# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stochastic wave equations with dissipative damping. (English) Zbl 1122.60056
For a stochastic wave equation with nonlinear damping in a bounded open set in $\Bbb R^n$ perturbed additively by a trace class Wiener process, growth conditions on the nonlinearity are given to ensure existence of an invariant measure for the associated Markov semigroup. A strengthened version of these conditions is shown to imply uniqueness of the invariant measure. The methods essentially rest upon employing suitable Lyapunov functions in order to obtain tightness and exponential decay properties, respectively.

##### MSC:
 60H15 Stochastic partial differential equations 37L40 Invariant measures (infinite-dimensional dissipative systems)
Full Text:
##### References:
 [1] Agmon, S. A.: Lectures on elliptic boundary value problems. (1965) · Zbl 0142.37401 [2] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. (1975) · Zbl 0335.47035 [3] Barbu, V.; Da Prato, G.: The stochastic nonlinear damped wave equation. Appl. math. Optim. 46, 125-141 (2002) · Zbl 1024.47025 [4] V. Barbu, G. Da Prato, L. Tubaro, Stochastic second order equations with nonlinear damping, 2005. Preprint · Zbl 05946901 [5] Carmona, R.; Nualart, D.: Random non-linear wave equation: smoothness of solutions. Probab. theory related fields 95, 87-102 (1993) [6] Crauel, H.; Debussche, A.; Flandoli, F.: Random attractors. J. dynam. Differential equations 9, 307-341 (1997) · Zbl 0884.58064 [7] S. Cerrai, M. Freidlin, On the Smoluchowski--Kramers approximation for a system with an infinite number of degrees of freedom, 2004. Preprint SNS · Zbl 1093.60036 [8] Dalang, R.; Frangos, N.: The stochastic wave equation in two spatial dimensions. Ann. probab. 26, No. 1, 187-212 (1998) · Zbl 0938.60046 [9] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions. (1992) · Zbl 0761.60052 [10] Da Prato, G.; Zabczyk, J.: Ergodicity for infinite dimensional systems. London mathematical society lecture notes 229 (1996) · Zbl 0849.60052 [11] M. Hairer, J. Mattingly, Ergodicity of the degenerate stochastic 2-D Navier--Stokes equations, 2004. Preprint · Zbl 1059.60073 [12] Kuksin, S.; Shirikyan, A.: Stochastic dissipative pdes and Gibbs measures. Commun. math. Phys. 213, No. 2, 291-330 (2000) · Zbl 0974.60046 [13] Masmoudi, N.; Yang, L. Sang: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes. Commun. math. Phys. 227, No. 3, 461-481 (2002) · Zbl 1009.37049 [14] Mattingly, J.: Ergodicity of 2-D Navier--Stokes equations with random forcing and large viscosity. Commun. math. Phys. 206, No. 2, 273-288 (1999) · Zbl 0953.37023 [15] Millet, A.; Morien, P. L.: On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution. Ann. appl. Probab. 11, 922-951 (2001) · Zbl 1017.60072 [16] Millet, A.; Sanz-Solé, M.: Approximation and support theorem for a wave equation in two space dimensions. Bernoulli 6, 887-915 (2000) · Zbl 0968.60059 [17] E. Pardoux, Equations aux derivées partielles stochastiques nonlinéaires monotones, Thèse, Université Paris XI, 1975 [18] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023 [19] Peszat, S.; Zabczyk, J.: Nonlinear stochastic wave and heat equations. Probab. theory related fields 116, 421-443 (2000) · Zbl 0959.60044