Regularity of solutions of linear stochastic equations in hilbert spaces

Prato, Giuseppe Da; Kwapień, Stanisław; Zabczyk, Jerzy

doi:10.1080/17442508708833480

Cited by 153 publications

(125 citation statements)

References 6 publications

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“…As in Proposition 3.2 this follows from the (stochastic) Fubini theorem (see [14]). Therefore, the above estimates imply (4.1).…”

Section: Stochastic Convolutionsmentioning

confidence: 61%

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

Veraar

2009

J. Evol. Equ.

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Abstract. In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:Here, (A(t)) t∈ [0,T ] (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapień, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second-order stochastic partial differential equation which has been studied by Sanz-Solé and Vuillermot. This leads to several improvements of their result.

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“…As in Proposition 3.2 this follows from the (stochastic) Fubini theorem (see [14]). Therefore, the above estimates imply (4.1).…”

Section: Stochastic Convolutionsmentioning

confidence: 61%

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

Veraar

2009

J. Evol. Equ.

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“…The choices of λ and δ are related to the constants in [2]. This extends results for autonomous equations in [13,14,19]. For completeness we note that under additional conditions on the Banach space E (see [32]) one can extend the results of Sections 3 and 4 to the case where B also depends on Ω in a suitable way.…”

mentioning

confidence: 53%

“…(2) the special case of additive noise. The short proof given below is an obvious adaption of the arguments in [13].…”

Section: Stochastic Integrationmentioning

confidence: 99%

“…[8,9,13,14,19,33] and references therein). The case where (A(t)) t∈[0,T ] depends on time has been studied for instance in [12] by Da Prato, Iannelli and Tubaro, in [41] by Seidler, and in [24] by Krylov. Working in a Hilbert space setting and with constant domains D(A(t)), in [12] time regularity of solutions is considered, whereas in [41] questions of space-time regularity are addressed in a special parabolic setting.…”

mentioning

confidence: 99%

“…In Section 3 the existence of mild and weak solutions of (1.1) is characterised and a factorisation method from Da Prato, Kwapień and Zabczyk [13], Millet and Smoleński [29], and [41] is extended to obtain sufficient conditions for path continuity of solutions.…”

mentioning

confidence: 99%

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Non-autonomous stochastic Cauchy problems in Banach spaces

Veraar

Zimmerschied

2008

Studia Math.

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Abstract. We study the non-autonomous stochastic Cauchy problem on a real Banach space E,Here, W H is a cylindrical Brownian motion on a real separable Hilbert space H, (B(t)) t∈ [0,T ] are closed and densely defined operators from a constant domain ] denotes the generator of an evolution family on E, and u 0 ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapień, and Zabczyk we obtain space-time regularity results for parabolic evolution families on Banach spaces. We apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.

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Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

Chen

Lei

2021

Math Methods in App Sciences

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This work is concerned with a class of stochastic partial differential equations with a fast random dynamical boundary condition. In the limit of fast diffusion, it derives an effective stochastic partial differential equation to describe the evolution of the dominant pattern. Using the multiscale analysis and the averaging principle, it then establishes deviation estimates of the original stochastic system towards the effective approximating system. A concrete example further illustrates the result on a large time scale.

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Regularity of solutions of linear stochastic equations in hilbert spaces

Cited by 153 publications

References 6 publications

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

Non-autonomous stochastic Cauchy problems in Banach spaces

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

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