Non-autonomous stochastic Cauchy problems in Banach spaces

Abstract

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.

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

“…This can be proved in a similar way as in [56,Lemma 4.1], by replacing the fractional domain spaces by E η . The only part of the proof of [56, Lemma 4.1] that requires a different argument is the estimate for…”

“…Condition (H ∞ ) has also appeared in [56] (with A(t) replaced by −A(t)). In the autonomous (H ∞ ) has been used in [19] to obtain maximal regularity for equations with additive noise in Banach spaces.…”

“…In the autonomous (H ∞ ) has been used in [19] to obtain maximal regularity for equations with additive noise in Banach spaces. This has been extended to the non-autonomous setting in [56]. We reformulate the sufficient conditions from Remark 2.6 in our situation here.…”

“…Recently in [37], van Neerven, Weis, and the author considered the autonomous case of (SE) in Banach spaces E which include all L p -spaces with p ∈ [1, ∞). In [56] Zimmerschied and the author studied (SE) with additive noise on a general Banach space, and some parts of the current paper build on these ideas.…”

Analytically weak solutions to linear SPDEs with unbounded time-dependent differential operators and an application

“…Prévôt and Röckner for coercive L(t) constructed variational solutions to (1.1), see [17]. Veraar and Zimmerschied in [19] considered the case where the L(t) are sectorial, uniformly in t ∈ [t 0 , T ].…”