This is the accepted version of the journal article: Anton, Rikard; Cohen, David; Quer i Sardanyons, Lluís. «A fully discrete approximation of the one-dimensional stochastic heat equation». IMA Journal of Numerical Analysis, Vol. 40, Issue 1 (January 2020), p. 247-284. DOI 10.1093/imanum/dry060 This version is available at https://ddd.uab.cat/record/274602 under the terms of the license A fully discrete approximation of the one-dimensional stochastic heat equation R IKARD A NTON∗, DAVID C OHEN† Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden AND L LUIS Q UER -S ARDANYONS‡ Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia November 22, 2017 Abstract A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in Lq (Ω), for all q ≥ 2, improv- ing some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are pre- sented to illustrate the theoretical results. Mathematics Subject Classification (2010): 60H15; 60H35. Keywords:stochastic heat equation; multiplicative noise; finite difference scheme; stochastic exponential integrator; Lq (Ω)-convergence. ∗ Email: rikard.anton@umu.se † Corresponding author. Email: david.cohen@umu.se ‡ Email: quer@mat.uab.cat 1 1 Introduction We study an explicit full numerical discretization of the one-dimensional stochastic heat equation ∂ ∂2 ∂2 u(t, x) = 2 u(t, x) + f (t, x, u(t, x)) + σ (t, x, u(t, x)) W (t, x) in (0, ∞) × (0, 1), ∂t ∂x ∂ t∂ x u(t, 0) = u(t, 1) = 0 for t ∈ (0, ∞), u(0, x) = u0 for x ∈ [0, 1], (1) where W is a Brownian sheet on [0, ∞) × [0, 1] defined on some probability space (Ω, F , P) satisfying the usual conditions, and u0 is a continuous function in [0, 1] such that u0 (0) = u0 (1) = 0. Assumptions on the coefficients f and σ will be specified below. As far as the spatial discretization is concerned, we use a standard finite dif- ference scheme, as in [20]. In order to discretize (1) with respect to the time variable, we consider an exponential method similar to the time integrators used in [9, 10, 3] for stochastic wave equations or in [2, 8] for stochastic Schrödinger equations. Our main aim is to improve the temporal rate of convergence that has been ob- tained by Gyöngy in the reference [21]. Indeed, in [21], the explicit as well as the semi-implicit Euler-Maruyama scheme have been applied for the time discretization of problem (1). When the functions f and σ are globally Lipschitz continuous in the third variable, a temporal convergence order of 18 − in the Lq (Ω)-norm, for all q ≥ 2, is ob- tained for these numerical schemes (see Theorem 3.1 in [21] for a precise statement). Our first objective is to see if an explicit exponential method can provide a higher rate of convergence. In the present work, we answer this question positively and obtain the temporal rate 41 − (see the first part of Theorem 2.3 below). We note that, as in [21], the latter estimate for the Lq (Ω)-error holds for any fixed t ∈ (0, T ] and uniformly in the spatial variable, where T > 0 is some fixed time horizon. On the other hand, we should also remark that, in [21], a rate of convergence 41 could be obtained only in the case where the initial condition u0 belongs to C3 ([0, 1]). Finally, as in [21], we also prove that the exponential scheme converges almost surely to the solution of (1), uniformly with respect to time and space variables (cf. Theorem 2.4). Our second objective consists in refining the above-mentioned