Stochastic maximal regularity for rough timedependent problems
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Abstract
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of timedependent 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 2mth order systems with VMO regularity in space, we obtain \(L^{p}(L^{q})\) estimates for all \(p>2\) and \(q\ge 2\), leading to optimal spacetime 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 spacetime 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.
Keywords
Stochastic PDEs Maximal regularity VMO coefficients Measurable coefficients Higher order equations Sobolev spaces \(A_p\)weightsMathematics Subject Classification
Primary 60H15 Secondary 35B65 42B37 47D061 Introduction
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 \(G(t,U(t))dW_H(t)\). Wellposedness of such nonlinear equations follows easily from these a priori estimates (see e.g. the proofs in [94]).
1.1 Deterministic maximal regularity
In the timedependent case, maximal regularity is far less understood. For abstract evolution equations, it has been established under regularity assumptions in time: continuity when D(A(t)) is constant (see [109] and the references therein), and Hölder regularity when D(A(t)) varies (see [106]). For concrete PDE with very general boundary conditions, it has been established under continuity assumptions on the coefficients in [28] and this was extended to equations with VMO coefficients in time and space in [31]. For equations with Dirichlet boundary conditions one can obtain maximal regularity when the coefficients satisfy a VMO condition in space and are measurable in time, see [32, 33, 72] and references therein.
In Sect. 4 we apply the results of [33] to obtain an \(L^p(L^q)\)theory for higher order systems. In Sect. 5 we consider second order systems and we use a more classical technique: we obtain an \(L^p(L^q)\) in the space independent case first, then for \(p=q\) we use standard localization arguments to reach the space dependent case under minimal regularity assumptions in the spatial variable.
For deterministic equations maximal \(L^p\)regularity can be used to obtain a local existence theory for quasilinear PDEs of parabolic type (see [21, 108, 111]). Moreover, it can sometimes be used to derive global existence for semilinear equations (see [103, 108]). In [65] maximal \(L^p\)regularity was used to study long time behavior of solutions to quasilinear equations. In [112] it was used to study critical spaces of initial values for which the quasilinear equation is wellposed.
At the moment it remains unclear which of the mentioned theories have a suitable version for stochastic evolution equations. In this paper we develop a maximal \(L^p\)regularity theory for (1.1) which extends several of existing known theories. In future works we plan to study consequence for concrete nonlinear SPDEs. In the next subsection we explain some of the known results, and then compare them to what is proved in the current paper.
1.2 SPDEs of second order
For second order elliptic operatorsA on \({{\mathbb {R}}}^d\) in nondivergence form, this theory was first developed by Krylov in a series of papers [66, 67, 68, 69] and was surveyed in [70, 71]. These works have been very influential. In particular they have led to e.g. [35, 55, 56, 57, 58, 60, 61, 63, 74] where also the case of smooth domains has been considered, and later to e.g. [18, 19, 20, 59, 81] where the case of nonsmooth domains is investigated. In the above mentioned results one uses \(L^p\)integrability in space, time and \(\Omega \). In [58, 69] \(p\ne q\) is allowed but only if \(q\le p\).
The above mentioned papers mostly deal with second order operators of scalar equations. In the deterministic setting higher order systems are considered as well (see e.g. [32, 33, 41, 42, 72]). In the stochastic case some \(L^p\)theory for second order systems has been developed in [62, 90] and an \(L^p(\Omega ;L^2)\)theory in [36], but in the last mentioned paper the main contribution is a \(C^{\alpha }\)theory.
1.3 The role of the \(H^\infty \)calculus assumptions

[29]: 2mth order elliptic systems with general boundary conditions and smooth coefficients

[37]: second order elliptic equations with VMO coefficients.

[38]: second order elliptic systems in divergence form on bounded Lipschitz domains, with \(L^{\infty }\) coefficients and mixed boundary conditions.

[76]: Stokes operator on a Lipschitz domain

[79]: Dirichlet Laplace operator on \(C^2\)domains with weights

[86]: Hodge Laplacian and Stokes operator with Hodge boundary conditions on very weakly Lipschitz domains
1.4 New results
Until now the approach based on functional calculus techniques was limited to equations where A was independent of time and \(\Omega \) (or continuous in time see [94]). We will give a simple method to also treat the case where the coefficients of the differential operator A only depend on time and \(\Omega \) in a progressive measurable way. The method is inspired by [73, Lemma 5.1] and [62] where it is used to reduce to the case of second order equations with constant coefficients.
Our paper extends and unifies the theories in [68] and [95] in several ways. Moreover, we introduce weights in time in order to be able to treat rough initial values. In the deterministic setting weight in time have been used for this purpose in [110]. In the stochastic case some result in this direction have been presented in [2], but not in a maximal regularity setting. Furthermore, we initiate a Lions’s type stochastic maximal regularity theory outside of Hilbert spaces, based on the \(L^2\) theory (see [75, 82, 83, 102]), [10, 114], and [9]. Our main abstract results can be found in Theorems 3.9 and 3.15 below. Our result in the Lions’s setting is Theorem 6.2.
Additionally we are able to give an abstract formulation of the stochastic parabolicity condition for A and B (see Sect. 3.5). It coincides with the classical one if A is a scalar second order operator on \({{\mathbb {R}}}^d\) and B consists of first order operators.
In the applications of our abstract results we will only consider equations on the full space \({{\mathbb {R}}}^d\), but in principle other situations can be considered as well. However, in order to include an operator B which satisfies an optimal abstract stochastic parabolicity condition, we require certain special group generation structure.

2mth order elliptic systems in nondivergence form with coefficients which are only progressively measurable (see Theorem 4.5). The main novelties are that, in space, the coefficients are assumed to be VMO, and we are able to give an \(L^p(L^q)\)theory for all \(p\in (2, \infty )\) and \(q\in [2, \infty )\) (\(p=q=2\) is allowed as well).

Second order elliptic systems in nondivergence form with coefficients which are only progressively measurable, with a diffusion coefficient that satisfies an optimal stochastic parabolicity condition (see Theorem 5.3). When the coefficients are independent of space we give an \(L^p(L^q)\)theory. Moreover, we give an \(L^p(L^p)\)theory if the coefficients are continuous in space.

Second order divergence form equations with coefficients which are only progressively measurable in both the time and the space variables, but satisfy the structural condition of being divergence free. We treat this problem in suitable tent spaces, and in the model case where \(B=0\), \(u_{0}=0\).
1.5 Other forms of maximal regularity
To end this introduction let us mention several other type of maximal \(L^p\)regularity results. In [13, 25] maximal \(L^p\)regularity for any analytic semigroup was established in the real interpolation scale. In [97] maximal regularity was obtained using \(\gamma \)spaces. In Banach function spaces variations of the latter have been obtained in [3].
1.5.1 Notation
We write \(A \lesssim _p B\) whenever \(A \le C_p B\) where \(C_p\) is a constant which depends on the parameter p. Similarly, we write \(A\eqsim _p B\) if \(A\lesssim _p B\) and \(B\lesssim _p A\). Moreover, C is a constant which can vary from line to line.
2 Preliminaries
2.1 Measurability
Let \((S,\Sigma , \mu )\) be a measure space. A function \(f:S\rightarrow X\) is called strongly measurable if it can be approximated by \(\mu \)simple functions a.e. An operator valued function \(f:S\rightarrow {{\mathcal {L}}}(X,Y)\) is called Xstrongly measurable if for every \(x\in X\), \(s\mapsto f(s) x\) is strongly measurable.
Let \((\Omega , {{\mathbb {P}}}, {{\mathcal {A}}})\) be a probability space with filtration \(({{\mathcal {F}}}_t)_{t\ge 0}\). A process \(\phi :{{\mathbb {R}}}_+\times \Omega \rightarrow X\) is called progressively measurable if for every fixed \(T\ge 0\), \(\phi \) restricted to \([0,T]\times \Omega \) is strongly \({{\mathcal {B}}}([0,T])\times {{\mathcal {F}}}_T\)measurable
An operator valued process \(\phi :{{\mathbb {R}}}_+\times \Omega \rightarrow {{\mathcal {L}}}(X,Y)\) will be called Xstrongly progressively measurable if for every \(x\in X\), \(\phi x\) is progressively measurable.
Let \(\bigtriangleup :=\{(s,t): 0\le s\le t<\infty \}\) and \(\bigtriangleup _T = \bigtriangleup \cap [0,T]^2\). Let \({{\mathcal {B}}}_T\) denotes the Borel \(\sigma \)algebra on \(\bigtriangleup _T\). A twoparameter process \(\phi :\bigtriangleup \times \Omega \rightarrow X\) will be called progressively measurable if for every fixed \(T\ge 0\), \(\phi \) restricted to \(\bigtriangleup _T\times \Omega \) is strongly \({{\mathcal {B}}}_T\times {{\mathcal {F}}}_T\)measurable.
