Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

Abstract

In this article we prove pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem

$$\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}$$

(1.1)

Here $A$ is the generator of an analytic $C_0$-semigroup on a umd Banach space $X,\,W_H$ is a cylindrical Brownian motion in a Hilbert space $H$, and the functions $F:[0,T]\times X\rightarrow X_{\theta _F}$ and $G:[0,T]\times X\rightarrow {\fancyscript{L}}(H,X_{\theta _G})$ satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise.

Appendices

Appendix A: Technical lemmas

Here we state and prove with two lemmas which give estimates for the $\gamma$-radonifying norm of stochastic and deterministic integral processes.

Lemma 9.1

Let $q\in [1,\infty ],\,\frac{1}{q}+\frac{1}{q^{\prime }}=1$, and let $(R,{\fancyscript{R}},\mu )$ be a finite measure space and $(S,{\fancyscript{S}},\nu )$ a $\sigma$-finite measure space. Let $Y_1$ and $Y_2$ be Banach spaces, and suppose $\Psi _1\in L^q(R,\gamma (S;Y_1))$ and $\Psi _2\in L^{q^{\prime }}(R,{\fancyscript{L}}(Y_1,Y_2))$ such that $(r,s)\mapsto \Psi _2(r)\Psi _1(r,s)$ defines an element of $L^1(R\times S;Y_2)$. Then:

$$\begin{aligned} \left| \!\left| s\mapsto \int _R \Psi _2(r)\Psi _1(r,s)\,d\mu (r) \right| \!\right| _{\gamma (S;Y_2)}\le |\!|{} \Psi _2 |\!|{} _{L^{q^{\prime }}(S;{\fancyscript{L}}(Y_1,Y_2))} |\!|{} \Psi _1|\!|{} _{L^q(R,\gamma (S;Y_1))}. \end{aligned}$$

Proof

We first consider the case $q\in [1,\infty )$. The $L^1$-assumption guarantees that the integral on the left-hand side exists as a Bochner integral in $Y_2$ for $\nu$-almost all $s\in S$. By (2.3) and the fact that $q<\infty$ we may identify $\Psi _1$ with an element in $\gamma (S;L^q(R;Y_1))$, and by the Hölder inequality $\Psi _2$ induces a bounded operator from $L^q(R;Y_1)$ to $Y_2$. Under these identifications, the expression inside the norm at left-hand side equals the operator $\Psi _2\circ \Psi _1 \in \gamma (S;Y_2)$ and the desired estimate is noting but the right ideal property for the $\gamma$-radonifying norm.

The case $q=\infty$ now follows by an approximation argument. Suppose first $\Psi _1\in L^1\cap L^{\infty }(R,\gamma (S;Y_1))$ and $\Psi _2\in L^1\cap L^{\infty }(R,{\fancyscript{L}}(Y_1,Y_2))$. By the above we have:

$$\begin{aligned}&\left| \!\left| s \mapsto \int _R \Psi _2(r)\Psi _1(r,s)\,d\mu (r)\right| \!\right| _{\gamma (S;Y_2)}\\&\quad \; \le \lim _{q\uparrow \infty ,q^{\prime }\downarrow 1}|\!|{} \Psi _2 |\!|{} _{L^{q^{\prime }}(S;{\fancyscript{L}}(Y_1,Y_2))} |\!|{} \Psi _1|\!|{} _{L^q(R,\gamma (S;Y_1))} \\&\quad \; = |\!|{} \Psi _2 |\!|{} _{L^{\infty }(S;{\fancyscript{L}}(Y_1,Y_2))} |\!|{} \Psi _1|\!|{} _{L^1(R,\gamma (S;Y_1))}. \end{aligned}$$

The result for general $\Psi _1\in L^{\infty }(R,\gamma (S;Y_1))$ and $\Psi _2\in L^{1}(R,{\fancyscript{L}}(Y_1,Y_2))$ follows by approximation. $\square$

The above lemma can be applied to prove the following generalization of [36, Proposition 4.5].

Lemma 9.2

Let $X_1$ and $X_2$ be umd Banach spaces. Let $(R,{\fancyscript{R}},\mu )$ be a finite measure space and $(S,{\fancyscript{S}},\nu )$ a $\sigma$-finite measure space. Let $\Phi _1:[0,T]\times \Omega \rightarrow {\fancyscript{L}}(H,X_1)$, let $\Phi _2\in L^1(R;{\fancyscript{L}}(X_1,X_2))$, and let $f\in L^{\infty }(R\times [0,T];L^2(S))$. If $\Phi _1$ is $L^p$-stochastically integrable for some $p\in (1,\infty )$, then

$$\begin{aligned}&\left| \!\left| s \mapsto \int _{0}^{T} \int _R f(r,u)(s)\Phi _2(r)\Phi _1(u)\, d\mu (r) \,dW_H(u)\right| \!\right| _{L^p(\Omega ;\gamma (S;X_2))}\\&\quad \; \lesssim {\mathop {\mathrm{ess\,sup}}\limits _{(r,u)\in R\times [0,T]}} |\!|{} f(r,u)|\!|{} _{L^2(S)} |\!|{} \Phi _2 |\!|{} _{L^1(R,{\fancyscript{L}}(X_1,X_2))} |\!|{} \Phi _1 |\!|{} _{L^p(\Omega ;\gamma (0,T;H,X_1))}, \end{aligned}$$

with implied depending only on $p,\,X_1,\,X_2$, provided the right-hand side is finite.