temporal rate of convergence in order to end up with a convergence order which is exactly 41 and with an estimate which is uniform both with respect to time and space variables. To this end, we assume that the initial condition u0 belongs to some fractional Sobolev space (see (12) for the precise definition). Indeed, as it can be deduced from the second part of Theorem 2.3 and well-known Sobolev embedding results, in order to have the rate 14 , the hypothesis on u0 implies that it is δ -Hölder continuous for all δ ∈ (0, 12 ). Eventually, as in [21], we remove the globally Lipschitz assumption on the coefficients f and σ in equation (1), and we prove convergence in probability for the proposed explicit exponential integrator (see Theorem 3.1 below). We should point out that there are also other important advantages with using the exponential method proposed here. Namely, first, it does not suffer a step size restric- tion (imposed by a CFL condition) as the explicit Euler-Maruyama scheme from [21]. 2 Secondly, it is an explicit scheme and therefore has implementation advantages over the implicit Euler-Maruyama scheme studied in [21]. These facts will be illustrated numerically. The numerical analysis of the stochastic heat equation (1) is an active research area. Without being too exhaustive, beside the above mentioned papers [20] and [21], we mention the following works regarding numerical discretizations of stochas- tic parabolic partial differential equations: [20, 55, 5, 47] (spatial approximations); [18, 22, 23, 1, 48, 45, 15, 17, 26, 44, 52, 39, 40, 31, 28, 11, 30, 29, 34, 38, 54, 6, 12, 33, 7, 53] (temporal and full discretizations); [49, 36] (stability). Observe that most of these references are concerned with an interpretation of stochastic partial differential equations in Hilbert spaces and thus error estimates are provided in the L2 ([0, 1]) norm (or similar norms). The reader is referred to the monographs [32, 35, 37] for a more comprehensive reference list. In the present publication, we follow a similar approach as in [10] and [21]. The main idea consists in establishing suitable mild forms for the spatial approximation uM and for the fully discretization scheme uM,N . The obtained mild equations, together with some auxiliary results and taking into account the hypotheses on the coefficients and initial data, will allow us to deal with the Lq (Ω)-error ( )1 q E[|uM (t, x) − uM,N (t, x)|q ] , for all q ≥ 2. The Lq (Ω)-error comparing uM with the exact solution of (1) has already been studied in [20]. The paper is organized as follows. In Section 2, we study the numerical approxi- mation of the solution to equation (1) in the case of globally Lipschitz continuous co- efficients. More precisely, we first recall the spatial discretization uM of (1) and prove some properties of uM needed in the sequel. Next, we introduce the full discretization scheme and prove that it satisfies a suitable mild form, and provide three auxiliary re- sults which will be invoked in the convergence results’ proofs. At this point, we state and prove the main result on Lq (Ω)-convergence along with some numerical experi- ments illustrating its conclusion. Section 2 concludes with the result on almost sure convergence, where we also provide some numerical experiments. Finally, Section 3 is devoted to deal with the convergence in probability of the numerical