2.2 Functional calculus
Let \(H^\infty (\Sigma _{\varphi })\) denote the space of all bounded holomorphic functions \(f:\Sigma _{\varphi }\rightarrow {{\mathbb {C}}}\) and let \(\Vert f\Vert _{H^\infty (\Sigma _{\varphi })} = \sup _{z\in \Sigma _{\varphi }} f(z)\). Let \(H^\infty _0(\Sigma _{\varphi })\subseteq H^\infty (\Sigma _{\varphi })\) be the set of all f for which there exists an \(\varepsilon >0\) and \(C>0\) such that \(f(z)\le C \frac{z^{\varepsilon }}{1+z^{2\varepsilon }}\).
2.3 Function spaces
In several cases the class of weight we will consider is the class of \(A_p\)weights \(w:{{\mathbb {R}}}^d\rightarrow (0,\infty )\). Recall that \(w\in A_p\) if and only if the Hardy–Littlewood maximal function is bounded on \(L^p({{\mathbb {R}}}^d,w)\).
Lemma 2.1
Lemma 2.2
The next result follows from [88, Proposition 7.4].
Proposition 2.3
Proposition 2.4
 (1)
\(W^{1,p}(I,w;X_0) = H^{1,p}(I,w;X_0)\).
 (2)
\([H^{s_0, p}(I,w;X_0), H^{s_1, p}(I,w;X_1)]_{\theta } = H^{s,p}(I,w;[X_0, X_1]_{\theta })\).
Proof
(1): This can be proved as in [80, Proposition 5.5] by using a suitable extension operator and a suitable extension of \(w_{I}\) to a weight on \({{\mathbb {R}}}\).
(2): For \(I = {{\mathbb {R}}}\), this follows from [79, Theorem 3.18]. The general case follows from an extension argument as in [80, Proposition 5.6]. \(\square \)
The following result follows from [89, Theorem 1.1] and standard arguments (see [1] for details). Knowing the optimal trace space is essential in the proof of Theorem 3.15.
Proposition 2.5
2.4 Stochastic integration
Let \(L^p_{{{\mathcal {F}}}}(\Omega ;L^q(I;X))\) denote the space of progressively measurable processes in \(L^p(\Omega ; L^q(I;X))\).
Theorem 2.6
The class of UMD Banach spaces includes all Hilbert spaces, and all \(L^q({\mathcal {O}};G)\) spaces for \(q \in (1,\infty )\), and G another UMD space. It is stable under isomorphism of Banach spaces, and included in the class of reflexive Banach spaces. Closed subspaces, quotients, and duals of UMD spaces are UMD. For more information on UMD spaces see [51] or [17].
Theorem 2.6 allows one to extend the stochastic integral, by density, to the closed linear span in \(L^p(\Omega ;\gamma (L^2({{\mathbb {R}}}_+;H),X))\) of all \({{\mathcal {F}}}\)adapted finite rank step processes in \(\gamma (H,X)\)) (see [93]). We denote this closed linear span by \(L^p _{{\mathcal {F}}} (\Omega ;\gamma (L^2({{\mathbb {R}}}_+;H),X))\). Moreover, this set coincides with the progressively measurable processes in \(L^p(\Omega ;\gamma (L^2({{\mathbb {R}}}_+;H),X))\).
If the UMD Banach space X has type 2 (and thus martingale type 2), then one has a continuous embedding \( L^2({{\mathbb {R}}}_+;\gamma (H,X))\hookrightarrow \gamma (L^2({{\mathbb {R}}}_+;H),X) \) (see [91, 113]). See [52] or [30, 104, 105] for a presentation of the notions of type and martingale type.
Note, however, that the sharp version of Itô’s isomorphism given in Theorem 2.6 is critical to prove stochastic maximal regularity, even in timeindependent situations. The weaker estimate (2.5) (where the right hand side would typically be \(L^{2}({{\mathbb {R}}}_{+};L^{p}({{\mathbb {R}}}^{d}))\) instead of \(L^{p}({{\mathbb {R}}}^{d};L^{2}({{\mathbb {R}}}_{+}))\)) does not suffice for this purpose (see [95]).
We end this subsection with a simple lemma which is applied several times. It will be stated for weights in the socalled \(A_q\) class in dimension one. In the unweighted case the lemma is simple and wellknown. Note that \(w(t) = t^{\alpha }\) is in \(A_q\) if and only if \(\alpha \in (1, q1)\).
Lemma 2.7
Proof
Remark 2.8
Fractional regularity of stochastic integrals in the vectorvalued setting is considered in many previous papers (see [14, 101, 107] and references therein). In particular, the unweighted case of Lemma 2.7 can be found in [107, Corollary 4.9] where it is a consequence of a regularity result on arbitrary UMD spaces. The weighted case appears to be new. Using Rubio de Francia extrapolation techniques one can extend Lemma 2.7 to a large class of Banach functions spaces E(I, w; X) instead of \(L^p(I,w;X)\) (see [24]).
3 Maximal regularity for stochastic evolution equations
In Sects. 3.1 and 3.2 we introduce the definitions of maximal \(L^p\)regularity for deterministic equations and stochastic equations respectively. This extends wellknown notions to the \((t,\omega )\)dependent setting. Moreover, we allow weights in time. In Sect. 3.3 we present a way to reduce the problem with timedependent operators to the timeindependent setting. In Sect. 3.4 we show that if one has maximal \(L^p\)regularity, then this implies wellposedness of semilinear initial value problems. Finally in Sect. 3.5 we explain a setting in which one can reduce to the case \(B=0\).
3.1 The deterministic case
Consider the following hypotheses.
Assumption 3.1
Let \(X_0\) and \(X_1\) be Banach spaces such that \(X_1\hookrightarrow X_0\) is dense. Let \(X_{\theta } = [X_0, X_1]_{\theta }\) and \(X_{\theta ,p} = (X_0, X_1)_{\theta ,p}\) denote the complex and real interpolation spaces at \(\theta \in (0,1)\) and \(p\in [1, \infty ]\), respectively.
Definition 3.2
Remark 3.3
Although we do allow \(T=\infty \) in the above definition, most result will be formulated for \(T\in (0,\infty )\) as this is often simpler and enough for applications to PDEs.
Note that \(A\in \mathrm{DMR}(p,\alpha ,T)\) implies that the solution u is unique (use (3.4)). Furthermore, it implies unique solvability of (3.2) on subintervals \(J = (a,b)\subseteq I\). In particular, \(\mathrm{DMR}(p,\alpha , T)\) implies \(\mathrm{DMR}(p,\alpha , t)\) for all \(t\in (0,T]\).
3.2 Hypothesis on A and B and the definition of SMR
Consider the following hypotheses.
Assumption 3.4
Definition 3.5
A variant of Remark 3.3 holds for \(\mathrm{SMR}\). In particular, any of the estimates (3.7) implies uniqueness.
Remark 3.6
Unlike in the deterministic case the stochastic case does not allow for an optimal endpoint \(H^{\frac{1}{2}, p}\), because already a standard Brownian motion does not have paths in this space a.s. Therefore, we need to quantify over \(\theta \in [0,\frac{1}{2})\) in the above definition.
In the case \(A\) is timeindependent and generates an analytic semigroup, some different type of endpoint results on the timeregularity in terms of Besov spaces have been obtained in [101] which even include regularity at exponent \(\frac{1}{2}\) which is known to be the optimal regularity of a standard Brownian motion.
In the timeindependent case, many properties of \(\mathrm{DMR}\) and \(\mathrm{SMR}\) are known such as independence of p, \(\alpha \) and T. For details we refer to [34, 110] for the deterministic case and [1, 84] for the stochastic case.
In the next two results we collect sufficient conditions for \(\mathrm{DMR}\) and \(\mathrm{SMR}\) in the time independent case. The first result follows from [53, Theorem 5.3 and (3.6)] and [116, Theorem 4.2] (in the latter \(\mathrm{DMR}\) was characterized in terms of Rboundedness).
Proposition 3.7
Suppose Assumption 3.1 is satisfied and assume \(X_0\) is a UMD space. Assume \(A\in {{\mathcal {L}}}(X_1, X_0)\).
If A has a bounded \(H^\infty \)calculus of angle \(<\pi /2\) and \(0\in \rho (A)\), then \(A\in \mathrm{DMR}(p,\alpha , T)\) for all \(p\in (1, \infty )\), \(\alpha \in (1,p1)\) and \(T\in (0,\infty ]\).
In the timeindependent setting the next result follows from [95] for \(\alpha = 0\) (also see [96, 98]). The case \(\alpha \ne 0\) was obtained in [1] by a perturbation argument.
Proposition 3.8
Suppose Assumption 3.1 is satisfied. Assume \(A\in {{\mathcal {L}}}(X_1, X_0)\). Let \(X_0\) be isomorphic to a 2convex Banach function space such that \((X^{1/2}_0)^*\) has the Hardy–Littlewood property (e.g. \(X_0 = L^q({\mathcal {O}};\ell ^2)\), where and \(q\in [2, \infty )\)).
