Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

Abstract

In this article we prove new results concerning the long-time behavior of random ®elds that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random ®elds eventually homogeneize with respect to the spatial variable and ®nally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random ®elds either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the ®rst case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random ®elds are in fact recurrent in that they can reach every point between the two asymptotic states in a ®nite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random ®elds spend in small neighborhoods of the asymptotic states. Our methods of proof

On the long-time behaviour of a class of parabolic SPDE's: monotonicity methods and exchange of stability

“…We end this work with concluding remarks which allow us to compare our results with those of Chueshov and Vuillermot [6], Chueshov and Vuillermot [7], Bergé et al [3] and Hetzer et al [9]. There are not many differences between the two formulations: on one hand, our evolution operator is more general than in the Laplacian operator and on the other hand, their drift coefficients may depend on the space variable.…”

“…It is worth noting that, since the function g may vanish, these four cases are possible. In Chueshov and Vuillermot [6], the case (C) may occur, but not the case (D) whereas in Bergé et al [3] and in Chueshov and Vuillermot [7], the case (D) appears but not the case (C). This shows that our framework encompasses all the cases of Bergé et al [3], Chueshov and Vuillermot [6] and Chueshov and Vuillermot [7].…”

“…This is one of the reason why this proposition seemed to be less important in Chueshov and Vuillermot [7] and was totally useless in Chueshov and Vuillermot [6] and Bergé et al [3]. Actually, in Chueshov and Vuillermot [6], Chueshov and Vuillermot [7] and Bergé et al [3], the asymptotic behaviour of (Qu ϕ (·, t)) t∈R + follows trivially from a martingale convergence theorem.…”

Variational solutions for a class of fractional stochastic partial differential equations

“…Nous nous intéressons à l'existence, à l'unicité et à l'équivalence de deux notions de solution variationnelle relatives au problème (2). Notre résultat principal est le suivant : Nous mettons ainsi en évidence l'existence, l'unicité et l'indiscernabilité des deux notions de solution variationnelle introduites ci-dessous.…”

“…Notre résultat principal est le suivant : Nous mettons ainsi en évidence l'existence, l'unicité et l'indiscernabilité des deux notions de solution variationnelle introduites ci-dessous. Notre démonstration de l'existence d'une solution variationnelle u I,ϕ repose sur l'existence et la convergence d'un schéma de Faedo-Galerkin convenable associé à (2). Nous pouvons ensuite démontrer que u I,ϕ est nécessairement une solution variationnelle de type II, ceci grâce à de nouvelles propriétés de continuité de l'intégrale stochastique que nous utilisons conjointement avec certaines propriétés d'approximation polynomiale des fonctions test admissibles.…”