Stochastic reaction - diffusion systems with multiplicative noise and non - Lipschitz reaction term
نویسندگان
چکیده
We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of R , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d = 1 and coloured in space if d > 1; in any case the covariance operator is never assumed to be Hilbert-Schmidt. The multiplication term in front of the noise is assumed to be Lipschitz-continuous and no restrictions are given either on its linear growth or on its degenaracy. Our results apply, in particular, to systems of stochastic Ginzburg-Landau equations with multiplicative noise.
منابع مشابه
Large Deviations for Stochastic Reaction–diffusion Systems with Multiplicative Noise and Non-lipschitz Reaction Term1 by Sandra Cerrai
Following classical work by Freidlin [Trans. Amer. Math. Soc. (1988) 305 665–657] and subsequent works by Sowers [Ann. Probab. (1992) 20 504–537] and Peszat [Probab. Theory Related Fields (1994) 98 113–136], we prove large deviation estimates for the small noise limit of systems of stochastic reaction–diffusion equations with globally Lipschitz but unbounded diffusion coefficients, however, ass...
متن کاملLarge deviations for invariant measures of stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term
In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of Rd , d 1, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitzcontinuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. This covers for example the case of Ginzburg–...
متن کاملContinuous Dependence on the Coefficients and Global Existence for Stochastic Reaction Diffusion Equations
We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H, dXn(t) = (AnX(t) + Fn(t,Xn(t))) dt+Gn(t,Xn(t)) dWH(t), Xn(0) = ξn, assuming that the operators An converge to A and the locally Lipschitz functions Fn and Gn converge to the locally Lipschitz functions F and G in an appropriate ...
متن کاملStabilization by noise for a class of stochastic reaction-diffusion equations
Abstract. We prove uniqueness, ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction-diffusion equations with multiplicative noise, in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable. To this purpose, we show that the L1-norm of the difference of two solution...
متن کاملAlmost sure exponential stability of stochastic reaction diffusion systems with Markovian jump
The stochastic reaction diffusion systems may suffer sudden shocks, in order to explain this phenomena, we use Markovian jumps to model stochastic reaction diffusion systems. In this paper, we are interested in almost sure exponential stability of stochastic reaction diffusion systems with Markovian jumps. Under some reasonable conditions, we show that the trivial solution of stocha...
متن کامل