If A has a bounded \(H^\infty \)calculus of angle \(<\pi /2\) and \(0\in \rho (A)\), then \(A\in \mathrm{SMR}(p,\alpha , T)\) for all \(p\in (2, \infty )\), \(\alpha \in (1,\frac{p}{2}1)\) and \(T\in (0,\infty ]\). Moreover, if \(X_0\) is a Hilbert space, then the result in the case \((p,\alpha ) = (2,0)\) holds as well.
3.3 \(\mathrm{SMR}\) for timedependent problems
The next result is a useful tool to derive \(A\in \mathrm{SMR}\) from \(A\in \mathrm{DMR}\) and \(A_0\in \mathrm{SMR}\) for a certain reference operator \(A_0\) which one is free to choose. It extends [73, Lemma 5.1] and [62] where the case with \(A_0 = \Delta \) on \(X_0 = L^p\) with \(\alpha =0\) was considered and where A(t) was a second order operator.
Theorem 3.9
 (i)
There exists a sectorial operator \(A_0\) with \(D(A_0) = X_1\), and \(X_{\frac{1}{2}} = D((\lambda +A_0)^{1/2})\) such that \(A_0\in \mathrm{SMR}(p,\alpha ,T)\).
 (ii)
Assume that there is a \(C>0\) such that for all \(\omega \in \Omega \), \(A(\cdot ,\omega )\in \mathrm{DMR}(p,\alpha ,T)\) and (3.4) holds with constant C.
Proof
Step 1: Progressive measurability and estimates for the deterministic part
Consider the mapping \(\Lambda _T:\Omega \rightarrow {{\mathcal {L}}}(\mathrm{MR}_T,L^p(0,T,w_{\alpha };X_0))\) given by \(\Lambda _T(\omega ) = d/dt + A(\cdot ,\omega )\). Then \(\Lambda _T\) is strongly \({{\mathcal {F}}}_T\)measurable and each \(\Lambda _T(\omega )\) is invertible. It is wellknown that its inverse mapping \(\omega \mapsto \Lambda _T(\omega )^{1}\) is strongly \({{\mathcal {F}}}_T\)measurable as well (see [99]). For convenience we include a short argument for this special case. Fix \(\omega _0\in \Omega \). Now \(\omega \mapsto \Lambda _T(\omega )\Lambda _T(\omega _0)^{1}\in {{\mathcal {L}}}(\mathrm{MR}_T)\) is strongly \({{\mathcal {F}}}_T\)measurable and takes values in the invertible operators. Since taking inverses is a continuous mapping on the open set of invertible mappings it follows that \(\omega \mapsto \Lambda _T(\omega _{0})\Lambda _T(\omega )^{1}\) is strongly \({{\mathcal {F}}}_T\)measurable as well. Clearly, the above holds with T replaced by any \(t\in (0,T]\) as well.
Step 2: Main step
Proposition 3.10
 (i)Assume that there is a constant C such that, for all \(\lambda \in [0,1]\), all \(f\in Z_0\), and all \(g\in Z_{\frac{1}{2}}^{\gamma }\), any strong solution to (3.11) \(U\in E_{\theta }\cap E_{0}\) satisfies$$\begin{aligned} \Vert U\Vert _{E_{\theta }}+\Vert U\Vert _{E_{0}}\le C( \Vert f\Vert _{Z_0} + \Vert g\Vert _{Z_{\frac{1}{2}}^{\gamma }}). \end{aligned}$$(3.12)
 (ii)
Assume that, for all \(f\in Z_0\) and all \(g\in Z_{\frac{1}{2}}^{\gamma }\), there exists a strong solution \(U\in E_{\theta } \cap E_{0}\) to (3.11) with \(\lambda = 0\).
In particular, the above result implies that if \(({\widetilde{A}},0)\in \mathrm{SMR}(p,\alpha ,T)\) and (i) holds for all \(\theta \in [0,\frac{1}{2})\), then \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\). Note that in (i) we only assume that, as soon as a solution \(U\in E_{\theta }\cap E_0\) to (3.11) exists, then (3.12) holds.
Proof
The proof is a generalization of a standard method (see [68, p. 218]). We include the details for completeness. Note that uniqueness follows from (3.12). Let \(\Lambda \subseteq [0,1]\) be the set of all points \(\lambda \) such that for all \(f\in Z_0\) and \(g\in Z_{\frac{1}{2}}^{\gamma }\) (3.11) has a strong solution \(U\in E_{\theta }\). It suffices to prove \(1\in \Lambda \). We claim that there exists an \(\varepsilon >0\) such that for every \(\lambda _0\in \Lambda \), \([\lambda _0\varepsilon , \lambda _0+\varepsilon ]\cap [0,1]\subseteq \Lambda \). Clearly, proving the claim would finish the proof.
3.4 Semilinear equations
In this section we show that our maximal regularity setup allows for simple perturbation arguments in order to include nonzero initial values and nonlinear functions F and G as in (3.1) on a fixed time interval \(I = (0,T)\) as soon as one knows that \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\). The results extend [94, Theorems 4.5 and 5.6] to a setting where we only assume measurability in time and where we can take rough initial values.
Consider the following conditions on A:
Assumption 3.11
Note that the constants \(K_{\mathrm{det}}\) and \(K_{\mathrm{st}}\) exists by the condition \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\). We introduce them in order to have more explicit bounds below.
Consider the following conditions on F and G.
Assumption 3.12
Assumption 3.13
Definition 3.14
 (i)
almost surely, \(U\in L^2(0,T;X_1)\);
 (ii)almost surely for all \(t\in [0,T]\), the following identity holds in \(X_0\):$$\begin{aligned} U(t) + \int _0^t A(s) U(s) \, ds&= u_0 + \int _0^t F(s,U(s)) \, ds \\&\quad + \int _0^t \Big (B(s)U(s) +G(s,U(s))\Big ) \, d W_H(s). \end{aligned}$$
It is straightforward to check that all integrals are welldefined by the assumptions.
Now we state the main result of this subsection:
Theorem 3.15
 (1)
Suppose Assumptions 3.11, 3.12, 3.13 hold, and \(u_0\in L^p(\Omega ,{{\mathcal {F}}}_0;X_{\delta ,p})\).
 (2)Assume the Lipschitz constants \(L_F\) and \(L_G\) satisfy$$\begin{aligned} K_{\mathrm{det}} L_F + K_{\mathrm{st}} L_G <1. \end{aligned}$$
 (3)
There exists a sectorial operator \(A_0\) on \(X_0\) with \(D(A_0)=X_1\) and angle \(<\pi /2\).
Proof
In the proof we use a variation of the arguments in [94, Theorems 4.5]. Let us assume, without loss of generality that \(L_{F},L_{G} \ne 0\), and that \(K_{\mathrm{det}} L_F + K_{\mathrm{st}} L_G = 1\nu \) for some \(\nu \in (0,1)\).
Step 3: Global existence and uniqueness.
3.5 Reduction to \(B = 0\)
Before we continue to applications to SPDEs we show that there is a setting in which one can reduce to the case where \(B = 0\) by using Itô’s formula. Such a reduction is standard (cf. [16, Theorem 3.1], [26, Section 6.6] and [68, Section 4.2]), but it seems that the general setting below has never been considered before. It leads to an abstract form of a socalled stochastic parabolicity condition. In the variational setting (\(p=2\)) a stochastic parabolicity condition appears in a more natural way (see [83, 114]). We refer to [15, 36, 62, 74] for situations in which one cannot reduce to \(B = 0\), but where one still is able to introduce a natural pdependent stochastic parabolicity condition.
Assumption 3.17
The adjoints \(A(t,\omega )^*\) are closed operators on \(X_0^*\) and have a constant domain \(D_{A^*}\) such that \(D_{A^*}\subseteq D((B_j^*)^2)\).
Next we show that the problems (3.5) and (3.26) are equivalent under the above commutation conditions.
Theorem 3.18
Suppose Assumptions 3.1, 3.4 and 3.17 hold. Let \(\widetilde{U}:[0,T]\times \Omega \rightarrow \gamma (H,X_{1})\) be progressively measurable and assume \(\widetilde{U}\in L^2(0,T;\gamma (H,X_{1})\) a.s. Let \({U}(t) = S_B(\zeta (t)) \widetilde{U}(t)\), where \(S_B\) and \(\zeta \) are as in (3.25). Then U is a strong solution to (3.5) if and only if \(\widetilde{U}\) is a strong solution to (3.26). Moreover, \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\) if and only if \(({\widetilde{A}}, 0)\in \mathrm{SMR}(p,\alpha ,T)\).
