 2014.
[18] Hairer M., Coupling stochastic PDEs, XIV International Congress on Mathematical
Physics, pp. 281–289, 2006.
[19] Liu W. and T ¨olle J., Existence and uniqueness of invariant measures for stochastic
evolution equations with weakly dissipative drifts, Electronic Communications in
Probability, vol. 16, pp. 447–457, 2011.
[20] Bianca C. and Dogbe C., On the existence and uniqueness of invariant measure for
multidimensional diffusion processes, Nonlinear Studies, vol. 24, no. 3, pp. 437–468,
2017.
[21] Krylov N., A relatively short proof of itos formula for SPDEs and its applications,
Stochastic Partial Differential Equations: Analysis and Computations, vol. 1, no. 1,
pp. 152–174, 2013.
[22] McKean H., Nagumo’s Equation, Advances in Mathematics, vol. 4, pp. 209–223.
[23] Dawson D., Stochastic evolution equations, Mathematical Biosciences, vol. 154, no.
3-4, pp. 187–316, 1972.
[24] Dirr N. and Yip N., Pinning and de-pinning phenomena in front propagation in heterogeneous media, Interfaces Free Bound., vol. 8, no. 1, pp. 79–109, 2006.
[25] Da Prato G. and Zabczyk J., Ergodicity for infinite-dimensional systems, London
Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press,
Cambridge, 1996.
[26] Eckmann J. and Hairer M., Invariant measures for stochastic partial differential equations in unbounded domains, Nonlinearity, vol. 14, no. 1, pp. 133–151, 2001.
[27] Tessitore G. and Zabczyk J., Invariant measures for stochastic heat equations, Probab.
Math. Statist., vol. 18, no. 2, pp. 271–287, 1998.
[28] Assing S. and Manthey R., Invariant measures for stochastic heat equations with
unbounded coefficients, Stochastic Processes and their Applications, vol. 103, no.2,
pp. 237–256, 2003.
[29] Manthey R. and Zausinger Th., Stochastic evolution equations in L_p^2v  ,  Stochastics
and Stochastic Reports, vol. 66, no. 1-2, pp.37–85, 1999.
[30] Da Prato G., Gatarek D., and Zabczyk J., Invariant measures for semilinear stochastic
equations, Stochastic Analysis and Applications, vol. 10, no. 4, pp. 387–408, 1992.
[31] Da Prato G. and Zabczyk J., Non-explosion, boundedness, and ergodicity for stochastic semilinear equations, Journal of Differential Equations, vol. 98, pp. 181–195,
1992.
[32] Chow P., Stochastic partial differential equations, Chapman & Hall/CRC Applied
Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL,
2007.
[33] Shiryaev A. and Prokhorov Yu., Probability Theory III: Stochastic Calculus, Springer,
1997.
[34] Zhikov V., Kozlov S., and Oleinik O., Homogenization of differential operators and
integral functionals, Springer-Verlag, 1994.
[35] Bensoussan A., Homogenization of a class of stochastic partial differential equations,
Progress in Nonlinear Differential Equations and Their Applications, vol. 5, pp. 47–
65, 1991.
[36] Ichihara N., Homogenization for stochastic partial differential equations derived from
nonlinear filtering with feedback, Journal of the Mathematical Society of Japan, vol.
57, no. 2, pp. 593–603, 2005.
[37] Wang W., Cao D., and Duan J., Effective macroscopic dynamics of stochastic partial
differential equations in perforated domains, SIAM Journal on Mathematical Analysis,
vol. 38, no. 5, pp. 1508–1527, 2007.
[38] Bessaih H., Efendiev Y., and Maris F., Homogenization of brinkman flows in heterogeneous dynamic media. Stochastics and Partial Differential Equations: Analysis and
Computations, vol. 3, pp. 479–505, 2015.
[39] Allaire G., Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, vol.6, pp. 1482–1518, 1991.
[40] Pr´ev ˆot C. and R¨ockner M., A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.
[41] Tung L., A bi-domain model for describing ischemic myocardial d-c potentials, Ph.D.
thesis, MIT, Cambridge, 1978.
[42] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in
Mathematics, 1981.
[43] Keener J. and Sneyd J., Mathematical physiology. Vol. I: Cellular physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer, New York, second edition, 2009.
[44] Kunisch K. and Rund A., Time optimal control of the monodomain model in cardiac
electrophysiology, IMA Journal of Applied Mathematics, vol. 80, no. 6, pp. 1664–
1683, 2015.
[45] Bourgault Y., Coudiere Y., and Pierre C., Existence and uniqeness of the solution for
the bidomain model used in cardiac electrophysiology, Nonlinear Anal: Real World
Applications, vol. 10, no. 1, pp. 458–482, 2009.
[46] Giga Y. and Kajiwara N., On a resolvent estimate for bidomain operators and its
applications, Journal of Mathematical Analysis and Applications, vol.459, no. 1, pp.
528–555, 2018