Stochastic evolution equations in UMD Banach spaces

Abstract

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equationwhere A generates an analytic C 0 -semigroup on a UMD Banach space E and W H is a cylindrical Brownian motion with values in a Hilbert space H . We prove that if the mappings F : [0, T ] × E → E and B : [0, T ] × E → L(H, E) satisfy suitable Lipschitz conditions and u 0 is F 0 -measurable and bounded, then this problem has a unique mild solution, which has trajectories in C λ ([0, T ]; D((−A) θ ))) provided λ 0 and θ 0 satisfy λ + θ < 1 2 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations. addresses: J.M.A.M.vanNeerven@tudelft.nl (J.M.A.M. van Neerven), mark@profsonline.nl (M.C. Veraar), Lutz.Weis@math.uni-karlsruhe.de (L. Weis).

Singular behavior of the solution to the stochastic heat equation on a polygonal domain

“…As a consequence, there is no explicit description of the singularities of the solution that are due to the shape of the domain. The situation is in a certain sense similar when considering regularity results that have been obtained in the framework of other approaches to stochastic PDEs, such as the semigroup approach; see, e.g., Da Prato and Zabzcyk [4], Jentzen and Röckner [15], Kruse and Larsson [23], van Neerven, Veraar and Weis [37], [38]. There, the spatial regularity of the solution is typically measured in terms of the domains of fractional powers of the governing linear operator; in our case, in tems of the spaces D (−∆ D O ) r/2 , r 0.…”

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

“…In higher dimensions, this seems to be possible if the order of the operator is larger than the dimension. This has been considered in [37] for the autonomous case (also see [9]). In the non-autonomous setting the case of Dirichlet boundary conditions has been studied in [55,Chapter 8].…”

“…This allows one to consider (SE) in L p -spaces with p ∈ [2, ∞). 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.…”

“…Proposition 2.8). Note that the theory of [36] was applied in [37] for general UMD spaces, but only for autonomous equations. In order to consider nonautonomous equations, it seems that one needs additional assumptions on A(t), and due to the extra technical difficulties we will not consider this situation here.…”

Stochastic delay fractional evolution equations driven by fractional Brownian motion

“…Systems are often subjected to random perturbations. Stochastic equations have been investigated by many authors, see, for example, Da Prato and Zabczyk [1], Liu [2], van Neerven et al [3], Taniguchi [4], Jentzen and Röckner [5], Ren et al [20], Ren and Sakthivel [24]. There has been some recent interest in studying evolution equations driven by fractional Brownian motion.…”