Parti Foundations 1 1 Why equations with Levy noise? 3 1.1 Discrete-time dynamical systems 3 1.2 Deterministic continuous-time systems ' ' ' 5 1.3 Stochastic continuous-time systems 6 1.4 Courrege's theorem 8 1.5 Ito's approach 9 1.6 Infinite-dimensional case 12 2 Analytic preliminaries 13 2.1 Notation ' 13 2.2 Sobolev and Holder spaces 13 2.3 L p-and C p-spaces " ' ' 15 2.4 Lipschitz functions and composition operators 16 2.5 Differential operators 17 3 Probabilistic preliminaries 20 "3.1 Basic definitions ~ 20 3.2 Kolmogorov existence theorem 22 3.3 Random elements in Banacti spaces 23 3.4 Stochastic processes in Banach spaces 25 3.5 Gaussian measures on Hilbert spaces 28 3.6 Gaussian 1 measures on topological spaces 30 ' 3.7 Submartingales , ; •. 31 3.8 Semimartingales • 36 3.9 Burkholder-Davies-Guhdy inequalities .

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We study nonlinear wave and heat equations on ‫ޒ‬ d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ‫ޒ‬ d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise. IntroductionThe paper is concerned with the following stochastic wave equationand heat equationIn (0.1) and (0.2), u 0 and v 0 are given functions, f, b : ‫ޒ‬ → ‫,ޒ‬ and W is a spatially homogeneous Wiener process defined on a filtered probability space ᑯ = ( , ᑠ, (ᑠ t ) t≥0 , ‫.)ސ‬ For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ‫ޒ‬ d -space. Process W takes values in the space of tempered distributions S ‫ޒ(‬ d ), and has the covariance of the following formwhere ϕ (s) (x) = ϕ(−x), x ∈ ‫ޒ‬ d , and is a positive-definite tempered distribution. Then has to be the Fourier transform of a positive symmetric tempered measure µ on ‫ޒ‬ d . We call and µ the space correlation and spectral measure of W. In Section

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