Proof

By [29, Corollary 2.17], for almost all $s\in S$ the family $\{T_{s,u}: \ u\in [0,T]\}$ is $\gamma$-bounded in ${\fancyscript{L}}(X_1,X_2)$, where

$$\begin{aligned} T_{s,u}x = \int _R f(r,u)(s)\Phi _2(r)x\, d\mu (r). \end{aligned}$$

Hence, by the $\gamma$-multiplier theorem (Theorem 2.2), for almost all $s\in S$ the function $u\mapsto \int _R f(r,u)(s)\Phi _2(r)\Phi _1(u)\, d\mu (r)$ belongs to $L^p_{\fancyscript{F}}(\Omega ;\gamma (0,T;H,X_2))$.

Moreover, by Theorem 2.2 in combination with Theorem 2.3 (note that umd Banach spaces have non-trivial cotype) we have, for almost all $r\in R$;

$$\begin{aligned} u\mapsto (s\mapsto f(r,u)(s)\Phi _1(u))\in L^p_{\fancyscript{F}}(\Omega ;\gamma (0,T;\gamma (S,X_1))). \end{aligned}$$

By the stochastic Fubini theorem, the isomorphism (2.3) and Lemma 9.1 (with $q=\infty ,\,Y_1= L^p(\Omega ;X_1)$ and $Y_2\!=\!L^p(\Omega ,X_2)$, to $\Psi (r,s)\!=\!\int _{0}^{T} f(r,u)(s)\Phi _1(u)\,dW_H(u)$ and $\Psi _2\!=\!\Phi _2$) we have:

$$\begin{aligned}&\left| \!\left| s \mapsto \int _{0}^{T} \int _R f(r,u)(s)\Phi _2(r)\Phi _1(u)\, d\mu (r) \,dW_H(u)\right| \!\right| _{L^p(\Omega ;\gamma (S;X_2))}\nonumber \\&\quad \;\eqsim \left| \!\left| s\mapsto \int _R \Phi _2(r) \int _{0}^{T} f(r,u)(s)\Phi _1(u)\, \,dW_H(u)d\mu (r)\right| \!\right| _{\gamma (S;L^p(\Omega ;X_2))}\nonumber \\&\quad \; \lesssim |\!|{} \Phi _2 |\!|{} _{L^1(R,{\fancyscript{L}}(X_1,X_2))} \left| \!\left| s\mapsto \int _{0}^{T} f(r,u)(s)\Phi _1(u)\,dW_H(u) \right| \!\right| _{L^{\infty }(R,\gamma (S;L^p(\Omega ;X_1)))}.\nonumber \\ \end{aligned}$$

(9.1)

By isomorphism (2.3), Theorem 2.1, and Theorem 2.2 in combination with Theorem 2.3 we have, for almost all $r\in R$ with implicit constants independent of $r$:

$$\begin{aligned}&\left| \!\left| s \mapsto \int _{0}^{T} f(r,u)(s)\Phi _1(u)\,dW_H(u) \right| \!\right| _{\gamma (S;L^p(\Omega ;X_1))} \\&\quad \; \eqsim |\!|{} u \mapsto (s\mapsto f(r,u)(s)\Phi _1(u)) |\!|{} _{L^p(\Omega ;\gamma (0,T;\gamma (S;X_1)))}\\&\quad \; \le {\mathop {\text{ ess } \text{ sup }}\limits _{u\in [0,T]}} |\!|{} f(r,u)|\!|{} _{L^2(0,T)} |\!|{} \Phi _1 |\!|{} _{L^p(\Omega ;\gamma (0,T;X_1))}. \end{aligned}$$

The result now follows by inserting the above estimate into (9.1). $\square$

We proceed with two lemmas on Besov embeddings. The proof of the first lemma is closely related to the proof of [36, Lemma 3.1].

Lemma 9.3

Suppose $Y$ is a Banach space with type $\tau \in [1,2)$, and let $\alpha \in [0,\frac{1}{2})$ and $q\in (2,\infty )$ satisfy $\frac{1}{q}<\frac{1}{\tau }-\alpha$. Let $\Phi \in B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,T;Y)\cap L^{\infty }(0,T;Y)$ and, for $t\in [0,T]$, define $\Phi _{\alpha ,t}:(0,t) \rightarrow Y$ by

$$\begin{aligned} \Phi _{\alpha ,t}(s)=(t-s)^{-\alpha }\Phi (s). \end{aligned}$$

Then there exists an $\varepsilon _0>0$ such that for all $T_0\in [0,T]$:

$$\begin{aligned} \sup _{0\le t\le T_0} |\!|{} \Phi _{\alpha ,t} |\!|{} _{B_{\tau ,\tau }^{\frac{1}{\tau }-\frac{1}{2}}(0,t;Y)} \lesssim T_0^{\varepsilon _0}|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)\cap B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,T_0;Y)}. \end{aligned}$$

(9.2)

Proof

We shall in fact prove the following stronger result, namely that there exists an $\varepsilon _0>0$ such that for all $T_0\in [0,T]$:

$$\begin{aligned} \sup _{0\le t\le T_0} |\!|{} \Phi _{\alpha ,t} |\!|{} _{B_{\tau ,\tau }^{\frac{1}{\tau }-\frac{1}{2}}(\mathbb{R };Y)} \lesssim T_0^{\varepsilon _0} |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)\cap B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,T_0;Y)}. \end{aligned}$$

(9.3)

On the left-hand side above, we think of $\Phi _{\alpha ,t}$ as being extended identically zero outside the interval $(0,t)$.

