Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems

We are concerned with the convergence of the splitting scheme for the stochastic linear Cauchy problem

were $A$ is the generator of a $C_{0}$-semigroup $S=\{S(t)\}_{t\ge 0}$ on a real Banach space $E$, $W=\{W(t)\}_{t\ge 0}$ is an $E$-valued Brownian motion on a probability space $(\Omega,{\mathbb{P}})$, and $x\in E$ is an initial value which is kept fixed throughout the paper. The concept of the scheme is to alternately add an increment of the Brownian motion $W$ and run the semigroup $S$ on a time interval of equal length. Taking time steps $\Delta t^{(n)}:=T/n$ and writing $t_{j}^{(n)}:=jT/n$ and $\Delta W_{j}^{(n)}:=W(t_{j}^{(n)})-W(t_{j-1}^{(n)})$, this generates a finite sequence $\{U_{x}^{(n)}(t_{j}^{(n)})\}_{j=0}^{n}$ defined by

We have the explicit formula

Assuming the existence of a unique solution $U_{x}$ of the problem (SCP_x) (see Proposition 7 below), we may ask for conditions ensuring the convergence of $U_{x}^{(n)}(T)$ to $U_{x}(T)$ in $L^{p}(\Omega;E)$ for some (all) $1\le p<\infty$ or even in $E$ almost surely. In order to describe our approach we start by noting that each $U_{x}^{(n)}(t_{j}^{(n)})$ can be represented as a stochastic integral of the discretised function

where $I_{0}^{(n)}=\{0\}$ and $I_{j}^{(n)}=(t_{j-1}^{(n)},t_{j}^{(n)}]$ for $j=1,\dots,n$. Indeed, defining the stochastic integral of a step function in the obvious way, we have

On the other hand, the exact solution of (SCP_x), if it exists, is given by the stochastic convolution integral

For the precise definition of the stochastic integral we refer to Section 3. Comparing (1) and (2) we see that the problem of convergence of the splitting scheme is really a problem of convergence of ‘Riemann sums’ for stochastic integrals. Let us henceforth put

The second formula interpolates the data in the identity (1) in a way that makes them easily accessible with continuous time techniques; other possible interpolations, such as piecewise linear interpolation, do not have this advantage. Needless to say, in Theorems 1 and 2 below we are primarily interested in what happens at the time points $t=t_{j}^{(n)}$. From $S^{(n)}(t_{j}^{(n)})x=S(t_{j}^{(n)})x$ we see that

for all $x\in E$, and therefore it suffices to analyse convergence of the splitting scheme with initial value $0$. In what follows, in order to simplify notations we shall write $U(t):=U_{0}(t)$ and $U^{(n)}(t):=U_{0}^{(n)}(t)$.

Our first result extends and simplifies previous work by Kühnemund and the second-named author [21, Theorems 4.3 and 5.2].

The class of spaces satisfying condition (a) includes all Hilbert spaces and the spaces $L^{p}(\mu)$ for $2\le p<\infty$. It follows from the results in [26] that condition (b) is satisfied if the transition semigroup associated with the solution process is analytic.

The main result of this article, Theorem 2 below, concerns actual convergence rates for the splitting scheme in the case that the semigroup $S$ is analytic on $E$. The convergence is considered in suitable Hölder norms in space and time, with explicit bounds for the convergence rate.

We denote by $E_{\alpha}$ the fractional power space of exponent $\alpha\ge 0$ associated with $A$ (see Section 4 for more details).

By a Borel-Cantelli argument, this result implies the almost sure convergence of $U^{(n)}$ to $U$ in $C^{\gamma}([0,T];E_{\alpha})$ with the same rates.

The proof of Theorem 2 heavily relies on the theory of $\gamma$-radonifying operators and $\gamma$-boundedness techniques. Standard techniques from stochastic analysis which are commonly used in connection with the problems considered here, such as Itô’s formula and the Burkholder-Davis-Gundy inequalities, are unavailable in the present general framework (unless one makes additional assumptions on $E$, such as martingale type $2$ or the UMD property). We also cannot use factorisation techniques (as introduced by Da Prato, Kwapień and Zabczyk [7]), the reason being that the semigroup property on which this technique relies fails for the discretised semigroup $S^{(n)}$.