Proof
Fix \(\psi \in D_{A^*}\) and let \(\phi :{{\mathbb {R}}}^J\rightarrow D_{A^*}\) be given by \(\phi (a) = S_B(a)^*\psi \). First assume \(\widetilde{U}\) is a strong solution to (3.26). The aim of this first step it to apply Itô calculus to find a formula for \(\langle U, \psi \rangle = \langle \widetilde{U}, \phi (\zeta )\rangle \).
Similarly, if U is a strong solution to (3.1) one sees that \(\widetilde{U}\) is a strong solution to (3.26) by applying Itô’s formula to \(\langle \widetilde{U}, \phi (\zeta )\rangle \), where now \(\phi :{{\mathbb {R}}}^J\rightarrow \Psi \) is given by \(\phi (a) = S_B(a)^*\psi \).
The final assertion is clear from the properties of \(S_B(\pm \zeta (t))\). \(\square \)
4 Parabolic systems of SPDEs of 2mth order
In this section we develop an \(L^p(L^q)\)theory for systems of SPDEs of order 2m. The case \(m=1\) (where more can be proven) will be considered in Sect. 5. A similar setting was considered in [94, Section 6] but under a regularity assumption on the coefficients of the operator A. At the same time it extends some of the results in [68]. For more discussion on this we refer to Sect. 5. Finally we mention that the temporal weights allow us to obtain an \(L^p(L^q)\)theory for a wider class of initial values than usually considered.
The main novelty in the results below are that the coefficients \(a_{\alpha \beta }\) are allowed to be matrixvalued, random, and in the timevariable we do not assume any smoothness of the coefficients. The precise assumptions are stated below.
Remark 4.1
We work in the nondivergence form case, but the divergence form case can be treated in the same manner. Indeed, we decompose the problem into a timedependent deterministic part, and a timeindependent stochastic part. We thus only need to use [33, Theorem 6.3] instead of [33, Theorem 5.2] for the timedependent deterministic part.
Remark 4.2
In (4.1) we have left out the Bterm which we did consider in (3.1) in the abstract setting. However, any operator B(t) of \((m1)\)th order with coefficients which are \(W^{1,\infty }\) in the space variable could be included as well. Moreover, no stochastic parabolicity is required as B is of lower order. We choose to leave this out as one can insert this into the nonlinearities \(g_n\). If one wishes to include operators B which are mth order, then this either requires a smallness condition in terms of the maximal regularity estimates, or in order to apply Sect. 3.5 the highest order terms needs to be a group generator. It follows from [12, Theorem 0.1] (and from [49, Theorem 1.14] if \(d=N=1\)) that if B is a differential operator of order \(\ge 2\) and B generates a strongly continuous group then necessarily \(q=2\).
Assumption 4.3
 (1)
The functions \(a_{\alpha \beta }:[0,T]\times \Omega \times {{\mathbb {R}}}^d\rightarrow {{\mathbb {C}}}^{N\times N}\) are strongly progressively measurable.
 (2)There exist \(\mu \in (0,1)\) and \(K>0\) such thatand \(a_{\alpha \beta }(t,\omega ,x)_{{{\mathbb {C}}}^{N\times N}}\le K\) for all \(\xi \in {{\mathbb {R}}}^d, \theta \in {{\mathbb {C}}}^N, x\in {{\mathbb {R}}}^d, t\in [0,T]\) and \(\omega \in \Omega \).$$\begin{aligned} \text {Re}\,\Big (\sum _{\alpha =\beta  = m} \xi ^{\alpha } \xi ^{\beta } (a_{\alpha \beta }(t,\omega ,x)\theta , \theta )_{{{\mathbb {C}}}^N}\Big ) \ge \mu \xi ^{2m} \theta ^2, \end{aligned}$$
 (3)Let \(\gamma \in (0,1)\). Assume there exists an \(R\in (0,\infty )\) such that for all \(\alpha , \beta =m\), \(r\in (0,R]\), \(x\in {{\mathbb {R}}}^d\), \(t\in [0,T]\) and \(\omega \in \Omega \),$$\begin{aligned} \text {osc}_{r,x}(a_{\alpha \beta }(t,\omega ,\cdot ))\le \gamma . \end{aligned}$$
Note that in \((t,\omega )\) only measurability is assumed.
For the \(\Omega \)independent setting, a slightly less restrictive condition appears in [33, Theorem 5.2]. We choose the above formulation assumption in order to make the assumptions easier to state. However, it is possible to extend the results of this section to their setting.
Concerning f and \(g_n\) we make the following assumptions:
Assumption 4.4
 (1)The function \(f:[0,T]\times \Omega \times H^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\rightarrow L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) is strongly progressively measurable, \(f(\cdot , \cdot ,0)\in L^p(\Omega ;L^p(I,w_{\alpha };H^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)))\), and there exist \(L_{f}\) and \({{\widetilde{L}}}_{f}\) such that for all \(t\in [0,T]\), \(\omega \in \Omega \), and \(u,v\in H^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\),$$\begin{aligned} \Vert f(t,\omega , u)  f(t,\omega ,v)\Vert _{L^q({{\mathbb {R}}}^d,{{\mathbb {C}}}^N)}&\le L_{f} \Vert D^{2m} uD^{2m} v\Vert _{L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)}\\&\quad +{{\widetilde{L}}}_{f} \Vert uv\Vert _{H^{2m1,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)}. \end{aligned}$$
 (2)The functions \(g_n:[0,T]\times \Omega \times H^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N) \rightarrow H^{m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) are strongly progressive measurable, \((g_n)_{n\ge 1} (\cdot , \cdot , 0)\in L^p(\Omega ;L^p(I,w_{\alpha };H^{m,q}({{\mathbb {R}}}^d;\ell ^2({{\mathbb {C}}}^N)))\) and there exist \(L_{g}\), \({{\widetilde{L}}}_{g}\) such that for all \(t\in [0,T]\), \(\omega \in \Omega \), and \(u,v\in H^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\),$$\begin{aligned} \Vert (g_n(t,\omega , u)  g_n(t,\omega ,v))_{n\ge 1}\Vert _{H^{m,q}({{\mathbb {R}}}^d;\ell ^2({{\mathbb {C}}}^N))}&\le L_{g} \Vert D^{2m}uD^{2m}v\Vert _{L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)}\\&\quad + {{\widetilde{L}}}_{g} \Vert uv\Vert _{H^{2m1,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)}. \end{aligned}$$
The nonlinearity f can depend on \(u, D^1 u, \ldots , D^{2m} u\) in a Lipschitz continuous way as long as the dependence on \(D^{2m} u\) has a small Lipschitz constant. One could allow lower order terms in A, but one can just put them into the function f. Similarly, g can depend on \(u, D^1 u, \ldots , D^m u\) in a Lipschitz continuous way as long as the dependence on \(D^m u\) has a small Lipschitz constant.
The main result of this section is as follows.
Theorem 4.5
Proof
Let \(X_0 =L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) and \(X_1 =W^{2m,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\). Since the coefficients \(a_{\alpha \beta }\) are uniformly bounded, Assumption 3.4 holds (with \(B=0\)). Let \(X_\theta =H^{2m\theta ,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) for \(\theta \in (0,1)\). Note that if \(\theta \in [0,1]\) and \(2m\theta \in {{\mathbb {N}}}\), then \(X_\theta = W^{2m\theta ,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) (see [115, Theorem 2.33]). Let \(X_{\theta ,p}:=(X_0, X_1)_{\theta ,p} = B^{2m\theta }_{q,p}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\). By [115, 2.4.2 (11) and (16)] Assumption 3.1 holds.
On \(X_0\) consider \(A_0 = (1\Delta )^{m} I_N\) with \(D(A_0) =X_1\), where \(I_N\) stands for the \(N\times N\) diagonal operator. Then by [52, Theorem 10.2.25] and [47, Corollary 5.5.5] \(0\in \rho (A_0)\) and the operator \(A_0\) has a bounded \(H^\infty \)calculus of angle 0. Now from Proposition 3.8 we obtain that condition (1) of Theorem 3.9 holds. From [33, Theorem 5.2 and Section 8] we deduce that the condition (ii) of Theorem 3.9 holds. Therefore, Theorem 3.9 shows that \(A\in \mathrm{SMR}(p,\alpha ,T)\).
Since \(\gamma (\ell ^2,X_{\frac{1}{2}}) = H^{m,q}({{\mathbb {R}}}^d;\ell ^2({{\mathbb {C}}}^N))\) isomorphically (use [52, Proposition 9.3.2] and the isomorphism \(A_0^{1/2}\)), in a similar way as above one sees that the function \(G = g\) satisfies Assumption 3.13 with \(L_G = L_g+\varepsilon \) and \({\widetilde{L}}_G = C_{\varepsilon }{\widetilde{L}}_g\).
