Stochastic maximal regularity for rough time-dependent problems

Abstract

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain L p (L q) estimates for all p > 2 and q ≥ 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain L p (L p) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces T p,2 σ of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free. Keywords Stochastic PDEs • Maximal regularity • VMO coefficients • Measurable coefficients • Higher order equations • Sobolev spaces • A p-weights

Stability properties of stochastic maximal L-regularity

“…Because of the weights in time one can treat rough initial data. This has already been demonstrated by Portal and the second author in [44] in the semilinear case.…”

“…[1,18,39]). Recently, extensions to the time and Ω-dependent setting have been obtained in [44]. The stochastic maximal regularity theory of the above mentioned papers provides an alternative approach and extension of a part of Krylov's L ptheory for stochastic PDEs (see [26] and the overview [27]).…”

“…Nowadays it is known that a large class of elliptic operators have a bounded H ∞ -calculus. For instances see [10], [39,Example 3.2], [44,Subsection 1.3], [20,Section 10.8] and in the reference therein.…”

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]).…”