For semi-linear (Stratonovich type) SPDEs governed by second order elliptic operators on ${\mathbb{R}}^{d}$ and driven by multiplicative noise, convergence in $E=L^{2}({\mathbb{R}}^{d})$ of splitting schemes like the one considered here has been proved by various authors [2, 3, 12, 14, 27]. Using techniques from PDE and stochastic analysis it is shown by Gyöngy and Krylov [14] that, with respect to the norm of $E=L^{2}({\mathbb{R}}^{d})$, for finite-dimensional noise and with sufficiently smooth coefficients one obtains the maximal estimate

Our result is valid in the full scale of spaces $L^{q}({\mathbb{R}}^{d})$ and infinite-dimensional noise, with a rate which (for smooth enough noise) is only slightly worse that $1/n$ and is independent of $1\le q<\infty$. More precisely, for $\beta\ge\frac{1}{2}+\alpha$ and taking $\gamma=0$ we obtain uniform convergence with rate $1/n^{\theta}$ for any $0\le\theta<1$. In addition to that we obtain Hölder regularity in both space and time. On the other hand, as we already mentioned, Gyöngy and Krylov [14] consider the semi-linear case and multiplicative noise.

The next example shows that by working in suitable fractional extrapolation spaces (this technique is explained in [10]; see also [4, 5]), the assumption that $W$ is a Brownian motion can be weakened to $W$ being a cylindrical Brownian motion (see, e.g., [30, 32] for the definition).

It is shown in [8] that any approximation scheme for a one-dimensional stochastic heat equation with additive space-time white noise which incorporates the contributions of the noise only by means of the terms $\Delta W_{k}$, $k=1,\ldots,n$, cannot have a convergence rate better than $1/n^{\frac{1}{4}}$. This shows that the exponent $\frac{1}{4}$ in Example 4 is the best possible.

The field of numerical approximation of stochastic partial differential equations (SPDEs) is a very active one; an up-to-date overview of the available results can be found in [19]. In [13] convergence rates are considered for various approximations schemes in space and time of a quasi-linear parabolic SPDE driven by white noise. The authors obtain a convergence rate $1/n^{\frac{1}{4}}$ in $L^{p}$ for an implicit Euler scheme. In [36] convergence in probability is proved (without rates) for the same SPDE with state-dependent dispersion. Rates for path-wise convergence are given for quasi-linear parabolic SPDEs in [15, 18, 24], albeit only for colored noise. It seems likely that the methods of this paper can be extended to the implicit Euler scheme and to semi-linear problems with multiplicative noise; we plan to address such extensions in a future paper.

The paper is organised as follows. Section 2 presents some preliminary material about spaces of $\gamma$-radonifying operators. The proofs of Theorems 1 and 2 are presented in Sections 3 (Theorems 9, 10) and 4 (Theorem 19), respectively.

It is known that each of the conditions in Theorems 1 and 2 implies that the solution process $U$ has continuous trajectories. In the final Section 5 we present an example which shows that without any additional assumptions on the space $E$ and/or the semigroup $S$ the splitting scheme may fail to converge even if a solution $U$ with continuous trajectories exists.

Let $\{\gamma_{j}\}_{j\ge 1}$ be a sequence of independent standard Gaussian random variables on a probability space $(\Omega,{\mathbb{P}})$, let $\mathscr{H}$ be a real Hilbert space (later we shall take $\mathscr{H}=L^{2}(0,T;H)$, where $H$ is another real Hilbert space) and $E$ a real Banach space. A bounded operator $R$ from $\mathscr{H}$ to $E$ is called $\gamma$-summing if

is finite, where the supremum is taken over all finite orthonormal systems $h=\{h_{j}\}_{j=1}^{k}$ in $\mathscr{H}$. It can be shown that $\|\cdot\|_{\gamma_{\infty}(\mathscr{H},E)}$ is indeed a norm which turns the space of $\gamma$-summing operators into a Banach space.

The space $\gamma(\mathscr{H},E)$ of $\gamma$-radonifying operators is defined to be the closure of the finite rank operators under the norm $\|\cdot\|_{\gamma_{\infty}}$; it is a closed subspace of $\gamma_{\infty}(\mathscr{H},E)$. A celebrated result of Kwapień and Hoffmann-Jørgensen [17, 23] implies that if $E$ does not contain a closed subspace isomorphic to $c_{0}$ then $\gamma(\mathscr{H},E)=\gamma_{\infty}(\mathscr{H},E)$.

Since convergence in $\gamma(\mathscr{H},E)$ implies convergence in $\mathscr{L}(\mathscr{H},E)$, every operator $R\in\gamma(\mathscr{H},E)$, being the operator norm limit of a sequence of finite rank operators from $\mathscr{H}$ to $E$, is compact.