solution to the ex- act solution of (1), in the case where the coefficients f and σ are non-globally Lipschitz continuous. Observe that, throughout this article, C will denote a generic constant that may vary from line to line. 2 Error analysis for globally Lipschitz continuous coef- ficients This section is divided into three subsections. We begin by stating the assumptions we will make and by recalling the mild solution of (1). The first subsection is dedicated to 3 recalling the finite difference approximation from [20] and some (new) results about it. In the second subsection, we numerically integrate the resulting semi-discrete system of stochastic differential equations in time to obtain a full approximation of (1). We also state and prove our main result about convergence in the 2p-th mean. Finally, in the third subsection, we prove almost sure convergence of the full approximation to the exact solution. In addition, numerical experiments are provided to illustrate the theoretical results of this section. In this section, we shall make the following assumptions on the coefficients of the stochastic heat equation (1): for a given positive real number T , there exist a constant C such that ( ) | f (t, x, u) − f (t, y, v)| + |σ (t, x, u) − σ (t, y, v)| ≤ C |x − y| + |u − v| , (L) for all t ∈ [0, T ], x, y ∈ [0, 1], u, v ∈ R, and | f (t, x, u)| + |σ (t, x, u)| ≤ C(1 + |u|), (LG) for all t ∈ [0, T ], x ∈ [0, 1], u ∈ R. Assume also that the initial condition u0 defines a continuous function on [0, 1] with u0 (0) = u0 (1) = 0. The assumptions (L) and (LG) imply existence and uniqueness of a solution u of equation (1) on the time interval [0, T ], see e.g. Theorem 3.2 and Exercise 3.4 in [51]. Let us recall that, for a stochas- tic basis (Ω, F , (Ft )t≥0 , P), a solution to equation (1) is an Ft -adapted continuous process {u(t, x), (t, x) ∈ [0, T ] × [0, 1]} satisfying that, for every Φ ∈ C∞ (R2 ) such that Φ(t, 0) = Φ(t, 1) = 0 for all t ≥ 0, we have ∫ 1 ∫ 1 u(t, x)Φ(t, x) dx = u0 (x)Φ(t, x) dx 0 0 ∫ t∫ 1 ( 2 ) ∂ Φ ∂Φ + u(s, x) (s, x) + (s, x) dx ds 0 0 ∂ x2 ∂s ∫ t∫ 1 + f (s, x, u(s, x))Φ(s, x) dx ds 0 0 ∫ t∫ 1 + σ (s, x, u(s, x))Φ(s, x)W (ds, dx), P-a.s., (2) 0 0 for all t ∈ [0, T ]. It is well-known that the above equation implies the following mild form for (1): ∫ 1 ∫ t∫ 1 u(t, x) = G(t, x, y)u0 (y) dy + G(t − s, x, y) f (s, y, u(s, y)) dy ds 0 0 0 ∫ t∫ 1 (3) + G(t − s, x, y)σ (s, y, u(s, y))W (ds, dy), P-a.s., 0 0 where G(t, x, y) is the Green function of the linear heat equation with homogeneous Dirichlet boundary conditions: ∞ ∑ e− j π t φ j (x)φ j (y), 2 2 G(t, x, y) = t > 0, x, y ∈ [0, 1], j=1 √ with φ j (x) := 2 sin( jπ x), j ≥ 1. Note that these functions form an orthonormal basis of L2 ([0, 1]). 4 2.1 Spatial discretization of the stochastic heat equation In this subsection we recall the finite difference discretization and some results obtained in [20]. In addition to this, we show new regularity results for the approximated Green function GM (t, x, y) defined below, and for the space discrete approximation, which will be needed in the sequel. Let M ≥ 1 be an integer and define the grid points xm = M m for m = 0, . . . , M, and the mesh size ∆x = M . We now use the standard finite difference scheme for the spatial 1 approximation of (1) from [20]. Let the process uM (t, ·) be defined as the solution of the system of stochastic