Now all the statements follow from Theorem 3.15. \(\square \)
Remark 4.6
To obtain the regularity result of Theorem 4.5 in the whole scale of spaces \(H^{s,q}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) with \(s\in {{\mathbb {R}}}\) one needs to assume smoothness on the coefficients \(a_{\alpha \beta }\), and change the assumptions on f, g and \(u_0\) appropriately. Indeed, as in [68, Section 5] this follows by applying \((1\Delta )^{s/2}\) to both sides of the equation and introducing \(V = (1\Delta )^{s/2} U\), where \(s\in {{\mathbb {R}}}\). The details are left to the reader.
In [68, Theorem 7.2] another method to derive spacetime regularity results is described. In the latter result one loses an \(\varepsilon \) of regularity. The sharp identifications and embeddings obtained in [95] and in the weighted case in [1], make it possible to avoid this loss in regularity.
Remark 4.7
A version of Theorem 4.5 with \(A_{\frac{p}{2}}\)weights in time and \(A_q\)weights in space holds as well. Here one can take any weight \(v\in A_{\frac{q}{2}}\) in time and \(w\in A_r\) in space. For details we refer to [33, Theorem 5.2 and Section 8] and [42, Section 4.4].
As a consequence we obtain the following Hölder continuity result in spacetime.
Corollary 4.8
Proof
Remark 4.9
5 Parabolic systems of SPDEs of second order
In this section we discuss \(L^p(L^q)\)theory for systems of second order SPDEs with rough initial values. The setting is the same as in Sect. 4. However, this time we will consider \(B \ne 0\). Related problems have been discussed in [62] in an \(L^p(L^p)\)setting with smooth initial values and in [36] in an \(L^p(\Omega ;L^2((0,T)\times {{\mathbb {R}}}^d))\)setting and a Hölder setting, with vanishing initial values.
Remark 5.1
The divergence form case could be treated in a similar manner. See Remark 4.1.
We make the following assumptions on the coefficients.
Assumption 5.2
 (1)
The functions \(a_{ij}:[0,T]\times \Omega \times {{\mathbb {R}}}^d\rightarrow {{\mathbb {C}}}^{N\times N}\) and \(\sigma _{jkn}:[0,T]\times \Omega \times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}\) are strongly progressively measurable.
 (2)There exist \(\mu \in (0,1)\) and \(K>0\) such that \(a_{ij}(t,\omega ,x)_{{{\mathbb {C}}}^{N\times N}}\le K\) and \(\Vert (\sigma _{jkn}(t,\omega ,\cdot ))_{n\ge 1}\Vert _{W^{1,\infty }({{\mathbb {R}}}^d;\ell ^2)}\le K\).for all \(\xi \in {{\mathbb {R}}}^d, \theta \in {{\mathbb {C}}}^N, x\in {{\mathbb {R}}}^d, t\in [0,T]\). Here for each fixed numbers \(i,j\in \{1, \ldots , d\}\), \(\Sigma _{ij}(t,\omega ,x)\) is the \(N\times N\) diagonal matrix with diagonal elements \((\frac{1}{2} \sum _{n\ge 1}\sigma _{ikn}(t,\omega ,x) \sigma _{jkn}(t,\omega ,x))_{k=1}^N\).$$\begin{aligned} \text {Re}\,\Big ( \sum _{i,j=1}^d \xi _{i} \xi _{j} ((a_{ij}(t,\omega ,x)\Sigma _{ij}(t,\omega ,x))\theta , \theta )_{{{\mathbb {C}}}^N}\Big ) \ge \mu \xi ^{2} \theta ^2, \end{aligned}$$
 (3)Assume there exists an increasing continuous function \(\zeta :[0,\infty )\rightarrow [0,\infty )\) with \(\zeta (0) = 0\) such that for all i, j, \(x,y\in {{\mathbb {R}}}^d\), \(t\in [0,T]\) and \(\omega \in \Omega \),$$\begin{aligned} a_{i,j}(t,\omega ,x)  a_{i,j}(t,\omega ,y) + \sum _{n\ge 1} \sigma _{jkn}(t,\omega ,x)\sigma _{jkn}(t,\omega ,y)^2\le \zeta (xy). \end{aligned}$$
We start with the xindependent case.
Theorem 5.3
Proof
In the xdependent case we obtain the following, where unlike in Theorems 4.5 and 5.3 we have to take \(p=q\).
Theorem 5.4
To prove this, arguing as in Theorem 5.3 it follows from Theorem 3.15 that it suffices to prove that \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\). We will use a variation of a localization argument of [68, Section 6]. For this we start with the following lemma.
Lemma 5.5
Proof
To be able to apply the freezing lemma, one also needs the following elementary Fubini result, which is trivial if \(I = {{\mathbb {R}}}\), but for bounded intervals can still be deduced from an interpolation argument.
Lemma 5.6
Proof
We also need the following simple commutator formula.
Lemma 5.7
Proof
Proof of Theorem 5.4
As already mentioned it suffices to prove that \((A,B)\in \mathrm{SMR}(p,\alpha ,T)\) (in particular, one only has to treat the problem with \(u_{0}=0\)). To prove this we will use Proposition 3.10 and Lemma 5.5.
Remark 5.8
To obtain regularity in the scale \(H^{s,p}({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)\) for \(s\in {{\mathbb {R}}}\) in Theorem 5.4, Remark 4.6 applies again.
Remark 5.9
 (1)
It would be natural to ask for an \(L^p(L^q)\)theory in Theorem 5.4. In [42] \(A_p\)weights in time, in combination with Rubio de Francia extrapolation techniques, have been used to derive the case \(p\ne q\) from \(p=q\), in the case of continuous coefficients in time. This was later extended to VMO coefficients in space in [33]. The extrapolation technique would be applicable here as well, but it only gives regularity in \(L^p(0,T,L^q(\Omega ;L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)))\) and not in \(L^p((0,T)\times \Omega ;L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^N)))\) as one would like.
 (2)
Another natural question is whether Theorem 5.4 holds if the coefficients are only VMO in the space variable. Some results in this direction have been found for equations in divergence form in [73].
Remark 5.10
Let us motivate that in the commuting case the assumption that the operators \(b_n\) as defined below (5.1) is not far from the general case. Indeed, assume that \(B_1, \ldots , B_J\), are differential operators of order one which generate commuting groups on \(L^q({{\mathbb {R}}}^d;{{\mathbb {C}}}^{N\times N})\) for all \(q\in (1, \infty )\). Then we can write \(B_j u = \sum _{k=1}^d M_{jk} \partial _k u + N_j u\), where \(M_{jk}, N_j\) are \(N\times N\) matrices. Then the \(M_{jk}\) have real eigenvalues since otherwise the Fourier symbol would be unbounded (use [44, Theorem 2.5.16] to reduce to one fixed direction k). Now by [12, Theorem 0.1] the matrices \((M_{jk})_{k=1}^d\) commute and are diagonalizable. Moreover, if the groups \((e^{\cdot B_j})_{j=1}^J\) are commuting, then the operators \((M_{jk})_{j=1,k=1}^{J,d}\) are commuting as well. Therefore, a standard result from linear algebra implies that \((M_{jk})_{j=1,k=1}^{J,d}\) are simultaneously diagonalizable. Hence by a coordinate transformation we could have assumed that \(M_{jk}\) are diagonal matrices with real entries.
Of course to reduce to this setting in a general setup the coordinate transformations would become \((t,\omega )\)dependent and then the reduction breaks down. Even in the \((t,\omega )\)independent case the coefficients of A change after a coordinate transformation, and more importantly the ellipticity conditions changes (unless all matrices \(M_{jk}\) are hermitian, in which case the transformation is orthogonal). On the other hand, if one does not assume the \(B_j\)’s generate commuting groups, then the above fails, and one needs to consider the case of general matrices. In general this leads to a pdependent stochastic parabolicity condition. See [36, 62] for results in this direction.
6 Divergence form equations of second order with measurable coefficients
We plan to develop the theory presented in this section in future work. We thus only include here the simplest situation that showcases how the method of proof used in this paper (particularly in Theorem 3.9), as well as the idea of using the time weights \(w_{\alpha }\) to vary regularity, can be combined with the tent space approach of [9]. For this reason, we choose to take zero initial data (although we could add data in appropriate fractional domains, see Remark 6.10 below), and keep the equation linear (although semilinearities could be treated through fixed point arguments).
Definition 6.1
Note that, by Fubini’s theorem, \(T^{2,2,\sigma } = L^{2}({{\mathbb {R}}}_{+}\times {{\mathbb {R}}}^{d}, \frac{dt dx}{t^{1+\sigma }})\).
Our main result is the following theorem, proven at the end of the section.