Let $q^{\prime }\in (1,\infty )$ be such that $\frac{1}{q}+\frac{1}{q^{\prime }}=\frac{1}{\tau }$. As we assumed $\frac{1}{q}<\frac{1}{\tau }-\alpha$ it follows that $\alpha q^{\prime }<1$. Thus we can pick $\varepsilon >0$ such that $\varepsilon <\min \{\frac{1}{2}-\alpha ,1-\alpha q^{\prime }\}$.

Fix $t\in [0,T_0]$. Let $\rho \in (0,1]$ and let $0<h<\rho$ (we only consider the case $h>0$; the case $h<0$ can be dealt with by observing that $|\!|{} T_h^\mathbb{R }f - f|\!|{} _{L^p(\mathbb{R },Y)}=|\!|{} T_{-h}^\mathbb{R }f - f|\!|{} _{L^p(\mathbb{R },Y)}$). First we consider the case that $h\le t$. In that case we have:

$$\begin{aligned}&|\!|{} T_h^\mathbb{R }(\Phi _{\alpha ,t})-\Phi _{\alpha ,t} |\!|{} _{L^{\tau }(\mathbb{R },Y)} \\&\quad \le |\!|{} s\mapsto [(t-s-h)^{-\alpha }1_{[-h,t-h]}(s) - (t-s)^{-\alpha }1_{[0,t-h]}(s)]\Phi (s+h)|\!|{} _{L^{\tau }(\mathbb{R },Y)}\\&\qquad + |\!|{} s\mapsto (t-s)^{-\alpha }1_{[0,t]}(s)[\Phi (s+h)1_{[0,t-h]}(s)-\Phi (s)]|\!|{} _{L^{\tau }(\mathbb{R },Y)}\\&\quad \le |\!|{} s\mapsto [(t-s-h)^{-\alpha }1_{[-h,t-h]}(s) - (t-s)^{-\alpha }1_{[0,t-h]}(s)]|\!|{} _{L^{\tau }(\mathbb{R })} |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)}\\&\qquad + |\!|{} s\mapsto (t-s)^{-\alpha }1_{[0,t]}(s)|\!|{} _{L^{q^{\prime }}(\mathbb{R },Y)}|\!|{} T_h^\mathbb{R }(\Phi )1_{[0,t-h]}-\Phi |\!|{} _{L^q(0,t;Y)}. \end{aligned}$$

As $\alpha q^{\prime } < 1$ we have:

$$\begin{aligned} |\!|{} s\mapsto (t-s)^{-\alpha }1_{[0,t]}(s)|\!|{} _{L^{q^{\prime }}(\mathbb{R },Y)} \lesssim T^{\frac{1}{q^{\prime }}-\alpha }. \end{aligned}$$

For $p\ge 1$ and $0\le b\le a$ one has $(a-b)^p\le a^p-b^p$ and thus:

$$\begin{aligned}&|\!|{} s\mapsto [(t-s-h)^{-\alpha }1_{[-h,t-h]}(s) - (t-s)^{-\alpha }1_{[0,t-h]}(s)]|\!|{} _{L^{\tau }(\mathbb{R })}\\&\quad =\left( \int _{-h}^{t-h} |(t-s-h)^{-\alpha } - (t-s)^{-\alpha }1_{[0,t-h]}(s)|^{\tau }\,ds \right) ^{\frac{1}{\tau }} \\&\quad \le \left( \int _{-h}^{t-h} [(t-s-h)^{-\alpha \tau } -(t-s)^{-\alpha \tau }1_{[0,t-h]}(s)] \,ds \right) ^{\frac{1}{\tau }}\\&\quad = (1-\alpha \tau )^{-\frac{1}{\tau }} h^{\frac{1}{\tau }-\alpha } \lesssim h^{\frac{1}{\tau }-\alpha -\varepsilon }T^{\varepsilon }, \end{aligned}$$

where the last inequality uses $h\le t\le T_0$.

Putting together these estimates,

$$\begin{aligned}&|\!|{} T_h^\mathbb{R }(\Phi _{\alpha ,t})-\Phi _{\alpha ,t} |\!|{} _{L^{\tau }(\mathbb{R },Y)}\\&\quad \le h^{\frac{1}{\tau }-\alpha -\varepsilon }T_0^{\varepsilon } |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)} + T_0^{\frac{1}{q^{\prime }}-\alpha }|\!|{} T_h^\mathbb{R }(\Phi )1_{[0,t-h]}-\Phi |\!|{} _{L^q(0,t;Y)}.\\&\quad = h^{\frac{1}{\tau }-\alpha -\varepsilon }T_0^{\varepsilon } |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)} + T_0^{\frac{1}{q^{\prime }}-\alpha }|\!|{} T_h^I(\Phi )-\Phi |\!|{} _{L^q(0,t;Y)}. \end{aligned}$$

Next suppose $h>t$. In that case:

$$\begin{aligned} |\!|{} T_h^\mathbb{R }(\Phi _{\alpha ,t})-\Phi _{\alpha ,t} |\!|{} _{L^{\tau }(\mathbb{R },Y)}&= 2|\!|{} \Phi _{\alpha ,t} |\!|{} _{L^{\tau }(\mathbb{R },Y)}\\&\lesssim t^{\frac{1}{\tau }-\alpha }|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)} \le h^{\frac{1}{\tau }-\alpha -\varepsilon }T_0^{\varepsilon }|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)}, \end{aligned}$$

this time using $t^{\frac{1}{\tau }-\alpha }\le t^{\frac{1}{\tau }-\alpha -\varepsilon }T_0^\varepsilon$ and $t\le h$.

