Maximal $L^p$-Regularity for Stochastic Evolution Equations

Abstract

Abstract. We prove maximal L p -regularity for the stochastic evolution equation dU (t) + AU (t) dt = F (t, U (t)) dt + B(t, U (t)) dW H (t), t ∈ [0, T ], U (0) = u 0 , under the assumption that A is a sectorial operator with a bounded H ∞ -calculus of angle less than 1. Introduction. Maximal L p -regularity techniques have been pivotal in much of the recent progress in the theory of parabolic evolution equations (see [2,22,25,54,76,86] and the references therein). Among other things, such techniques provide a systematic and powerful tool to study nonlinear and time-dependent parabolic problems.For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed. A first step towards such a theory has been taken in our recent paper [68], where it was shown that if A is a sectorial operator with a bounded H ∞ -calculus of angle <

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.…”

“…By Lemma 5.8 and Lemma 5.9 we have, for P-almost all ω ∈ Ω, 38) where the constant C > 0 depends only on s, O and the cut-off function η in (3.1), and where M(ω) is defined by (5.30). We redefine c(ω, z) := 0 for z ∈ [0, ∞) + iR and all ω ∈ Ω such that (5.37) and (5.38) does not hold.…”

Stability properties of stochastic maximal L-regularity

“…These results have been applied in several other papers (e.g. [1,18,39]). Recently, extensions to the time and Ω-dependent setting have been obtained in [44].…”

Schauder-type estimates for higher-order parabolic SPDEs

“…For higher-order SPDEs, Krylov and Rozovskii [26] applied their abstract result to obtain the existence and uniqueness of solutions in the Sobolev space W m 2 (R n ). Recently, van Neeven et al [35] and Portal and Veraar [32] obtained some maximal L p -regularity results for strong solutions of abstract stochastic parabolic time-dependent problems, which can also apply to higher-order SPDEs with proper conditions. Another approach to the regularity problem of SPDEs is based on some Hölder spaces, corresponding to the celebrated Schauder theory for classical elliptic and parabolic PDEs (see [13] and references therein).…”

“…However, things may change when one considers L p -integrability (p > 2) rather than square-integrability; more specifically, the coercivity condition (1.2) being adequate for L 2 -theory seems not to be sufficient for L p -integrability of solutions or their derivatives when m ≥ 2. An indirect evidence is that, when the abstract maximal L p -regularity results obtained in [35,32] applied to higher-order SPDEs of type (1.1) the coefficients B α with |α| = m were required to either be sufficiently small or have some additional analytic properties (see [32] for details). Similar phenomena have been found also in complex valued SPDEs (see [2]) and systems of second-order SPDEs (see [18,12]).…”