We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded H^∞-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded C^1,λ-domains with λ∈[0,1], revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable.

In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal ℓp-regularity in terms of its continuous counterpart, thereby establishing a unified framework that yields numerous new discrete regularity results. Moreover, as a consequence of the continuous-time theory, we establish several important properties of discrete stochastic maximal regularity such as extrapolation in the exponent p and with respect to a power weight. Furthermore, employing the H∞-functional calculus, we derive a powerful discrete maximal estimate in the trace space norm DA(1-1p,p) for p∈[2,∞).

Using the weak convergence approach, we prove the large deviation principle (LDP) for solutions to quasilinear stochastic evolution equations with small Gaussian noise in the critical variational setting, a recently developed general variational framework. No additional assumptions are made apart from those required for well-posedness. In particular, no monotonicity is required, nor a compact embedding in the Gelfand triple. Moreover, we allow for flexible growth of the diffusion coefficient, including gradient noise. This leads to numerous applications for which the LDP was not established yet, in particular equations on unbounded domains with gradient noise. Since our framework includes the 2D Navier–Stokes and Boussinesq equations with gradient noise and unbounded domains, our results resolve an open problem that has remained unsolved for over 15 years.

We consider solutions to linear parabolic SPDEs of the form du(t) + Au(t)dt = g (t)dβ, u(0) = 0, where A is a positive, invertible, and self-adjoint operator on a Hilbert space X, β is a one-dimensional Brownian motion, and g (t) ≡ x ∈ X . We show that, for all α ∈^[0,¹2), u ∈ L²⁽Ω;W^α,2(0,T;D(A^1/2)⁾⁾ if and only if x ∈ D(A^α). In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient g to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [Comptes Rendus. Mathématique 354 (2016)].

In this paper we consider the variational setting for SPDE on a Gelfand triple (V,H,V∗). Under the standard conditions on a linear coercive pair (A, B), and a symmetry condition on A we manage to extrapolate the classical L2-estimates in time to Lp-estimates for some p>2 without any further conditions on (A, B). As a consequence we obtain several other a priori regularity results of the paths of the solution. Under the assumption that V embeds compactly into H, we derive a universal compactness result quantifying over all (A, B). As an application of the compactness result we prove global existence of weak solutions to a system of second order quasi-linear equations.

In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical spaces, which, when applied to nonlinear SPDEs, coincides with the concept of scaling-invariant spaces. This framework leads to several sharp blow-up criteria and enables one to obtain instantaneous regularization results. Additionally, we refine and unify our previous results, while also presenting several new contributions. In the second part of the survey, we apply the abstract results to several concrete SPDEs. In particular, we give applications to stochastic perturbations of quasi-geostrophic equations, Navier-Stokes equations, and reaction-diffusion systems (including Allen–Cahn, Cahn–Hilliard and Lotka–Volterra models). Moreover, for the Navier–Stokes equations, we establish new Serrin-type blow-up criteria. While some applications are addressed using L2-theory, many require a more general Lp(Lq)-framework. In the final section, we outline several open problems, covering both abstract aspects of stochastic evolution equations, and concrete questions in the study of linear and nonlinear SPDEs.

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. We prove that this operator admits a bounded H^∞-calculus on Sobolev spaces with power weights measuring the distance to the boundary. These weights do not necessarily belong to the class of Muckenhoupt A_p weights. We additionally study the corresponding Dirichlet and Neumann heat semigroup. It is shown that these semigroups, in contrast to the L^p-case, have polynomial growth. Moreover, maximal regularity results for the heat equation are derived on inhomogeneous and homogeneous weighted Sobolev spaces.

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

We obtain polynomial decay rates for C₀-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups, we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces.

In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen-Cahn equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for d \in \{1, 2, 3, 4\} and the Brusselator for d \in \{1, 2, 3\}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on It\^ o's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new L^\zeta-coercivity/dissipativity conditions and the development of an L^p(L^q)framework for systems of reaction-diffusion equations, which are needed when treating dimensions d \in \{2, 3\} in the case of cubic or higher order nonlinearities.

In this paper we give growth estimates for ‖Tⁿ‖ for n→∞ in the case T is a strongly Kreiss bounded operator on a UMD Banach space X. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and L^p-spaces, but also intermediate UMD spaces such as non-commutative L^p-spaces and variable Lebesgue spaces.

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error (Formula presented.) where p∈[2,∞), U is the mild solution, U^j is obtained from a time discretisation scheme, k is the step size, and N_k=T/k for final time T>0. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error (Formula presented.) which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

This paper introduces a new p-dependent coercivity condition through which (Formula presented.) -moments for solutions can be obtained for a large class of SPDEs in the variational framework. If p = 2, our condition reduces to the classical coercivity condition, which only yields second moments for the solution. The abstract result is shown to be optimal. Moreover, the results are applied to obtain (Formula presented.) -moments of solutions for several classical SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burgers' equation and the Navier–Stokes equations in two spatial dimensions. Furthermore, we can recover recent results for systems of SPDEs and higher-order SPDEs using our unifying coercivity condition.

In this paper, we present counterexamples to maximal L^p-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L²-regularity on H^-1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal L^p-regularity on H^-1(R^d) or L²-regularity on L²(R^d).

In this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator is the generator of a strongly continuous semigroup on a Hilbert space, and the focus is on nonparabolic problems. The main results are optimal bounds for the uniform strong errorwhere, is the mild solution, is obtained from a time discretization scheme, is the step size and. The usual schemes such as the exponential Euler (EE), the implicit Euler (IE), the Crank-Nicolson (CN) method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show (linear equation, additive noise, general) (nonlinear equation, multiplicative noise, contractive) (nonlinear wave equation, multiplicative noise), for a large class of time discretization schemes. The logarithmic factor can be removed if the EE method is used with a (quasi)-contractive. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong errorApplications to Maxwell equations, Schrödinger equations and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the IE and the CN method.

In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω), and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove L^p((0,T)× Ω,t^κ dt;H^σ,q(T^d))-estimates. The main novelty is that we do not require p = q. Moreover, we allow arbitrary σ ∈ R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case B_q,p^d/q-1 for q∈[2,2d) and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2. Moreover, for d=2, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

We develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent p ∈ (0, 1]. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.

In this chapter we address a couple of topics in the theory of H^∞-calculus centering around the question what can be said about an operator of the form A+B when A and B have certain “good” properties such as being (R-)sectorial or admitting a bounded H^∞-calculus.