Summary
We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.
References
Baklan, V.V.: On the existence of solutions of stochastic equations in Hilbert space. Depov. Akad. Nauk. Ukr. USR.10, 1299–1303 (1963)
Barlow, M.T., Yor, M.: Semi-martingale inequalities via the Garsia-Rodemich-Rumsey Lemma, and applications to local times. J. Funct. Anal.49, 198–229 (1982)
Bensoussan, A., Temam, R.: Équations aux dérivées partielle stochastiques non linéaires. Isr. J. Math.11, 95–129 (1972)
Chojnowska-Michalik, A.: Stochastic differential equations in Hilbert spaces and their applications, Ph.D. Thesis, Institute of Mathematics, Polish Academy of Science, 1976
Chow, P.L.: Function-space differential equations associated with a stochastic partial differential equation Indiana Univ. Math. J.25, 609–627 (1976)
Chow, P.L.: Stochastic PDE's in turbulence related problems. In: Bharucha-Reid, A.T. (ed.) Probability Analysis and Related Topics, vol. I pp. 1–43 New York: Academic Press, 1978
Chow, P.L.: Stability of nonlinear stochastic-evolution equations. J. Math. Anal. Appl.89, 400–419 (1982)
Chow, P.L.: Large deviation problem for some parabolic Itô equations. Commun. Pure and Appl. Math.65, 97–120 (1992)
Da Prato, G., Iannelli, M., Tubaro, L.: Semi-linear stochastic Differential Equations in Hilbert spaces. Bollettino U.M.I (5)16 A, 168–185 (1979)
Dawson, D.A.: Stochastic evolution equations. Math. Biosci.15, 287–316 (1972)
Dawson, D.A.: Stochastic evolution equations and related measured process. J. Multivariate Anal.5, 1–52 (1975)
Fleming, W.H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J.28, 817–843 (1979)
Freidlin, M.I.: Random perturbations of reaction-diffusion equations: the quasi-deterministic approximation. Trans. Am. Math. Soc.305, 665–697 (1988)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood cliffs, N.J., 1964.
Ichikawa, A.: Linear stochastic evolution equations in Hilbert space. J. Differ. Equations28, 266–277 (1978)
Kotelenez, P.: On the semigroup approach to stochastic evolution equations. In: Arnold, L., Kotelenez, P. (eds.) Stochastic space-time models and limit theorems. Reidel: Dordrecht and Boston, 1985
Krylov, N.N., Rozovskii, B.L.: Stochastic evolution equations J. Sov. Math.14, 1233–1277 (1981)
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge: Cambridge University Press, 1990
Kuo, H.H.: Gaussian Measures in Banach Spaces (Lect. Notes in Math., vol. 463) Berlin Heidelberg New York: Springer, 1975
Ladyženskaja, O.A., Solonnikov, V.A., Ural'cera, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, 1968
Markus, R.: Parabolic Itô equations. Trans. Am. Math. Soc.198, 177–190 (1979)
Metivier, M., Pellaumail, J.: Stochastic Integration. New York London Academic Press, 1980
Pardoux, E.: Equations aux dérivées partielles stochastiques non linéaires monotones. Thesis, Université Paris, XI, 1975
Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics3, 127–167 (1979)
Sowers, R.B.: Large deviations for a reaction-diffusion equation with non-Gaussian perturbation. Ann. Probab.20, 504–537 (1992)
Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole d'Eté de Probabilités de Saint Flour XIV (Lect. Notes Math., vol. 1180, pp. 265–435) Berlin Heidelberg New York: Springer, 1984
Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.11, 230–243 (1969)
Author information
Authors and Affiliations
Additional information
The work of the first author was supported in part by the NSF grant DMS-91-01360
Rights and permissions
About this article
Cite this article
Chow, PL., Jiang, JL. Stochastic partial differential equations in Hölder spaces. Probab. Th. Rel. Fields 99, 1–27 (1994). https://doi.org/10.1007/BF01199588
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01199588
Mathematics Subject Classification (1991)
- 60H05
- 60H15
- 35R60