Invariant measures for stochastic heat equations

• Autorzy: Tessitore G. , Zabczyk J.
• Języki publikacji: EN

Abstrakty

EN

The paper is concerned with the asymptotic behaviour of solutions to the nonlinear stochastic heat equations, with spatially homogeneous noise, in the whole space. Sufficient conditions for the existence of invariant measures, in weighted spaces of locally square-integrable functions, are given. For linear equations with multiplicative noise an invariant measure, supported by positive functions, is constructed. The existence of a stationary solution to the vector Burgers equations is obtained as an application of the general theory.

Słowa kluczowe

EN

nonlinear stochastic heat equations; linear equation; asymptotic solutions

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1998
• Tom: Vol. 18, Fasc. 2
• Strony: 271--287
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Tessitore G.

• Universita di Firenze, Dipartimento di Matematica Applicata, Firenze, Italy

autor

Zabczyk J.

• Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

Bibliografia

• [1] L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, preprint.
• [2] G. Da Prato, D. Gatarek and J. Zabczyk, Invariant measures for semilinear stochastic equations, Stochastic Anal. Appl. 10 (1992), pp. 387-408.
• [3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge 1992.
• [4] — Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge 1996,
• [5] D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), pp. 141-180.
• [6] I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions, Vol. 4, Academic Press, 1964.
• [7] Y. Kifer, The Burgers equation with a random force and a general model for directed polymers in random environments, preprint.
• [8] P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equation, Stochastics Stochastics Rep. 41 (1992), pp. 177-199.
• [9] H. Кunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge 1990.
• [10] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.
• [11] J. Nobel, Evolution equations with gaussian potential, Stochastic Anal. Appl., to appear.
• [12] E. Pardoux, Equations aux derives partielles stochastiques nonlinéaires monotones, PhD Thesis, Université Paris XI, 1975.
• [13] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, preprint SNS 1995.
• [14] M. Reed and B. Sim on, Methods of Modem Mathematical Physics, Academic Press, 1975.
• [15] G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Preprint Scuola Normale Superiore di Pisa No. 15, March 1996. To appear in Stochastic Process. Appl.
• [16] — Invariant measures for the stochastic heat equation, Preprint Scuola Normale Superiore di Pisa No. 28, May 1996.