differential equations (for m = 1, . . . , M − 1) ( ) duM (t, xm ) = M 2 uM (t, xm+1 ) − 2uM (t, xm ) + uM (t, xm−1 ) dt + f (t, xm , uM (t, xm )) dt (4) + M σ (t, xm , uM (t, xm )) d(W (t, xm+1 ) −W (t, xm )) with Dirichlet boundary conditions uM (t, 0) = uM (t, 1) = 0, and initial value uM (0, xm ) = u0 (xm ), for m = 1, . . . , M − 1. For x ∈ [xm , xm+1 ) we define uM (t, x) := uM (t, xm ) + (Mx − m)(uM (t, xm+1 ) − uM (t, xm )). (5) √ We use the notations uM m (t) := u (t, xm ) and Wm (t) := M(W (t, xm+1 ) − W (t, xm )), M M for m = 1, . . . , M − 1 and write the system (4) as M−1 duM m (t) = M 2 ∑ Dmi uMi (t) dt + f (t, xm , uMm (t)) dt i=1 √ + M σ (t, xm , uM M m (t)) dWm (t), with initial value uM m (0) = u0 (xm ), for m = 1, . . . , M − 1, where D = (Dmi )m,i is a square matrix of size M − 1, with ele- ments Dmm = −2, Dmi = 1 for |m − i| = 1, Dmi = 0 for |m − i| > 1. Also W M (t) := m=1 is an M − 1 dimensional Wiener process. Observe that the matrix M D (WmM (t))M−1 2 has eigenvalues ( ) jπ λ jM := −4 sin2 M 2 = − j 2 π 2 cM j , 2M where ( ) 2 jπ 4 sin 2M ≤ cM j := ( )2 ≤ 1, π2 jπ 2M for j = 1, 2, . . . , M − 1 and every M ≥ 1. 5 Using the variation of constants formula, the exact solution to (4) reads 1 M−1 M−1 uM (t, xm ) = ∑ ∑ exp(λ jM t)φ j (xm )φ j (xl )u0 (xl ) M l=1 j=1 ∫ t 1 M−1 M−1 + 0 M ∑ ∑ exp(λ jM (t − s))φ j (xm )φ j (xl ) f (s, xl , uM (s, xl )) ds l=1 j=1 ∫ t 1 M−1 M−1 + √ ∑ ∑ exp(λ jM (t − s))φ j (xm )φ j (xl )σ (s, xl , uM (s, xl )) dWlM (s), 0 M l=1 j=1 √ (6) where we recall that φ j (x) := 2 sin( jxπ ) for j = 1, . . . , M − 1. We next define the discrete kernel GM (t, x, y) by M−1 GM (t, x, y) := ∑ exp(λ jM t)φ Mj (x)φ j (κM (y)), (7) j=1 [My] where κM (y) := M , φM j (x) := φ j (xl ) for x = xl and φM j (x) := φ j (xl ) + (Mx − l) (φ j (xl+1 ) − φ j (xl )) , if x ∈ (xl , xl+1 ]. With these definitions in hand, one sees that the semi-discrete solution uM satisfies the mild equation: ∫ 1 uM (t, x) = GM (t, x, y)u0 (κM (y)) dy 0 ∫ t∫ 1 + GM (t − s, x, y) f (s, κM (y), uM (s, κM (y))) dy ds (8) 0 0 ∫ t∫ 1 + GM (t − s, x, y)σ (s, κM (y), uM (s, κM (y))) dW (s, y) 0 0 P-a.s., for all t ≥ 0 and x ∈ [0, 1]. Next, we proceed by collecting some useful results for the error analysis of the fully discrete numerical discretization presented in the next subsection. The following two results are proved in [20]. Recall that uM is the space discrete approximation given by (8) and that u is the exact solution given by equation (3). Proposition 2.1 (Proposition 3.5 in [20]). Assume that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0, and that the functions f and σ satisfy the condition (LG). Then, for every p ≥ 1, there exists a constant C such that sup sup E[|uM (t, x)|2p ] ≤ C. M≥1 (t,x)∈[0,T ]×[0,1] Theorem 2.1 (Theorem 3.1 in [20]). Assume that f and σ satisfy the conditions (L) and (LG), and that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0. Then, for every 0 < α < 14 , p ≥ 1 and for every t > 0, there is a constant C = C(α , p,t) such that ( )1 ≤ C(∆x)α . 