Lower and upper bounds forstrong approximation errors for numericalapproximations of stochastic heat equations

Lower and upper bounds for
strong approximation errors for numerical
approximations of stochastic heat equations

Sebastian Becker, Benjamin Gess,
Arnulf Jentzen, and Peter E. Kloeden

Mathematics Institute, Goethe University Frankfurt,
60325 Frankfurt am Main, Germany, e-mail: becker@math.uni-frankfurt.de
Max Planck Institute for Mathematics in the Sciences,
04103 Leipzig, Germany, e-mail: bgess@mis.mpg.de
SAM, Department of Mathematics, ETH Zurich,
8092 Zurich, Switzerland, e-mail: arnulf.jentzen@sam.math.ethz.ch
Mathematics Institute, Goethe University Frankfurt,
60325 Frankfurt am Main, Germany, e-mail: kloeden@math.uni-frankfurt.de
Abstract

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have recently been obtained in [Becker, S., Gess, B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. arXiv:1711.02423 (2017)].

1 Introduction

In this work we consider space-time discrete numerical methods for linear stochastic heat partial differential equations of the type

(1)

with zero Dirichlet boundary conditions for , where is the time horizon under consideration and where is a space-time white noise on a probability space . In particular, we analyse strong rates of convergence of a full-discrete exponential Euler method, proving optimal upper and lower estimates on the strong rate of convergence. The next result, Theorem 1.1 below, summarizes the main findings of this article.

Theorem 1.1.

Let , , , , let be an orthonormal basis of , let be the Laplacian with Dirichlet boundary conditions on , let be a probability space, let be an -cylindrical Wiener process, let and , , be stochastic processes which satisfy that for all , , it holds -a.s. that and , and assume for all , that and . Then there exist such that

  1. we have for all that

    (2)

    and

  2. we have for all that

    (3)

Theorem 1.1 above is an immediate consequence of Corollary 2.4 below, Lemma 2.6 below, and Da Prato & Zabczyk [6, Lemma 7.7]. The recent article [1] establishes strong convergence rates for suitable space-time discrete approximation methods for stochastic Allen-Cahn equations of the type

(4)

with zero Dirichlet boundary conditions for , where are real numbers. Roughly speaking, in [1, Theorem 1.1] a spatial convergence rate of the order and a temporal convergence rate of the order have been established. More precisely, [1, Theorem 1.1] shows that for every there exists such that for all we have

(5)

where denotes the nonlinearity-truncated approximation scheme in [1] applied to (4). The results of this article, that is, inequalities (2) and (3), prove that these rates are essentially (up to an arbitrarily small polynomial order of convergence) optimal. We also refer, e.g., to [9, 25, 10, 8, 23, 20, 21, 7, 13, 14, 11, 2, 22, 15, 4, 3, 24, 19] for further research articles on explicit approximation schemes for stochastic differential equations with superlinearly growing non-linearities. Furthermore, related lower bounds for approximation errors in the linear case (i.e., in the case in (4)) can, e.g., be found in Müller-Gronbach, Ritter, & Wagner [17, Theorem 1], Müller-Gronbach & Ritter [16, Theorem 1], Müller-Gronbach, Ritter, & Wagner [18, Theorem 4.2], Conus, Jentzen, & Kurniawan [5, Lemma 6.2], and Jentzen & Kurniawan [12, Corollary 9.4].

1.1 Acknowledgments

This work has been partially supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.

2 Lower and upper bounds for strong approximation errors of numerical approximations of linear stochastic heat equations

2.1 Setting

Let , , the functions which satisfy for all , that , for every measure space , every measurable space , every set , and every function let the set given by , let , , , satisfy for all , , that and , let be the Laplacian with Dirichlet boundary conditions on times the real number , let be a probability space, let be an -cylindrical Wiener process, and let and , , be stochastic processes which satisfy for all , , that and .

2.2 Lower and upper bounds for Hilbert-Schmidt norms of Hilbert-Schmidt operators

Lemma 2.1.

Assume the setting in Section 2.1 and let , with . Then

  1. we have that

    (6)

    and

  2. we have that

    (7)
Proof of Lemma 2.1.

Throughout this proof let satisfy for all that . Next observe that

(8)

This establishes (i). Moreover, note that

(9)

The proof of Lemma 2.1 is thus completed. ∎

Lemma 2.2.

Assume the setting in Section 2.1 and let , . Then

(10)
Proof of Lemma 2.2.

Observe that

(11)

This and the integral transformation theorem imply that

(12)

Moreover, note that

(13)

The fact that

(14)

the fact that

(15)

and the integral transformation theorem hence yield that

(16)

Again the fact that

(17)

therefore ensures that

(18)

Combining this and (12) completes the proof of Lemma 2.2. ∎

2.3 Lower and upper bounds for strong approximation errors of temporal discretizations of linear stochastic heat equations

Lemma 2.3.

Assume the setting in Section 2.1 and let , . Then

(19)
Proof of Lemma 2.3.

Throughout this proof let satisfy for all that and let , , be the functions which satisfy for all , that . Observe that Lemma 2.1 (i) ensures for all that

(20)

Therefore, we obtain that

(21)

Next note that Itô’s isometry implies for all that

(22)

This, the fact that , and Lemma 2.1 (ii) ensure that

(23)

Inequality (21) hence proves that

(24)

Lemma 2.2 therefore implies that

(25)

In the next step observe that (22) and Lemma 2.1 (ii) assure that