If $\mathscr{H}$ is separable with orthonormal basis $\{h_{j}\}_{j\ge 1}$, then an operator $R:\mathscr{H}\to E$ is $\gamma$-radonifying if and only if the Gaussian sum $\sum_{j\ge 1}\gamma_{j}Rh_{j}$ converges in $L^{2}(\Omega;E)$, and in this situation we have

The general case may be reduced to the separable case by observing that for any $R\in\gamma(\mathscr{H},E)$ there exists a separable closed subspace $\mathscr{H}_{R}$ of $\mathscr{H}$ such that $R$ vanishes on the orthogonal complement $\mathscr{H}_{R}^{\perp}$.

If $R\in\gamma(H,E)$ is given and $\{h_{j}\}_{j\ge 1}$ is an orthonormal basis for $\mathscr{H}_{R}$, the sum $\sum_{j\ge 1}\gamma_{j}Rh_{j}$ defines a centred $E$-valued Gaussian random variable. Its distribution $\mu$ is a centred Gaussian Radon measure on $E$ whose covariance operator equals $RR^{\ast}$. We will refer to $\mu$ as the Gaussian measure associated with $R$. In the reverse direction, if $Y$ is a centred $E$-valued Gaussian random variable with reproducing kernel Hilbert space $\mathscr{H}$, then $\mathscr{H}$ is separable, the natural inclusion mapping $i:\mathscr{H}\hookrightarrow E$ is $\gamma$-radonifying, and we have

Below we shall need the following simple continuity result.

We start with a brief discussion of stochastic integrals of operator-valued functions. Let $H$ be a Hilbert space and fix $T>0$. An $H$-cylindrical Brownian motion, indexed by $[0,T]$ and defined on a probability space $(\Omega,\mathscr{F},{\mathbb{P}})$, is a mapping $W_{H}:L^{2}(0,T;H)\rightarrow L^{2}(\Omega)$ with the following properties:

• •

  for all $h\in L^{2}(0,T;H)$ the random variable $W_{H}(h)$ is Gaussian;

• •

  for all $h_{1},h_{2}\in L^{2}(0,T;H)$ we have ${\mathbb{E}}W_{H}(h_{1})W_{H}(h_{2})=\langle h_{1},h_{2}\rangle$.

Formally, an $H$-cylindrical Brownian motion can be thought of as a standard Brownian motion taking values in the Hilbert space $H$. One easily checks that $W_{H}$ is linear and that for all $h_{1},\ldots,h_{n}\in L^{2}(0,T;H)$ the random variables $W_{H}(h_{1}),\ldots,W_{H}(h_{n})$ are jointly Gaussian. These random variables are independent if and only if $h_{1},\ldots,h_{n}$ are orthogonal in $H$. For further details see [28, Section 3].

A finite rank step function is function of the form $\sum_{n=1}^{N}1_{(a_{n},b_{n}]}\otimes B_{n}$ where each operator $B_{n}:H\to E$ is of finite rank. The stochastic integral with respect to $W_{H}$ of such a function is defined by setting

and this definition is extended by linearity. A function $\Psi:(0,T)\to\mathscr{L}(H,E)$ is said to be stochastically integrable with respect to $W_{H}$ if there exists a sequence of finite rank step functions $\Psi_{n}:(0,T)\to\mathscr{L}(H,E)$ such that:

1. (i)

   for all $h\in H$ we have $\lim_{n\to\infty}\Psi_{n}h=\Psi h$ in measure on $(0,T)$;

2. (ii)

   the limit $Y:=\lim_{n\to\infty}\int_{0}^{T}\Psi_{n}\,dW_{H}$ exists in probability.

In this situation we write

and call $Y$ the stochastic integral of $\Psi$ with respect to $W_{H}$.

As was shown in [32], for finite rank step functions $\Psi$ one has the isometry

where $R_{\Psi}:L^{2}(0,T;H)\to E$ is the bounded operator represented by $\Psi$, i.e.,

As a consequence, a function $\Psi:(0,T)\to\mathscr{L}(H,E)$ is stochastically integrable on $(0,T)$ with respect to $W_{H}$ if and only if $\Psi^{\ast}x^{\ast}\in L^{2}(0,T;H)$ for all $x^{\ast}\in E^{\ast}$ and there exists an operator $R_{\Psi}\in\gamma(L^{2}(0,T;H),E)$ such that

The isometry (3) extends to this situation. The following simple observation [10, Lemma 2.1] will be used frequently:

For the remainder of this section we fix an $E$-valued Brownian motion $W=\{W(t)\}_{t\ge 0}$ and $T>0$. Let $H$ be the reproducing kernel Hilbert space associated with the Gaussian random variable $W(1)$ and let $i:H\hookrightarrow E$ be the natural inclusion mapping. Then $W$ induces an $H$-cylindrical Brownian motion $W_{H}$ by putting

This motivates us to call a function $\Psi:(0,T)\to\mathscr{L}(E)$ stochastically integrable with respect to $W$ if the function $\Psi\circ i:(0,T)\to\mathscr{L}(H,E)$ is stochastically integrable with respect to $W_{H}$, in which case we put

It is easy to check that for all $S\in\mathscr{L}(E)$ the indicator function $1_{(a,b]}\otimes S$ is stochastically integrable with respect to $W$ and

This shows that the definition is consistent with (1) and (2).

