Stochastic evolution equations

Крылов, Н. В.; Rozovskiĭ, B. L.

doi:10.1007/bf01084893

Cited by 463 publications

(291 citation statements)

References 36 publications

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“…While our proof of the existence and of the uniqueness of u u can be traced down to the variational arguments of ( [37], [41]), the proofs of the remaining statements rest upon the combination of an approximation procedure of the Wong-ZakaõÈ -type with a generalization of some results of ( [12], [14]) which we prove in the appendix. For all this we need several preparatory results.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…It turns out that the methods of ( [37], [41]) are directly applicable to Problem (3.1). In fact, let r 1YÃ h be the strong dual of r 1 h and write h Y i r 1 Yr 1YÃ for the corresponding duality pairing.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Owing to the properties of the three operators h j stated above and to Lemma 3.2, it follows from the Faedo-Galerkin approximation schemes of ( [37], [41]) that there exists a unique v 2 h-valued, measurable random ®eld u u which satis®es Properties (1) and (2) of De®nition 2.1 and which solves Problem (3.1), in that relation (2.1) holds with u u replacing u u , u u À u 0 replacing u and g replacing g. The result then follows by setting u u u u u 0 , which implies that g u u gu u whereg is the nonlinearity of Lemma 3.1. (…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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

Čhuešhov

Vuillermot

1998

Probab Theory Relat Fields

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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

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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

Čhuešhov

Vuillermot

1998

Probab Theory Relat Fields

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“…Existence of weak solutions under appropriate conditions was proved by Skorokhod [13]. Of course, several results on existence of weak (strong) solutions under coercivity condition are available, starting with the work of Viot [14], (see also Pardoux [12], Krylov and Rozovskii [11]). In addition, considerable literature on mild solutions can be found (see Da Prato and Zabczyk [2] for references).…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Gawarecki

Mandrekar²,

Richard

1999

Random Operators and Stochastic Equations

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Several results concerning the existence and uniqueness of solutions of Ito SDE's in a real separable Hubert space have recently been reported. In this work we first obtain the existence and uniqueness of strong solutions to (not necessarily) Ito SDE's in a Hubert space under Lipschitz-type conditions on the coefficients. We assume usual continuity and linear growth conditions. For non-Lipschitz coefficients an approximation technique of Gikhman and Skorokhod is then used to prove the existence of weak solutions taking values in a larger Hubert space H-\. This result depends on an assumption that H can be compactly embedded in H-\, such that the coefficients satisfy regularity conditions with respect to H-\. This assumption is not a limitation to our method as it is necessary even in the deterministic case. Additionally, we prove the existence of martingale solutions to infinite dimensional semilinear SDE's. In both cases, coefficients F(£, ·) and J9(£, ·) may depend on the entire past of X £ C([0, T], H) and not on the value of x(t) alone.Brought to you by | provisional account Unauthenticated Download Date | 7/2/15 1:26 AM

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“…In the "diffusion type" class there is a further distinction on whether the noise is white only with respect to the time variable (see e.g. [74,114,139]) or with respect to both space and time variables (see e.g. [8,9,12,22,98,122,137]).…”

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confidence: 99%

Glauber Dynamics for Quantum Lattice Systems

et al. 2001

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Models of quantum mechanical anharmonic lattice systems ("anharmonic crystals") are described. Temperature quantum Gibbs states are represented by classical Gibbs measures for lattice systems of loop-valued spin variables. These Gibbs measures are also obtained as invariant (equilibrium) measures of a system of stochastic differential equations ("stochastic dynamics", "stochastic quantization"). Existence and uniqueness results for these equations are established and a construction of the solution via a finite volume approximation is given. The Markov property of this solution is also exhibited and properties of the Gibbs distributions (existence, a prioiri estimates, regularity of support) are characterized in terms of the stochastic dynamics. Ergodicity and uniqueness of the Gibbs distributions are also discussed.

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Stochastic evolution equations

Cited by 463 publications

References 36 publications

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

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

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Glauber Dynamics for Quantum Lattice Systems

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