2p sup E[|uM (t, x) − u(t, x)|2p ] (9) x∈[0,1] 6 We recall that ∆x = 1/M is the mesh size in space. Moreover, uM (t, x) converges to u(t, x) almost surely as M → ∞, uniformly in t ∈ [0, T ] and x ∈ [0, 1], for every T > 0. If u0 is sufficiently smooth (e.g. u0 ∈ C3 ([0, 1])) then for every T > 0, estimate (9) holds with α = 12 and with the same constant C for all t ∈ [0, T ] and integer M ≥ 1. We will also make use of the following estimates on the discrete Green function. Lemma 2.1. There is a constant C such that the following estimates hold: (i) For all 0 < s < t ≤ T : ∫ s∫ 1 sup sup |GM (t − r, x, y) − GM (s − r, x, y)|2 dy dr ≤ C(t − s)1/2 . (10) M≥1 x∈[0,1] 0 0 (ii) For all t ∈ (0, T ]: ∫ 1 1 sup sup |GM (t, x, y)|2 dy ≤ C √ . M≥1 x∈[0,1] 0 t (iii) For all 0 < s < t ≤ T and α ∈ ( 12 , 25 ): ∫ 1 |GM (t, x, y) − GM (s, x, y)|2 dy ≤ Cs−α (t − s)α − 2 . 1 sup sup M≥1 x∈[0,1] 0 Proof. Recall that M−1 GM (t, x, y) = ∑ exp(λ jM t)φ Mj (x)φ j (κM (y)), j=1 (l) where κM (y) = [My] M , φ j (x) = φ j M for x = M and M l ( ) ( ( ) ( )) ( ] l l +1 l l l +1 φMj (x) = φ j + (Mx − l) φ j − φ j , if x ∈ , . M M M M M We first prove (i). Observe that a general version of this result is used in the proof of [20, Lem. 3.6] (see the term A2p 1 therein). Using the definition of the discrete Green function, we have ∫ s∫ 1 |GM (t − r, x, y) − GM (s − r, x, y)|2 dy dr 0 0 ∫ s ∫ 1 M−1 2 = 0 0 ∑ (exp(λ jM (t − r)) − exp(λ jM (s − r)))φ Mj (x)φ j (κM (y)) dy dr. j=1 At this point, we use the fact that the vectors (√ ( ) ) 2 k ej = sin j π , k = 1, . . . , M − 1 , j = 1, . . . , M − 1, M M 7 form an orthonormal basis of RM−1 , which implies that ∫ 1 φ j (κM (y))φl (κM (y)) dy = δ{ j=l} . (11) 0 Hence, using also the definitions of φ M j and λ j , M ∫ s∫ 1 |GM (t − r, x, y) − GM (s − r, x, y)|2 dy dr 0 0 ∫ s M−1 = 0 ∑ | exp(λ jM (t − r) − exp(λ jM (s − r))|2 |φ Mj (x)|2 dr j=1 M−1 ∫ s ≤C ∑ 0 exp(λ jM (s − r))2 dr|1 − exp(λ jM (t − s))|2 j=1 M−1 ∫ s ≤C ∑ 0 exp(−2 j2 π 2 cM j (s − r)) dr(1 − exp(− j π c j (t − s))) 2 2 M 2 j=1 ∞ ≤ C ∑ j−2 ( j4 (t − s)2 ∧ 1). j=1 [ ] Here we have used that 1 − exp(−x) ≤ x, and that (cM )−1 is bounded. Let N := √ 1 , j t−s where [·] denotes the integer part, and observe that (by comparing sums with integrals) ∞ N ∞ ∑ j−2 ( j4 (t − s)2 ∧ 1) = ∑ j2 (t − s)2 + ∑ j−2 j=1 j=1 j=N+1 ≤ C(t − s) (N + 1)3 + (N + 1)−1 2 ([ √ ])3 t −s+1 ≤ C(t − s)2 √ + (N + 1)−1 t −s ([ ])3 ( )−1 1 1 ≤ C(t − s)2 √ + √ t −s t −s 1 ≤ C(t − s) 2 . This proves part (i). The proof of (ii) follows by similar arguments as those used in the proofs of [52, Lem. 8.1, Thm 8.2]. First note that, as above, we have ∫ 1 ∫ 1 M−1 2 0 |G (t, x, y)| dy = M 2 0 ∑ exp(λ jM t)φ M j (x)φ j (κM (y)) dy j=1 M−1 ≤C ∑ exp(−2 j2 π 2 cMj t). j=1 The estimate in (ii) now follows from the inequality   M−1 1 ∑ exp(−2 j2 π 2 cMj t) ≤ C M ∧ √ √ , j=1 2cM j π t 8 which is proved in [52, Lem. 8.1]. We now prove (iii). Using the definition of the discrete Green function, properties of φ j , and the definition of λ jM , we have ∫ 1 M−1 0 |GM (t, x, y) − GM (s, x, y)|2 dy ≤ ∑ | exp(λ jM t) − exp(λ jM s)|2 j=1 M−1 ≤ ∑ | exp(− j2 π 2 cMj s)|2 |1 − exp(− j2 π 2 cMj (t − s))|2 . j=1 Since 1 − exp(−x) ≤ x and exp(−x2 ) ≤ Cα |x|−α , for all α ∈ R, it follows that ∫ 1 M−1 0 |GM (t, x, y) − GM (s, x, y)|2 dy ≤ Cα ∑ j−2α s−α (1 ∧ j4 (t − s)2 ) j=1 N ∞ ≤ C̃1 (t − s)2 s−α ∑ j4−2α + C̃2 s−α ∑ j−2α , j=1 j=N+1 [ ] where N = √1 and C̃1 and C̃2 are independent of t and s. We now estimate these t−s two terms as we did in the proof of part (i). Namely, whenever α < 5 2 we have that N (t − s)2 s−α ∑ j4−2α ≤ C(t − s)2 s−α (N + 1)5−2α j=1 ≤ C(t − s)α −1/2 s−α , √ 1+ √ t−s using the fact that N + 1 ≤ t−s ≤ √CT . t−s For the second term, if α > 1 2 we obtain ∞ ∞ s−α ∑ j−2α = s−α (N + 1)−2α + s−α ∑ j−2α ≤ Cs−α (N + 1)1−2α j=N+1 j=N+2 ≤ (t − s)α −1/2 s−α . Collecting these two estimates leads to the conclusion of the theorem. For the numerical analysis of the exponential method applied to the nonlinear stochastic heat equation (1) presented in the next subsection, the initial data u0 will be in the space H α ([0, 1]), which we now define. For α ∈ R, we define the space H α ([0, 1]) to be the set of functions g : [0, 1] → R such that ( )1/2 ∞ ⟨ ⟩ ∥g∥α = ∑ (1 + j ) | g, φ j |2 2 α < ∞, (12) j=1 √ where we recall that φ j (x) = 2 sin( jxπ ), for j ≥ 1. The inner product in the above sum stands for the usual L2 ([0, 1]) inner product. Further restrictions on α will be made 9 in the results below. For the sake of simplicity, the space H α ([0, 1]) will be denoted by H α . Note that this space is a subspace of the fractional Sobolev space of fractional order α and integrability order p = 2 (see [50]). Moreover, for any α > 21 , the space H α is continuously embedded in the space of δ -Hölder-continuous functions for all δ ∈ (0, α − 12 ) (see, e.g., [16, Thm. 8.2]). Finally, we need the following regularity results for the finite difference approxi- mation uM given by (8). Proposition 2.2. Assume that f and σ satisfy the condition (LG). 1. Assume that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0. For any 0 < s ≤ t ≤ T , any p ≥ 1, and 12 < α < 52 , we have sup sup E[|uM (t, x) − uM (s, x)|2p ] ≤ Cs−α p (t − s)ν p , M≥1 x∈[0,1] where ν = 12 ∧ (α − 12 ). 2. Assume that u0 ∈ H β ([0, 1]), with u0 (0) = u0 (1) = 0, for some β > 12 . For any 0 ≤ s ≤ t ≤ T and any p ≥ 1, we have sup sup E[|uM (t, x) − uM (s, x)|2p ] ≤ C(t − s)τ p , M≥1 x∈[0,1] where τ = 21 ∧ (β − 12 ). Proof. For ease of presentation, we consider functions f (u) and σ (u) depending only on u. Let us first define ∫ t∫ 1 F M (t, x) := GM (t − s, x, y) f (uM (s, y)) dy ds 0 0 ∫ t∫ 1 H M (t, x) := GM (t − s, x, y)σ (uM (s, y)) dW (s, y). 0 0 Then we have ∫ 1 uM (t, x) − uM (s, x) = (GM (t, x, y) − GM (s, x, y))u0 (κM (y)) dy 0 + F M (t, x) − F M (s, x) + H M (t, x) − H M (s, x). By [20, Lem. 3.6], the last two terms can be estimated by p E[|F M (t, x) − F M (s, x)|2p ] + E[|H M (t, x) − H M (s, x)|2p ] ≤ C|t − s| 2 . (13) It remains to estimate the term involving u0 . 10 Assume first that u0 ∈ C([0, 1]). We use the third part of Lemma 2.1 to get the following estimate: ( [∫ ])1/p 1 2p E (GM (t, x, y) − GM (s, x, y))u0 (κM (y)) dy 0 ∫ 1 2 = (GM (t, x, y) − GM (s, x, y))u0 (κM (y)) dy 0 ∫ 1 ≤C |GM (t, x, y) − GM (s, x, y)|2 |u0 (κM (y))|2 dy 0 −α (t − s)α − 2 . 1 ≤ Cs Collecting the above estimates and taking into account that s−α p ≥ T −α p in (13), we get sup sup E[|uM (t, x) − uM (s, x)|2p ] ≤ Cs−α p (t − s)ν p , M≥1 x∈[0,1] where ν = 12 ∧ (α − 12 ). Assume now that u0 ∈ H β ([0, 1]) for some β > 12 . Using the explicit expression of G , Cauchy-Schwarz inequality and that 1 − exp(−x) ≤ x, we have M ∫ 1 2p (GM (t, x, y) − GM (s, x, y))u0 (κM (y)) dy 0 2p M−1 = ∑ (exp(λ jM t) − exp(λ jM s))⟨u0 (κM (y)), φ j (κM (y))⟩φ M j (x) j=1 ( )2p M−1 ≤ ∑ | exp(λ jM t) − exp(λ jM s)||⟨u0 , φ j ⟩| j=1 ( )p ( )p M−1 ∞ ≤C ∑ j −2β | exp(λ jM t) − exp(λ jM s)|2 ∑j 2β |⟨u0 , φ j ⟩| 2 j=1 j=1 ( )p M−1 ≤C ∑ j −2β exp(2λ jM s)| exp(λ jM (t − s)) − 1|2 ∥u0 ∥β2p j=1 ( )p ∞ ≤C ∑j −2β ( j (t − s) ∧ 1) 4 2 . j=1 Here we have used that ⟨u0 (κM (y)), φ j (κM (y))⟩ = ⟨u0 , φ j ⟩, which can be verified by a simple calculation (see equation (21) in [46]). Furthermore, for β > 52 , we have ∞ ∑ j−2β ( j4 (t − s)2 ∧ 1) ≤ C(t − s)2 . j=1 11 On the other hand, if β ∈ ( 12 , 25 ], ∞ N ∞ ∑ j−2β ( j4 (t − s)2 ∧ 1) = (t − s)2 ∑ j4−2β + ∑ j−2β , j=1 j=1 j=N+1 [ ] 1 where N = √t−s , and [·] denotes the integer part. Note that N (t − s)2 ∑ j4−2β ≤ C(t − s)β − 2 1 j=1 and ∞ ∑ j−2β ≤ C(t − s)β − 2 . 