It follows that

$$\begin{aligned} |\!|{} \Phi _{\alpha ,t} |\!|{} _{B_{\tau ,\tau }^{\frac{1}{\tau }\!-\!\frac{1}{2}}(\mathbb{R },Y)}\!\!&= \!\! |\!|{} \Phi _{\alpha ,t}|\!|{} _{L^{\tau }(\mathbb{R },Y)} \!+\! \left( \int _{0}^{1}\rho ^{-1+\frac{\tau }{2}}\sup _{|h|<\rho }|\!|{} T_h^\mathbb{R }(\Phi _{\alpha ,t})\!-\!\Phi _{\alpha ,t} |\!|{} _{L^{\tau }(\mathbb{R },Y)}^{\tau }\frac{d\rho }{\rho }\right) ^{\frac{1}{\tau }}\\&\lesssim T_0^{\frac{1}{\tau }-\alpha } |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)} +T_0^{\frac{1}{q^{\prime }}-\alpha }\\&\times \left( \int _{0}^{1}\rho ^{-1+\frac{\tau }{2}}\sup _{|h|<\rho }|\!|{} T_h^I(\Phi )-\Phi |\!|{} _{L^{q}(0,t;Y)}^{\tau }\frac{d\rho }{\rho }\right) ^{\frac{1}{\tau }}\\&+ T_0^{\varepsilon } \left( \int _{0}^{1} \rho ^{\tau (\frac{1}{2} -\alpha -\varepsilon )} \frac{d\rho }{\rho }\right) ^{\frac{1}{\tau }}|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)}\\&\lesssim (T_0^{\frac{1}{\tau }-\alpha }\vee T_0^{\frac{1}{q^{\prime }}-\alpha } \vee T_0^\varepsilon )\left( |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;Y)} + |\!|{} \Phi |\!|{} _{B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,t;Y)}\right) . \end{aligned}$$

This gives the result, noting that $T_0^{\alpha }\vee T_0^{\beta } \le T_0^{\min \{\alpha ,\beta \}}(1\vee T^{|\alpha -\beta |})$ for all $T_0\in [0,T]$.

$\square$

This lemma will be used to deduce the following estimate.

Lemma 9.4

Let $X$ be a umd Banach space, $H$ a Hilbert space, and suppose $G:[0,T]\times \Omega \rightarrow {\fancyscript{L}}(H,X)$ satisfies $(\mathbf{G}^{\prime })$ of Sect. 5. For all $0\le \alpha <\frac{1}{2}$ and $\varepsilon >0$ there for any $T>0$ there exists a constant $C>0$ such that for any $n\in \mathbb{N }$, and any sequence $(B_j)_{j=0}^{n}$ in $L^p(\Omega ;X),\,p\in [2,\infty )$, we have:

$$\begin{aligned} \sup _{0\le t \le T} |\!|{} s&\mapsto (t-s)^{-\alpha }\left[ G(s,B_{\underline{s}n/T}))-G(\underline{s},B_{\underline{s}n/T}))\right] |\!|{} _{L^p(\Omega ;\gamma (0, t;H,X_{\theta _G}))} \\&\le Cn^{-\zeta _{\max }+\varepsilon }\left( 1+\sup _{0\le j\le n}|\!|{} B_j |\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

Proof

Without loss of generality we may assume $\varepsilon < \frac{1}{2}-\alpha$. Set $q=(\frac{1}{\tau }-\frac{1}{2}+\varepsilon )^{-1}$, so that $\frac{1}{\tau }-\frac{1}{2}<\frac{1}{q}<\frac{1}{\tau }-\alpha$. For $s\in [0,T)$ define

$$\begin{aligned} \Phi (s):= G(s,B_{\underline{s}n/T}))-G(\underline{s},B_{\underline{s}n/T}). \end{aligned}$$

Note that as $p\ge 2$, the type of $L^p(\Omega ,X_{\theta _G})$ is the same as the type of $X$. By embedding (2.6), Lemma 9.3 and isomorphism (2.3) we have:

$$\begin{aligned}&\sup _{0\le t \le T} |\!|{} s\mapsto (t-s)^{-\alpha }\Phi (s)|\!|{} _{\gamma (0,t;H,L^p(\Omega ;X_{\theta _G}))} \nonumber \\&\quad \lesssim |\!|{} \Phi |\!|{} _{B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))} + |\!|{} \Phi |\!|{} _{L^{\infty }(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))}. \end{aligned}$$

(9.4)

By the Hölder assumption of $(\mathbf{G}^{\prime })$ we have:

$$\begin{aligned} |\!|{} \Phi |\!|{} _{L^{\infty }(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))}&\lesssim n^{-\zeta _{\max }-\frac{1}{\tau }+\frac{1}{2}} \left( 1 + \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^{p}(\Omega ;X)}\right) \nonumber \\&= n^{-\zeta _{\max }-\frac{1}{q}+\varepsilon }\left( 1 + \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^{p}(\Omega ;X)}\right) .\qquad \qquad \end{aligned}$$