Theorem 6.2
Remark 6.3
(2) On the other hand, we estimate \(\Vert U\Vert _{T^{p,2}_{\sigma +2}}\) in terms of \(\Vert \nabla g\Vert _{T^{p,2}_{\sigma }(H^{d})}\). A more natural generalisation of Lions’s maximal regularity result would be to estimate \(\Vert \nabla U\Vert _{T^{p,2}_{\sigma }(H^{d})}\) in terms of \(\Vert g\Vert _{T^{p,2}_{\sigma }}\) or \(\Vert U\Vert _{T^{p,2}_{\sigma }}\) in terms of \(\Vert G\Vert _{T^{p,2}_{\sigma }(H^{d})}\) for \(g =div G\). At this stage, we would need to know that \(\{(ts)^{\frac{1}{2}}\nabla \Gamma (t,s) \;;\; t>s\}\) is uniformly bounded in \({\mathcal {L}}(L^{2}({{\mathbb {R}}}^{d}),L^{2}({{\mathbb {R}}}^{d};{{\mathbb {C}}}^{d}))\) to study such maximal regularity. The fact that we only know uniform boundedness of \(\{\Gamma (t,s) \;;\; t>s\}\) is what led us to the notion of maximal regularity used here. Note, however, that this uniform boundedness of the evolution family is, in Lions’s theory, a consequence of the energy estimates that also give the \(L^{2}(W^{1,2})L^{2}(W^{1,2})\) maximal regularity. So, while \(L^{2}(\frac{dt}{t^{\sigma }};L^2)L^{2}(\frac{dt}{t^{2+\sigma }};L^2)\) maximal regularity is trivial in the time independent case, it is not so in the time dependent case, where generation of a bounded evolution family is not a substantially easier question than maximal regularity. Nevertheless, we plan to return to the question of estimating \(\Vert \nabla U\Vert _{T^{p,2}_{\sigma }(H^{d})}\) in future work.
To prove Theorem 6.2, we proceed as in Theorem 3.9, and decompose the problem into a timeindependent stochastic part, and a timedependent deterministic part. Our key technical tool to estimate both parts is extrapolation in tent spaces, as developed in [8] and [9]. In particular, we need some simple variations of [9, Proposition 5.1], proven below (we include the details for the convenience of the reader). These results make extensive use of the notion of \(L^2  L^2\) offdiagonal decay.
Definition 6.4
Proposition 6.5
Proof
In a very similar way, we have the following stochastic version:
Proposition 6.6
Proof
6.1 Deterministic timedependent maximal regularity in \(T^{p,2}_{\sigma }\).
Proposition 6.7
The operator \({\mathcal {M}}\), initially defined on \(L^{2}({{\mathbb {R}}}_{+}\times {{\mathbb {R}}}^{d})\), extends to a bounded linear operator from \(T^{2,2}_{\sigma }\) to \(T^{2,2}_{\sigma +2}\) for all \(\sigma \ge 0\).
Proof
Theorem 6.8
Let \(\sigma \ge 0\) and \(p>\min (1,\frac{2d}{d+2\sigma +4})\) The deterministic maximal regularity operator \({\mathcal {M}}\) extends to a bounded linear operator from \(T^{p,2}_{\sigma }\) to \(T^{p,2}_{\sigma +2}\).
Proof
The family \(\{\Gamma (t,s) \;;\; t>s\}\) has \(L^2  L^2\) offdiagonal decay of any order by [10, Proposition 3.19]. The result thus follows from Propositions 6.7 and 6.5. \(\square \)
Remark 6.9
As in [8, 9, 10], we could also exploit \(L^{p}L^{2}\) offdiagonal decay for \(p<2\) (and even \(p=1\) if the coefficients are real valued). This would give a wider range of p (and the full range \((1,\infty )\) if the coefficients are real valued). We leave this technical improvement for future work.
6.2 Stochastic timeindependent maximal regularity in \(T^{\,\,p,2}_{\sigma }\)
6.3 Stochastic timedependent maximal regularity in \(T^{p,2}_{\sigma }\).
Assumption 6.11
Remark 6.12
Assumption 6.11 can be satisfied by coefficients that do not have any regularity in space or time. Indeed, it holds for all divergence free coefficients, i.e. coefficients such that \(\sum \limits _{i=1} ^{d} a_{i,j}\partial _{i} = 0\) (almost surely) in the sense of distributions for all \(j=1,..,d\). This was first remarked (to the best of our knowledge) in [39, Lemma 4.4]. Example of divergence free coefficients include, for \(d=3\), matrices for which columns are of the form curlF for some Lipschitz vector field F. Since Assumption 6.11 is also satisfied when the coefficients a are Lipschitz continuous in space (by the product rule and Riesz transform boundedness), we have that Assumption 6.11 holds for all coefficients of the form \(b+c\) where \(b \in L^{\infty }(\Omega \times {{\mathbb {R}}}_{+};W^{1,\infty }({{\mathbb {R}}}^{d}))\) and \(c \in L^{\infty }(\Omega \times {{\mathbb {R}}}_{+} \times {{\mathbb {R}}}^{d})\) is divergence free.
Lemma 6.13
Under Assumption (6.11), we have that \(\{tL(t,\omega )(It\Delta )^{1} \;,\; t>0\}\) has \(L^{2}L^{2}\) offdiagonal decay of any order, uniformly in \(\omega \in \Omega \).
Proof
We can now prove Theorem 6.2, which statement we recall here.
Theorem 6.14
Proof
Notes
Acknowledgements
This work was started during a visit of Veraar at the Australian National University in 2016, and concluded during a visit of Portal at the TU Delft in 2018. The authors would like to express their gratitude to these institutions for providing an excellent environment for their collaboration. We would like to thank Antonio Agresti for carefully reading the manuscript.
References
 1.Agresti, A., Veraar, M.C.: Stability properties of stochastic maximal \(L^p\)regularity. Preprint arXiv:1901.08408 (2019)
 2.Andersson, A., Jentzen, A., Kurniawan, R.: Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. Preprint arXiv:1512.06899 (2015)
 3.Antoni, M.: Regular random field solutions for stochastic evolution equations. Ph.D. thesis (2017)Google Scholar
 4.Auscher, P.: Change of angle in tent spaces. C. R. Math. Acad. Sci. Paris 349(5–6), 297–301 (2011)MathSciNetzbMATHGoogle Scholar
 5.Auscher, P., Axelsson, A.: Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems I. Invent. Math. 184(1), 47–115 (2011)MathSciNetzbMATHGoogle Scholar
 6.Auscher, P., Stahlhut, S.: Functional calculus for first order systems of Dirac type and boundary value problems. Mém. Soc. Math. Fr. (N.S.) 144, vii+164 (2016)Google Scholar
 7.Auscher, P., Hofmann, S., Martell, J.M.: Vertical versus conical square functions. Trans. Am. Math. Soc. 364(10), 5469–5489 (2012)MathSciNetzbMATHGoogle Scholar
 8.Auscher, P., Kriegler, C., Monniaux, S., Portal, P.: Singular integral operators on tent spaces. J. Evol. Equ. 12(4), 741–765 (2012)MathSciNetzbMATHGoogle Scholar
 9.Auscher, P., van Neerven, J., Portal, P.: Conical stochastic maximal \(L^p\)regularity for \(1\le p<\infty \). Math. Ann. 359(3–4), 863–889 (2014)MathSciNetzbMATHGoogle Scholar
 10.Auscher, P., Monniaux, S., Portal, P.: On existence and uniqueness for nonautonomous parabolic Cauchy problems with rough coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear) (2018). Preprint arXiv:1511.05134
 11.Axelsson, A., Keith, S., McIntosh, A.: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163(3), 455–497 (2006)MathSciNetzbMATHGoogle Scholar
 12.Brenner, P.: The Cauchy problem for systems in \(L_{p}\) and \(L_{p,\alpha }\). Ark. Mat. 11, 75–101 (1973)MathSciNetGoogle Scholar
 13.Brzeźniak, Z.: Stochastic partial differential equations in Mtype 2 Banach spaces. Potential Anal. 4(1), 1–45 (1995)MathSciNetzbMATHGoogle Scholar
 14.Brzeźniak, Z.: On stochastic convolution in Banach spaces and applications. Stoch. Stoch. Rep. 61(3–4), 245–295 (1997)MathSciNetzbMATHGoogle Scholar
 15.Brzeźniak, Z., Veraar, M.C.: Is the stochastic parabolicity condition dependent on \(p\) and \(q\)? Electron. J. Probab. 17(56), 24 (2012)MathSciNetzbMATHGoogle Scholar
 16.Brzeźniak, Z., van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differ. Equ. 245(1), 30–58 (2008)zbMATHGoogle Scholar
 17.Burkholder, D.L.: Martingales and singular integrals in Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the geometry of Banach spaces, vol. I, pp. 233–269. NorthHolland, Amsterdam (2001)zbMATHGoogle Scholar
 18.Cioica, P.A., Kim, K.H., Lee, K., Lindner, F.: On the \(L_q(L_p)\)regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains. Electron. J. Probab. 18(82), 41 (2013)zbMATHGoogle Scholar
 19.CioicaLicht, P.A., Kim, K.H., Lee, K.: On the regularity of the stochastic heat equation on polygonal domains in \({{\mathbb{R}}}^{2}\). Preprint arXiv:1809.00429 (2018)