Now let $S=\{S(t)\}_{t\ge 0}$ denote a $C_{0}$-semigroup of bounded linear operators on $E$, with generator $A$. We will be interested in the case where the function to be integrated against $W_{H}$ is one of the following:

We may define bounded operators $R_{\Phi^{(n)}}$ and $R_{\Phi}$ from $L^{2}(0,T;H)$ to $E$ by the formula (4). Being associated with $\gamma(H,E)$-valued step functions, the operators $R_{\Phi^{(n)}}$ belong to $\gamma(L^{2}(0,T;H),E)$ by Proposition 6. Concerning the question whether the operator $R_{\Phi}$ is in $\gamma(L^{2}(0,T;H),E)$ we have the following result [32, Theorem 7.1].

In [32] an example is presented showing even for rank one Brownian motions $W$ in $E$ the equivalent conditions need not always be satisfied for all $C_{0}$-semigroups $S$ on $E$. The conditions are satisfied, however, if one of the following additional conditions holds:

1. (a)

   $E$ is a type 2 Banach space,

2. (b)

   $S$ restricts to a $C_{0}$-semigroup on $H$,

3. (c)

   $S$ is an analytic $C_{0}$-semigroup on $E$.

We refer to [10, 32] for the easy proofs.

We are now in a position to state the main result of this section. We use the notations introduced above, and let $\mu$ and $\mu^{(n)}$ denote the Gaussian measures on $E$ associated with the operators $R_{\Phi}$ and $R_{\Phi^{(n)}}$, respectively.

The assertions (1), (2), (3) are equivalent to the validity of a Lie–Trotter product formula for the Ornstein-Uhlenbeck semigroup ${\mathscr{P}}=\{\mathscr{P}(t)\}_{t\ge 0}$ associated with the problem (SCP_x), which is defined on the space $C_{\rm b}(E)$ of all bounded real-valued continuous functions on $E$ by the formula

where $U_{x}$ is the solution of (SCP_x). In order to explain the precise result, let us denote by $\mathscr{S}=\{{\mathscr{S}}(t)\}_{t\ge 0}$ and $\mathscr{T}=\{{\mathscr{T}}(t)\}_{t\ge 0}$ the semigroups on $C_{\rm b}(E)$ corresponding to the drift term and the diffusion term in (SCP_x). Thus,

Each of the semigroups $\mathscr{P}$, $\mathscr{S}$ and $\mathscr{T}$ is jointly continuous in $t$ and $x$, uniformly on $[0,T]\times K$ for all compact sets $K\subseteq E$. It was shown in [21] that if condition (3) of Theorem 8 holds, then for all $f\in C_{\rm b}(E)$ we have the Lie–Trotter product formula

with convergence uniformly on $[0,T]\times K$ for all compact sets $K\subseteq E$. Conversely it follows from the proof of this result that (6) with $x=0$ implies condition (3) of Theorem 8. In the same paper it was shown that (6) holds if at least one of the next two conditions is satisfied:

1. (a)

   $E$ is isomorphic to a Hilbert space;

2. (b)

   $S$ restricts to a $C_{0}$-semigroup on $H$.

Thus, either of these conditions implies the convergence $\lim_{n\to\infty}U_{x}^{(n)}(t)=U_{x}(t)$ in $L^{p}(\Omega;E)$ for all $x\in E$ and $t\in[0,T]$ of the splitting scheme. The proofs in [21] are rather involved. A simple proof for case (b) has been subsequently obtained by Johanna Tikanmäki (personal communication). In Theorems 9 and 10 below we shall give simple proofs for both cases (a) and (b), based on the Proposition 5 and an elementary convergence result for $\gamma$-radonifying operators from [30], respectively. Moreover, case (a) is extended to Banach spaces with type $2$. Recall that a Banach space is said to have type $1\le p\le 2$ if there exists a constant $C\ge 0$ such that for all finite choices $x_{1},\dots,x_{k}\in E$ we have

Hilbert spaces have type $2$ and $L^{p}$-spaces ($1\le p<\infty$) have type $\min\{p,2\}$. We refer to [1] for more information.