(9.5)

In order to estimate the Besov norm on the right-hand side of (9.4) we fix $\rho \in (0,1)$, and let $|h|< \rho$. We have, with $I = [0,T]$,

$$\begin{aligned} \begin{aligned}&|\!|{} T_h^I \Phi (s) - \Phi (s) |\!|{} _{L^p(\Omega ;\gamma (H,\theta _G))}\\&\le \left\{ \begin{aligned}&|\!|{} G(s+h,B_{\underline{s}n/T}) - G(s,B_{\underline{s}n/T})|\!|{} _{L^p(\Omega ;\gamma (H,X_{\theta _G}))},&\underline{s+h}=\underline{s}, \ s+h\in [0,T],\\&2|\!|{} \Phi |\!|{} _{L^{\infty }(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))},&\mathrm{otherwise}. \end{aligned}\right. \end{aligned} \end{aligned}$$

For $|h|\ge \frac{T}{n}$ one never has $\underline{s+h}=\underline{s}$ and thus it follows from the above and (9.5) that

$$\begin{aligned} |\!|{} T_h^I\Phi -\Phi |\!|{} _{L^q(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))}&\lesssim n^{-\zeta _{\max }-\frac{1}{q}+\varepsilon }\left( 1 + \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) \\&\lesssim |h|^{\frac{1}{q}}n^{-\zeta _{\max }+\varepsilon }\left( 1 + \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

On the other hand, for $h< \frac{T}{n}$ and $\underline{s+h} = \underline{s}$ we obtain, by $(\mathbf{G}^{\prime })$:

$$\begin{aligned}&|\!|{} G(s+h,B_{\underline{s+h}n/T}) - G(s,B_{\underline{s}n/T})|\!|{} _{L^p(\Omega ;\gamma (H,X_{\theta _G}))} \\&\quad \lesssim |h|^{\zeta _\mathrm{max} + \frac{1}{\tau }-\frac{1}{2}} (1+ |\!|{} B_{\underline{s}n/T}|\!|{} _{L^p(\Omega ;X)}) \\&\quad \le |h|^{\frac{1}{q}}\left( \tfrac{T}{n}\right) ^{\zeta _\mathrm{max} + \frac{1}{\tau }-\frac{1}{2}-\frac{1}{q}} \left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) \\&\quad \lesssim |h|^{\frac{1}{q}} n^{-\zeta _\mathrm{max} + \varepsilon }\left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

For $|h|<\frac{T}{n}$ observe that $|\{ s\in [0,T]: \underline{s+h} \ne \underline{s}\}|=n|h|$. Thus for $|h|<\frac{T}{n}$ we have, by the above estimate and (9.5):

$$\begin{aligned}&|\!|{} T_h^I\Phi -\Phi |\!|{} _{L^q(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G}))} \\&\quad \lesssim (T-n|h|)^{\frac{1}{q}}|h|^{\frac{1}{q}} n^{-z\eta _\mathrm{max} + \varepsilon } \left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) \\&\qquad +(n|h|)^{\frac{1}{q}}n^{-\zeta _{\max }-\frac{1}{q}+\varepsilon } \left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) \\&\quad \lesssim |h|^{\frac{1}{q}} n^{-\zeta _\mathrm{max} + \varepsilon } \left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

Collecting these estimates we find:

$$\begin{aligned} \sup _{|h|< \rho } |\!|{} T_h^I\Phi -\Phi |\!|{} _{L^q(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G}))} \lesssim \rho ^{\frac{1}{q}}n^{-\zeta _{\max }+\varepsilon } \left( 1 + \sup _{0\le j\le n}|\!|{} B_j |\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

Because $\frac{1}{q}>\frac{1}{\tau }-\frac{1}{2}$ it follows that

$$\begin{aligned}&|\!|{} \Phi |\!|{} _{B^{\frac{1}{\tau }-\frac{1}{2}}_{q,\tau }(0,T;L^p(\Omega ;\gamma (H,X_{\theta _G})))}\\&\quad \quad \lesssim |\!|{} \Phi |\!|{} _{L^q(0,T;\gamma (H,L^p(\Omega ;X_{\theta _G})))} + n^{-z\eta _{\max }+\varepsilon } \left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) \\&\quad \quad \lesssim n^{-z\eta _{\max }+\varepsilon }\left( 1+ \sup _{0\le j\le n}|\!|{} B_j|\!|{} _{L^p(\Omega ;X)}\right) . \end{aligned}$$

Inserting the above and (9.5) into (9.4) gives the required result. $\square$

The final lemma is an elementary calculus fact.