 20.CioicaLicht, P.A., Kim, K.H., Lee, K., Lindner, F.: An \(L_p\)estimate for the stochastic heat equation on an angular domain in \({\mathbb{R}}^2\). Stoch. Partial Differ. Equ. Anal. Comput. 6(1), 45–72 (2018)MathSciNetzbMATHGoogle Scholar
 21.Clément, Ph., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3(Special Issue), 17–32 (1993–1994)Google Scholar
 22.Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985)MathSciNetzbMATHGoogle Scholar
 23.Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \(H^\infty \) functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996)MathSciNetzbMATHGoogle Scholar
 24.CruzUribe, D.V., Martell, J.M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio de Francia, volume 215 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel (2011)zbMATHGoogle Scholar
 25.Da Prato, G., Lunardi, A.: Maximal regularity for stochastic convolutions in \(L^p\) spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9(1), 25–29 (1998)MathSciNetzbMATHGoogle Scholar
 26.Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and Its Applications, 2nd edn. Cambridge University Press, Cambridge (2014)Google Scholar
 27.Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Springer, Berlin (1992) (Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig)Google Scholar
 28.Denk, R., Hieber, M., Prüss, J.: \(R\)boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii + 114 (2003)MathSciNetzbMATHGoogle Scholar
 29.Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R. T. Seeley. Math. Ann. 328(4), 545–583 (2004)MathSciNetzbMATHGoogle Scholar
 30.Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
 31.Dong, H., Gallarati, C.: Higherorder elliptic and parabolic equations with VMO assumptions and general boundary conditions. J. Funct. Anal. 274(7), 1993–2038 (2018)MathSciNetzbMATHGoogle Scholar
 32.Dong, H., Kim, D.: On the \(L_p\)solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Ration. Mech. Anal. 199(3), 889–941 (2011)MathSciNetzbMATHGoogle Scholar
 33.Dong, H., Kim, D.: On \(L_p\)estimates for elliptic and parabolic equations with \(A_p\) weights. Trans. Am. Math. Soc. 370(7), 5081–5130 (2018)zbMATHGoogle Scholar
 34.Dore, G.: Maximal regularity in \(L^p\) spaces for an abstract Cauchy problem. Adv. Differ. Equ. 5(1–3), 293–322 (2000)zbMATHGoogle Scholar
 35.Du, K.: \(W^{2,p}\)solutions of parabolic SPDEs in general domains. Stoch. Process. Appl. (2018)Google Scholar
 36.Du, K., Liu, J., Zhang, F.: Stochastic continuity of random fields governed by a system of stochastic PDEs. Preprint arXiv:1706.01588 (2017)
 37.Duong, X.T., Yan, L.X.: Bounded holomorphic functional calculus for nondivergence form differential operators. Differ. Integr. Equ. 15(6), 709–730 (2002)MathSciNetzbMATHGoogle Scholar
 38.Egert, M.: \(L^p\)estimates for the square root of elliptic systems with mixed boundary conditions. J. Differ. Equ. 265(4), 1279–1323 (2018)MathSciNetGoogle Scholar
 39.ter Elst, A.F.M., Robinson, D., Sikora, A.: On secondorder periodic elliptic operators in divergence form. Math. Z. 238(3), 569–637 (2001)MathSciNetzbMATHGoogle Scholar
 40.Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetzbMATHGoogle Scholar
 41.Gallarati, C., Veraar, M.C.: Evolution families and maximal regularity for systems of parabolic equations. Adv. Differ. Equ. 22(3–4), 169–190 (2017)MathSciNetzbMATHGoogle Scholar
 42.Gallarati, C., Veraar, M.C.: Maximal regularity for nonautonomous equations with measurable dependence on time. Potential Anal. 46(3), 527–567 (2017)MathSciNetzbMATHGoogle Scholar
 43.Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). (Reprint of the 1998 edition)zbMATHGoogle Scholar
 44.Grafakos, L.: Classical Fourier Analysis, volume 86 of Graduate Texts in Mathematics. Springer, New York (2008)Google Scholar
 45.Grafakos, L.: Modern Fourier Analysis, volume 250 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2009)Google Scholar
 46.Grisvard, P.: Espaces intermédiaires entre espaces de Sobolev avec poids. Ann. Scuola Norm. Sup. Pisa 3(17), 255–296 (1963)zbMATHGoogle Scholar
 47.Haase, M.H.A.: The Functional Calculus for Sectorial Operators, volume 169 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2006)Google Scholar
 48.Hofmann, S., Kenig, C., Mayboroda, S., Pipher, J.: Square function/nontangential maximal function estimates and the Dirichlet problem for nonsymmetric elliptic operators. J. Am. Math. Soc. 28(2), 483–529 (2015)MathSciNetzbMATHGoogle Scholar
 49.Hörmander, L.: Estimates for translation invariant operators in \(L^{p}\) spaces. Acta Math. 104, 93–140 (1960)MathSciNetzbMATHGoogle Scholar
 50.Hytönen, T.P., van Neerven, J.M.A.M., Portal, P.: Conical square function estimates in UMD Banach spaces and applications to \(H^\infty \)functional calculi. J. Anal. Math. 106, 317–351 (2008)MathSciNetzbMATHGoogle Scholar
 51.Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces. Vol. I. Martingales and LittlewoodPaley Theory, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Cham (2016)zbMATHGoogle Scholar
 52.Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces. Vol. II. Probabilistic Methods and Operator Theory, volume 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Cham (2017)Google Scholar
 53.Kalton, N.J., Weis, L.W.: The \(H^\infty \)calculus and sums of closed operators. Math. Ann. 321(2), 319–345 (2001)MathSciNetzbMATHGoogle Scholar
 54.Kalton, N.J., Weis, L.W.: The \(H^{\infty }\)calculus and square function estimates. In: Gesztesy, F., Godefroy, G., Grafakos, L., Verbitsky, I. (eds.) Nigel J. Kalton Selecta, vol. 1, pp. 715–764. Springer (2016)Google Scholar
 55.Kim, K.H.: On \(L_p\)theory of stochastic partial differential equations of divergence form in \(C^1\) domains. Probab. Theory Relat. Fields 130(4), 473–492 (2004)zbMATHGoogle Scholar
 56.Kim, K.H.: On stochastic partial differential equations with variable coefficients in \(C^1\) domains. Stoch. Process. Appl. 112(2), 261–283 (2004)zbMATHGoogle Scholar
 57.Kim, K.H.: \(L_p\) estimates for SPDE with discontinuous coefficients in domains. Electron. J. Probab. 10(1), 1–20 (2005)MathSciNetzbMATHGoogle Scholar
 58.Kim, K.H.: Sobolev space theory of SPDEs with continuous or measurable leading coefficients. Stoch. Process. Appl. 119(1), 16–44 (2009)MathSciNetzbMATHGoogle Scholar
 59.Kim, K.H.: A weighted Sobolev space theory of parabolic stochastic PDEs on nonsmooth domains. J. Theor. Probab. 27(1), 107–136 (2014)MathSciNetzbMATHGoogle Scholar
 60.Kim, I.: A BMO estimate for stochastic singular integral operators and its application to SPDEs. J. Funct. Anal. 269(5), 1289–1309 (2015)MathSciNetzbMATHGoogle Scholar
 61.Kim, I., Kim, K.H.: A regularity theory for quasilinear stochastic PDEs in weighted Sobolev spaces. Stoch. Process. Appl. 128(2), 622–643 (2018)MathSciNetzbMATHGoogle Scholar
 62.Kim, K.H., Lee, K.: A note on \(W_p^\gamma \)theory of linear stochastic parabolic partial differential systems. Stoch. Process. Appl. 123(1), 76–90 (2013)Google Scholar
 63.Kim, I., Kim, K.H., Kim, P.: Parabolic Littlewood–Paley inequality for \(\phi (\Delta )\)type operators and applications to stochastic integrodifferential equations. Adv. Math. 249, 161–203 (2013)MathSciNetzbMATHGoogle Scholar
 64.Koch, H., Tataru, D.: Wellposedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 (2001)MathSciNetzbMATHGoogle Scholar
 65.Köhne, M., Prüss, J., Wilke, M.: On quasilinear parabolic evolution equations in weighted \(L_p\)spaces. J. Evol. Equ. 10(2), 443–463 (2010)MathSciNetzbMATHGoogle Scholar
 66.Krylov, N.V.: A generalization of the Littlewood–Paley inequality and some other results related to stochastic partial differential equations. Ulam Q. 2(4):16 ff., approx. 11 pp. (electronic) (1994)Google Scholar
 67.Krylov, N.V.: On \(L_p\)theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27(2), 313–340 (1996)MathSciNetzbMATHGoogle Scholar
 68.Krylov, N.V.: An analytic approach to SPDEs. In: Carmona, R., Rozovskii, B. (eds.) Stochastic Partial Differential Equations: Six Perspectives, volume 64 of Mathematical Surveys Monographs, pp. 185–242. American Mathematical Society, Providence, RI (1999)Google Scholar
