Abstract
We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures μ, solvability of the associated Kolmogorov equation in L1(μ) is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aliprantis, C.D., Border, K.C.: Infinite dimensional analysis, 3rd edn. Springer, Berlin (2006). MR 2378491
Ambrosetti, A., Prodi, G.: A primer of nonlinear analysis. Cambridge University Press, Cambridge (1995). MR 1336591 (96a:58019)
Barbu, V.: Nonlinear differential equations of monotone types [sic] in Banach spaces. Springer, New York (2010). MR 2582280
Barbu, V., Da Prato, G.: Ergodicity for nonlinear stochastic equations in variational formulation. Appl. Math. Optim. 53(2), 121–139 (2006). MR 2172782 (2007d:60030)
Bourbaki, N.: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés. Hermann, Paris (1969). MR 0276436
Cerrai, S.: Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, vol. 1762. Springer-Verlag, Berlin (2001). MR 2002j:35327
Da Prato, G.: Kolmogorov equations for stochastic PDEs. Birkhäuser Verlag, Basel (2004). MR 2111320 (2005m:60002)
Da Prato, G., Zabczyk, J.: Ergodicity for infinite-dimensional systems. Cambridge University Press, Cambridge (1996). MR 1417491 (97k:60165)
Eberle, A.: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Lecture Notes in Mathematics, vol. 1718. Springer-Verlag, Berlin (1999). MR 1734956 (2001c:60122)
Es-Sarhir, A., Stannat, W.: Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications. J. Evol. Equ. 8(1), 129–154 (2008). MR 2383485
Es-Sarhir, A., Stannat, W.: Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications. J. Funct. Anal. 259(5), 1248–1272 (2010). MR 2652188
Krylov, N.V., Rozovskiı̆, B.L.: Stochastic evolution equations, Current problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 71–147, 256. MR 570795 (81m:60116)
Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968). MR 0247243
Marinelli, C., Prévôt, C., Röckner, M.: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 258(2), 616–649 (2010). MR 2557949
Marinelli, C., Scarpa, L.: Refined existence and regularity results for a class of semilinear dissipative SPDEs. arXiv:1711.11091
Marinelli, C., Scarpa, L.: Well-posedness of monotone semilinear SPDEs with semimartingale noise. arXiv:1805.07562
Marinelli, C., Scarpa, L.: A variational approach to dissipative SPDEs with singular drift. Ann. Probab. 46(3), 1455–1497 (2018). MR 3785593
Marinelli, C., Ziglio, G.: Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise. Dyn. Partial Differ. Equ. 7(1), 1–23 (2010). MR 2656416
Pardoux, E.: Equations aux derivées partielles stochastiques nonlinéaires monotones. Ph.D. thesis, Université Paris XI (1975)
Stannat, W.: Stochastic partial differential equations: Kolmogorov operators and invariant measures. Jahresber. Dtsch. Math.-Ver. 113(2), 81–109 (2011). MR 2768734
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Marinelli, C., Scarpa, L. Ergodicity and Kolmogorov Equations for Dissipative SPDEs with Singular Drift: a Variational Approach. Potential Anal 52, 69–103 (2020). https://doi.org/10.1007/s11118-018-9731-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-018-9731-5