Lemma 9.5

For all $0\le \delta ,\theta <\frac{1}{2}$ there exists a constant $C$, depending only on $\delta$ and $\theta$, such that for all $0\le u \le t$, all $T>0$ and all $n\in \mathbb{N }$:

$$\begin{aligned} \int _{\underline{u}}^{t} (t-s)^{-2\theta }(\overline{s}-\underline{u})^{-2\delta } \,ds \le C^2 \left( t+\tfrac{T}{n}-u\right) ^{1-2\delta -2\theta }. \end{aligned}$$

Proof

If $t-\underline{u}\le \tfrac{T}{n}$, then for $s\in [\underline{u},t)$ one has $\overline{s}-\underline{u}=\overline{u}-\underline{u}=\tfrac{T}{n}$ so

$$\begin{aligned} \int _{\underline{u}}^{t} (t-s)^{-2\theta }(\overline{s}-\underline{u})^{-2\delta }\,ds = (1-2\theta )^{-1} \left( \tfrac{T}{n}\right) ^{-2\delta }(t-\underline{u})^{1-2\theta }. \end{aligned}$$

Note that $\tfrac{T}{n}\ge \frac{1}{2}\left( t+\tfrac{T}{n}-u\right)$ and $(t-\underline{u})^{1-2\theta }\le \left( t+\tfrac{T}{n}-u\right) ^{1-2\theta }$. Thus:

$$\begin{aligned} \int _{\underline{u}}^{t} (t-s)^{-2\theta }(\overline{s}-\underline{u})^{-2\delta }\,ds \le 2^{2\delta } (1-2\theta )^{-1} \left( t+\tfrac{T}{n}-u\right) ^{1-2\delta -2\theta }. \end{aligned}$$

On the other hand if $t-\underline{u}>\frac{T}{n}$ then ${t-\underline{u}} < {t+T/n-u} < 2({t-\underline{u}})$. Moreover, $\overline{s}-\underline{u} \ge s-\underline{u},$ and the substitution $v={(s-\underline{u})}/{(t-\underline{u})}$ gives:

$$\begin{aligned} \int _{\underline{u}}^{t} (t-s)^{-2\theta }(\overline{s}-\underline{u})^{-2\delta }\,ds&\le \int _{\underline{u}}^{t}(t-s)^{-2\theta }(s-\underline{u})^{-2\delta }\,ds\\&\le (t-\underline{u})^{1-2\delta -2\theta } \int _{0}^{1} (1-v)^{-2\theta }v^{-2\delta }\,dv \\&\le 2^{(2\delta +2\theta -1)^+}\left( t+\tfrac{T}{n}-u\right) ^{1-2\delta -2\theta }\int _{0}^{1} (1-v)^{-2\theta }v^{-2\delta }\,dv. \end{aligned}$$

$\square$

Appendix B: Estimates for stochastic convolutions

We shall present two estimates for stochastic convolutions. Throughout this section, $Y$ is a umd Banach space and $\tau \in (1,2]$ denotes its type. Moreover, $S$ is an analytic semigroup on $Y$.

Roughly speaking, Lemma 10.1 is contained in Step 2 of the proof of [36, Proposition 6.1], but there the space $V^{\alpha ,p}_\mathrm{c}([0,T]\times \Omega ;X)$ is considered [see (2.16)]. For completeness we give the proof below.

Lemma 10.1

Let $\delta \in (-\frac{3}{2}+\frac{1}{\tau },\infty ),\,\alpha \in [0,\frac{1}{2})$, and $p\in [2,\infty )$. For all $\Phi \in L^{\infty }(0,T;L^p(\Omega ;Y_{\delta }))$, the convolution $S*\Phi$ belongs to ${\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;Y)$, and for all $T_0\in [0,T]$ we have:

$$\begin{aligned} |\!|{} S*\Phi |\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;Y)} \lesssim (T_0^{1+(\delta \wedge 0)}+T_0^{\frac{1}{2}-\alpha })|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;L^p(\Omega ;Y_{\delta }))}. \end{aligned}$$

Proof

By analyticity of the semigroup [Eq. (2.7)] we have, for $t\in [0,T_0]$:

$$\begin{aligned} |\!|{} (S*\Phi )(t) |\!|{} _{L^p(\Omega ;Y)}&\lesssim \int _{0}^{t} (t-s)^{\delta \wedge 0}\,ds |\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;L^p(\Omega ;Y_{\delta }))}\\&\le T_0^{1+(\delta \wedge 0)}|\!|{} \Phi |\!|{} _{L^{\infty }(0,T_0;L^p(\Omega ;Y_{\delta }))}. \end{aligned}$$

Taking the supremum over $t\in [0,T_0]$ gives the estimate in $L^{\infty }(0,T_0;L^p(\Omega ,Y))$.

It remains to prove the estimate in the weighted $\gamma$-norm. Fix $t\in [0,T_0]$. As $p\ge 2$, it follows that $L^p(\Omega ,Y)$ has type $\tau \in [1,2]$ whenever $Y$ has type $\tau$. Moreover, if we interpret $A$ as an operator on $L^p(\Omega ,Y)$ acting pointwise, then $(L^p(\Omega ,Y))_{\delta }=L^p(\Omega ,Y_{\delta })$. Thus by [36, Proposition 3.5] with $E=L^p(\Omega ,Y),\,\eta =0$, and $\theta =-\delta$ we have, as $\delta >-\frac{3}{2}+\frac{1}{\tau }$;

$$\begin{aligned} |\!|{} s\mapsto (t-s)^{-\alpha }(S*\Phi )(s)|\!|{} _{\gamma (0,t;L^p(\Omega ,Y))} \lesssim T_0^{\frac{1}{2}-\alpha }|\!|{} \Phi |\!|{} _{L^\infty (0,T_0;L^p(\Omega ;Y_{\delta }))}. \end{aligned}$$

Taking the supremum over $t\in [0,T_0]$ gives the desired estimate. $\square$

We proceed with the second Lemma.