 69.Krylov, N.V.: SPDEs in \(L_q((0,\tau ]\!],L_p)\) spaces. Electron. J. Probab. 5(Paper no. 13), 29 pp. (electronic) (2000)Google Scholar
 70.Krylov, N.V.: On the foundation of the \(L_p\)theory of stochastic partial differential equations. In: Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications—VII, volume 245 of Lecture Notes on Pure Applied Mathematics, pp. 179–191. Chapman & Hall/CRC, Boca Raton, FL (2006)Google Scholar
 71.Krylov, N.V.: A brief overview of the \(L_p\)theory of SPDEs. Theory Stoch. Process. 14(2), 71–78 (2008)MathSciNetzbMATHGoogle Scholar
 72.Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, vol. 96. AMS, Providence, RI (2008)zbMATHGoogle Scholar
 73.Krylov, N.V.: On divergence form SPDEs with VMO coefficients. SIAM J. Math. Anal. 40(6), 2262–2285 (2009)MathSciNetzbMATHGoogle Scholar
 74.Krylov, N.V., Lototsky, S.V.: A Sobolev space theory of SPDEs with constant coefficients in a half space. SIAM J. Math. Anal. 31(1), 19–33 (1999)MathSciNetzbMATHGoogle Scholar
 75.Krylov, N.V., Rozovskiĭ, B.L.: Stochastic evolution equations. In Current Problems in Mathematics, vol. 14 (Russian), pp. 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)Google Scholar
 76.Kunstmann, P.C., Weis, L.: New criteria for the \(H^\infty \)calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. 17(1), 387–409 (2017)MathSciNetzbMATHGoogle Scholar
 77.Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)MathSciNetzbMATHGoogle Scholar
 78.Ladyženskaja, O.A., Solonnikov, V.A., Ural\(^{\prime }\)ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, RI (1968)Google Scholar
 79.Lindemulder, N., Veraar, M.C.: The heat equation with rough boundary conditions and holomorphic functional calculus. Preprint arXiv:1805.10213 (2018)
 80.Lindemulder, N., Meyries, M., Veraar, M.: Complex interpolation with Dirichlet boundary conditions on the half line. Math. Nachr. 291(16), 2435–2456 (2018)MathSciNetzbMATHGoogle Scholar
 81.Lindner, F.: Singular behavior of the solution to the stochastic heat equation on a polygonal domain. Stoch. Partial Differ. Equ. Anal. Comput. 2(2), 146–195 (2014)MathSciNetzbMATHGoogle Scholar
 82.Lions, J.L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer, Berlin (1961)Google Scholar
 83.Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Universitext. Springer, Cham (2015)zbMATHGoogle Scholar
 84.Lorist, E., Veraar, M.C.: Singular stochastic integral operators. In preparation (2019)Google Scholar
 85.McIntosh, A.: Operators which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), volume 14 of Proc. Centre Math. Anal. Austral. Nat. Univ., pp. 210–231. The Australian National University, Canberra (1986)Google Scholar
 86.McIntosh, A., Monniaux, S.: Hodge–Dirac, Hodge–Laplacian and Hodge–Stokes operators in \(L^p\) spaces on Lipschitz domains. Rev. Mat. Iberoam. 34(4), 1711–1753 (2018)MathSciNetzbMATHGoogle Scholar
 87.Meyries, M.: Maximal regularity in weighted spaces, nonlinear boundary conditions, and global attractors. Ph.D. thesis (2010)Google Scholar
 88.Meyries, M., Veraar, M.: Sharp embedding results for spaces of smooth functions with power weights. Studia Math. 208(3), 257–293 (2012)MathSciNetzbMATHGoogle Scholar
 89.Meyries, M., Veraar, M.C.: Traces and embeddings of anisotropic function spaces. Math. Ann. 360(3–4), 571–606 (2014)MathSciNetzbMATHGoogle Scholar
 90.Mikulevicius, R., Rozovskii, B.: A note on Krylov’s \(L_p\)theory for systems of SPDEs. Electron. J. Probab. 6(12), 35 (2001)zbMATHGoogle Scholar
 91.van Neerven, J.M.A.M., Weis, L.W.: Stochastic integration of functions with values in a Banach space. Studia Math. 166(2), 131–170 (2005)MathSciNetzbMATHGoogle Scholar
 92.van Neerven, J.M.A.M., Veraar, M.C., Weis L.W.: Conditions for stochastic integrability in UMD Banach spaces. In: Banach Spaces and Their Applications in Analysis (in honor of Nigel Kalton’s 60th birthday), pp. 127–146. De Gruyter Proceedings in Mathematics, De Gruyter (2007)Google Scholar
 93.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35(4), 1438–1478 (2007)MathSciNetzbMATHGoogle Scholar
 94.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Maximal \(L^p\)regularity for stochastic evolution equations. SIAM J. Math. Anal. 44(3), 1372–1414 (2012)MathSciNetzbMATHGoogle Scholar
 95.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Stochastic maximal \(L^p\)regularity. Ann. Probab. 40(2), 788–812 (2012)MathSciNetzbMATHGoogle Scholar
 96.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Stochastic integration in Banach spaces–a survey. In: Dalang, R.C., Dozzi, M., Flandoli, F., Russo, F. (eds.) Stochastic Analysis: A Series of Lectures, volume 68 of Progress in Probability. Birkhäuser, Basel (2015)zbMATHGoogle Scholar
 97.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Maximal \(\gamma \)regularity. J. Evol. Equ. 15(2), 361–402 (2015)MathSciNetzbMATHGoogle Scholar
 98.van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: On the \(R\)boundedness of stochastic convolution operators. Positivity 19(2), 355–384 (2015)MathSciNetzbMATHGoogle Scholar
 99.Nashed, M.Z., Salehi, H.: Measurability of generalized inverses of random linear operators. SIAM J. Appl. Math. 25, 681–692 (1973)MathSciNetzbMATHGoogle Scholar
 100.Neidhardt, A.L.: Stochastic integrals in 2uniformly smooth Banach spaces. Ph.D. thesis, University of Wisconsin (1978)Google Scholar
 101.Ondreját, M., Veraar, M.C.: On temporal regularity for SPDEs in Besov–Orlicz spaces. Preprint arXiv:1901.01018 (2018)
 102.Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(2), 127–167 (1979)MathSciNetzbMATHGoogle Scholar
 103.Pierre, M.: Global existence in reactiondiffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)MathSciNetzbMATHGoogle Scholar
 104.Pisier, G.: Martingales with values in uniformly convex spaces. Isr. J. Math. 20(3–4), 326–350 (1975)MathSciNetzbMATHGoogle Scholar
 105.Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Probability and analysis (Varenna, 1985), vol. 1206 of Lecture Notes in Mathematics, pp. 167–241. Springer, Berlin (1986)Google Scholar
 106.Portal, P., Štrkalj, Ž.: Pseudodifferential operators on Bochner spaces and an application. Math. Z. 253(4), 805–819 (2006)MathSciNetzbMATHGoogle Scholar
 107.Pronk, M., Veraar, M.C.: Forward integration, convergence and nonadapted pointwise multipliers. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18(1), 1550005 (2015)MathSciNetzbMATHGoogle Scholar
 108.Prüss, J.: Maximal regularity for evolution equations in \(L_p\)spaces. Conf. Semin. Mat. Univ. Bari 285(2003), 1–39 (2002)Google Scholar
 109.Prüss, J., Schnaubelt, R.: Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl. 256(2), 405–430 (2001)MathSciNetzbMATHGoogle Scholar
 110.Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted \(L_p\)spaces. Arch. Math. (Basel) 82(5), 415–431 (2004)MathSciNetzbMATHGoogle Scholar
 111.Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations, volume 105 of Monographs in Mathematics. Birkhhäuser/Springer, Cham (2016)zbMATHGoogle Scholar
 112.Prüss, J., Simonett, G., Wilke, M.: Critical spaces for quasilinear parabolic evolution equations and applications. J. Differ. Equ. 264(3), 2028–2074 (2018)MathSciNetzbMATHGoogle Scholar
 113.Rosiński, J., Suchanecki, Z.: On the space of vectorvalued functions integrable with respect to the white noise. Colloq. Math. 43(1), 183–201 (1981, 1980)MathSciNetzbMATHGoogle Scholar
 114.Rozovskiĭ, B.L.: Stochastic Evolution Systems, volume 35 of Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1990) (Linear theory and applications to nonlinear filtering. Translated from the Russian by A, Yarkho)Google Scholar
 115.Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
 116.Weis, L.W.: Operatorvalued Fourier multiplier theorems and maximal \(L_p\)regularity. Math. Ann. 319(4), 735–758 (2001)MathSciNetzbMATHGoogle Scholar
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