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 qge 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}_{sigma } 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.
Highlights
On X0 (typically X0 = Lr (O; CN ) where r ∈ [2, ∞)), we consider the following stochastic evolution equation: dU (t) + A(t)U (t)dt = F(t, U (t))dt + B(t)U (t) + G(t, U (t)) d WH (t ), (1.1)U (0) = u0, where H is a Hilbert space, WH a cylindrical Brownian motion, A : R+ × → L(X1, X0) and B : R+ × → L(X1, γ (H, X 1 )) are progressively measurable, the functions F and G are suitable nonlinearities, and the initial value u0 : → X0 is F0-measurable
We are interested in maximal L p-regularity results for (1.1)
In deterministic parabolic PDE, maximal regularity is routinely used without identifying it as a specific property
Summary
On X0 (typically X0 = Lr (O; CN ) where r ∈ [2, ∞)), we consider the following stochastic evolution equation: dU (t) + A(t)U (t)dt = F(t, U (t))dt +. We are interested in maximal L p-regularity results for (1.1) This means that we want to investigate well-posedness together with sharp L p-regularity estimates given the data F, G and u0. Knowing these sharp regularity results for equations such as (1.1), gives a priori estimates to nonlinear equations involving suitable nonlinearities F(t, U (t))dt and G(t, U (t))d WH (t). Well-posedness of such non-linear equations follows from these a priori estimates (see e.g. the proofs in [94])
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