Lemma 10.2

Let $\delta \in (-\frac{1}{2},\infty )$ and $\alpha \in [0,\frac{1}{2})$. Suppose $\Phi : [0,T]\times \Omega \rightarrow {\fancyscript{L}}(H,Y_\delta )$ is strongly measurable and adapted and satisfies

$$\begin{aligned} \sup _{0\le t\le T}|\!|{} s\mapsto (t-s)^{-\alpha } \Phi (s) |\!|{} _{L^p(\Omega ;\gamma (0,T;H,Y_{\delta }))}<\infty , \end{aligned}$$

(10.1)

for some $p\in (1,\infty )$.

• (i) If $0\le \beta < \min \{\frac{1}{2}-\alpha ,\frac{1}{2}+\delta \}$, then there exists an $\epsilon >0$ such that for all $T_0\in [0,T]$:

  $$\begin{aligned}&\sup _{0\le t\le T_0}\left| \!\left| s \mapsto (t-s)^{-\alpha -\beta } \int _0^{s} S(s-u)\Phi (u)\, dW_H(s)\right| \!\right| _{L^p(\Omega ;\gamma (0,t;Y))}\\&\lesssim T_0^{\epsilon }\sup _{0\le t\le T_0}|\!|{} s\mapsto (t-s)^{-\alpha } \Phi (s) |\!|{} _{L^p(\Omega ;\gamma (0,t;H,Y_{\delta }))}. \end{aligned}$$

• (ii) If, moreover, $\alpha >-\delta$, then there exists an $\epsilon >0$ such that for all $T_0\in [0,T]$:

  $$\begin{aligned}&\left| \!\left| s \mapsto \int _{0}^{s} S(s-u)\Phi (u)\,dW_H(u) \right| \!\right| _{{\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }([0,T_0]\times \Omega ;Y)} \\&\quad \; \lesssim T_0^{\epsilon }\sup _{0\le t\le T_0}|\!|{} s\mapsto (t-s)^{-\alpha } \Phi (s) |\!|{} _{L^p(\Omega ;\gamma (0,t;H,Y_{\delta }))}. \end{aligned}$$

Proof

Fix $t\in [0,T_0]$. Let $\epsilon >0$ be such that $\epsilon <\frac{1}{2}-\delta ^- -\beta$. Here $\delta ^-= (-\delta )\vee 0$. We apply Lemma 9.2 with $X_1=Y_\delta ,\,X_2=Y,\,R=S= [0,t]$, and the functions $\Phi _1(u)=(t-u)^{-\alpha }\Phi (u),\,\Phi _2(r) = \frac{d}{dr}[r^{\delta ^{-}+\epsilon }S(r)]$, and $f(r,u)(s)= (t-s)^{-\alpha -\beta }(s-u)^{-\delta ^- -\epsilon }(t-u)^{\alpha }1_{0\le r\le s-u}$. By (2.7) we have $|\!|{} \Phi _2(r)|\!|{} _{{\fancyscript{L}}(X_{\delta },X)} \lesssim r^{-1+\varepsilon }$ for $r\in [0,T]$. From the lemma it follows that:

$$\begin{aligned}&\left| \!\left| s \mapsto (t-s)^{-\alpha -\beta }\int _{0}^{s} S(s-u)\Phi (u)\,dW_H(u)\right| \!\right| _{L^p(\Omega ;\gamma (0, t;Y))} \\&\quad \; \lesssim t^{\frac{1}{2}-\beta -\delta ^-}|\!|{} s\mapsto (t-s)^{-\alpha }\Phi (s)|\!|{} _{L^p(\Omega ;\gamma (0,t;H,Y_\delta ))}. \end{aligned}$$

Taking the supremum over $t\in [0,T_0]$ we obtain (i).

For the estimate in ${\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }$-norm it remains, by part (i), to prove the estimate in $L^{\infty }(0,T_0;L^p(\Omega ,Y_{\delta }))$. Let $\epsilon < \min \{\alpha +\delta ,\frac{1}{2}-\delta ^{-}-\beta \}$. By Lemma 2.4 (apply part (1) if $\delta \in (-\frac{1}{2}, 0]$ and part (2) if $\delta \in [0,\infty )$) the operators $r^\alpha S(r),\,r\in [0,t]$, are $\gamma$-bounded from $Y_\delta$ to $Y$, with $\gamma$-bound at most $C t^{\alpha +\delta }$ with $C$ independent of $t\in [0,T]$. Hence, by the $\gamma$-multiplier theorem, for all $t\in [0,T]$,

$$\begin{aligned} \left| \!\left| \int _{0}^{t} S(t-s)\Phi (s)\,dW_H(s)\right| \!\right| _{L^p(\Omega ;Y)} \lesssim t^{\alpha +\delta }|\!|{} s\mapsto (t-s)^{-\alpha } \Phi (s) |\!|{} _{L^p(\Omega ;\gamma (0,t;H,Y_{\delta }))}. \end{aligned}$$

The norm estimate in $L^{\infty }(0,T;L^p(\Omega ;Y_{\delta }))$ is obtained by taking the supremum over $t\in [0,T]$. $\square$

Appendix C: Existence and uniqueness

The aim of this section is to outline the proof of Theorem 2.7. The setting is always that of Sect. 2.

Proof of Theorem 2.7

Assume first that $\alpha \in [0,\frac{1}{2})$ is so large that $\alpha +\theta _G>\eta$. Let $p\in [2,\infty )$ and $T_0\in [0,T]$ be fixed. For $\Phi \in {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )$ define

$$\begin{aligned} L(\Phi )(t):= S(t)x_0 + \int _{0}^{t}S(t-s)F(s,\Phi (s))\,ds + \int _{0}^{t}S(t-s)G(s,\Phi (s))\,dW_H(s). \end{aligned}$$

Copying Step 1 of the proof of [36, Proposition 6.1] without changes, and substituting Steps 2 and 3 by the Lemmas 10.1 and 10.2 above, we find that there exists an $\varepsilon _0>0$ and a $C>0$ such that $L:{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta ) \rightarrow {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )$ and

$$\begin{aligned} |\!|{} L(\Phi )|\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )}&\le C|\!|{} x_0 |\!|{} _{X} + CT^{\varepsilon _0}|\!|{} F(\cdot ,\Phi (\cdot ))|\!|{} _{L^{\infty }(0,T_0;L^p(\Omega ;X_{\theta _F}))}\\&+ CT^{\varepsilon _0}\sup _{0\le t\le T_0}|\!|{} s\mapsto (t-s)^{-\alpha } G(s,\Phi (s))|\!|{} _{L^p(\Omega ;\gamma (0,t;X_{\theta _G}))}\\&\le C|\!|{} x_0 |\!|{} _{X} \\&+ C (M(F) + M(G)) T_0^{\varepsilon _0}(1+|\!|{} \Phi |\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )}), \end{aligned}$$

where in the last line we used (F) and (2.12). Moreover,

$$\begin{aligned}&|\!|{} L(\Phi _1)-L(\Phi _2)|\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )}\\&\quad \le C (\mathrm{Lip}(F) + \mathrm{Lip}_{\gamma }(G)) T_0^{\varepsilon _0}|\!|{} \Phi _1 - \Phi _2 |\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )}, \end{aligned}$$

where again we used (F) and (2.11).

Thus by a fixed-point argument, for sufficiently small $T_0$ there exists a unique process $\Phi \in {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T_0]\times \Omega ;X_\eta )$ satisfying (2.13) on the interval $[0,T_0]$. By repeating this construction a finite number of times, each time taking the final value of the previous step as the initial value of the next, we obtain a solution on $[0,T]$.

So far, we have proved existence and uniqueness under the additional assumption $\alpha +\theta _G>\eta$. Existence in ${\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X_\eta )$ for arbitrary $\alpha \in [0,\frac{1}{2})$ follows from (2.9). It remains to prove uniqueness for arbitrary $\alpha \in [0,\frac{1}{2})$.

Let $\alpha \in [0,\frac{1}{2})$ and let $\Phi \in {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X_{\eta })$. Viewing $F$ as a mapping from $[0,T]\times X_\eta$ to $X_{\theta _F}$ (as $\eta \ge 0$), we have $F(\cdot ,\Phi (\cdot )) \in {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X_{\theta _F})$. Then, by Lemma 10.1 with $\delta =\theta _F-\eta$ and $Y = X_\eta$ (and $\tilde{\alpha }= \alpha +\beta$ ), we find $S*F(\cdot ,\Phi (\cdot ))\in {\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }([0,T]\times \Omega ;X_{\eta })$ for all $\beta \in [0,\frac{1}{2}-\alpha )$.

By part (i) in Lemma 10.2 (with $Y=X_{\eta }$ and $\delta = \theta _G-\eta$) and (2.12) we have, for all $\beta \in [0,\frac{1}{2}-\alpha )$ such that $\beta <\frac{1}{2}+\theta _G-\eta$:

$$\begin{aligned}&\sup _{0\le t\le T} \left| \!\left| s \mapsto (t-s)^{-\alpha -\beta } \int _{0}^{s} S(s-u)G(u,\Phi (u))\,dW_H(u)\right| \!\right| _{L^p(\Omega ,\gamma (0,t;X_\eta ))}\\&\quad \lesssim \sup _{0\le t\le T} |\!|{} s\mapsto (t-s)^{-\alpha } G(s,\Phi (s))|\!|{} _{L^p(\Omega ;\gamma (0,t;H,X_{\theta _G}))}\\&\quad \le |\!|{} \Phi |\!|{} _{{\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X_\eta )}. \end{aligned}$$

Since also $S(\cdot )x_0 \in {\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }([0,T]\times \Omega ;X_\eta )$ for all $\beta \in [0,\frac{1}{2}-\alpha )$, we see that if $\alpha \in [0,\frac{1}{2})$ and $\Phi \in {\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X_{\eta })$ satisfies (2.13), then $\Phi \in {\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }([0,T]\times \Omega ;X)$ for all $\beta \in [0,\frac{1}{2}-\alpha )$ such that $\beta <\frac{1}{2}+\theta _G-\eta$. Repeating this argument a finite number of steps if necessary, we obtain that $\Phi \in {\fancyscript{V}}^{\alpha +\beta ,p}_{\infty }([0,T]\times \Omega ;X)$ for all $\beta \in [0,\frac{1}{2}-\alpha )$. As uniqueness of a process in ${\fancyscript{V}}^{\alpha ,p}_{\infty }([0,T]\times \Omega ;X)$ satisfying (2.13) has been established for $\alpha >\eta -\theta _G$ this completes the proof. $\square$

Remark 11.1

Inspection of the proofs of the main theorems reveals that uniqueness is only used for large $\alpha \in [0,\frac{1}{2})$. As a consequence, the last part of the above proof is not needed for our purposes. It has been included for completeness reasons.