1 j=N+1 Hence, we arrive at the estimate [∫ ] 1 2p E (G (t, x, y) − G (s, x, y))u0 (κM (y)) dy M M ≤ C(t − s)γ p , (14) 0 where γ = 2 ∧ (β − 12 ), for β > 12 . By the estimates (13) and (14) we have p sup sup E[|uM (t, x) − uM (s, x)|2p ] ≤ C(|t − s| 2 + |t − s|γ p ) M≥1 x∈[0,1] ≤ C|t − s|τ p , where τ = 21 ∧ (β − 21 ), for β > 21 . 2.2 Full discretization: L2p (Ω)-convergence This section is devoted to introduce the time discretization of the semi-discrete problem presented in the previous subsection, which will be denoted by uM,N . Next we prove properties of uM,N which will be needed in the sequel and we will state and prove the main result of the present section (cf. Theorem 2.2 below). Finally, some numerical experiments will be performed in order to illustrate the theoretical results obtained so far. We start by discretizing the space discrete solution (6) in time using an exponential integrator. For an integer N ≥ 1 and some fixed final time T > 0, let ∆t = NT and define the discrete times tn = n∆t for n = 0, 1, . . . , N. For simplicity of presentation, we consider that the functions f and σ only depend on the third variable. Let us now consider the mild equation (6) on the small time interval [tn ,tn+1 ] written in a more compact form (recall the notation uM M m (t) = u (t, xm )), as follows: ∫ tn+1 ∫ tn+1 uM (tn+1 ) = eA∆t uM (tn )+ eA(tn+1 −s) F(uM (s)) ds+ eA(tn+1 −s) Σ(uM (s)) dW M (s), tn tn with the finite difference matrix A := M 2 D, the vector F(uM (s)) with entries √ f (um M (s)) for m = 1, 2, . . . , M − 1, and the diagonal matrix Σ(u (s)) with elements M σ (uM M m (s)) 12 for m = 1, 2, . . . , M − 1. The matrix D has been defined in Section 2.1. We next dis- cretize the integrals in the above mild equation by freezing the integrands at the left endpoints of the intervals, so we obtain the explicit exponential integrator (omitting the explicit dependence on M for clarity) U 0 := uM (0), ( ) (15) U n+1 := eA∆t U n + F(U n )∆t + Σ(U n )∆W n , where the terms ∆W n := W M (tn+1 ) −W M (tn ) denote the (M − 1)-dimensional Wiener increments. The above formulation of the exponential integrator will be used for the practical computations presented below. Remark 2.1. In some particular situations, alternative approximations of the integrals in the mild equations are possible, see for instance [27, 31, 38]. This could possibly lead to better numerical schemes or improved error estimates, which will be investi- gated in future works. For the theoretical parts presented below, we will make use of the discrete Green function GM (see (7)) in order to write the numerical scheme in a more suitable form. We thus obtain the approximation Umn+1 ≈ u(tn+1 , xm ) given by (with a slight abuse of notations for the functions f and σ ) 1 M−1 M−1 Umn+1 = ∑ ∑ exp(λ jM ∆t)φ j (xm )φ j (xl )Uln M l=1 j=1 1 M−1 M−1 + ∆t ∑ ∑ exp(λ jM ∆t)φ j (xm )φ j (xl ) f (Uln ) M l=1 j=1 1 M−1 M−1 + √ ∑ ∑ exp(λ jM ∆t)φ j (xm )φ j (xl )σ (Uln )(WlM (tn+1 ) −WlM (tn )). M l=1 j=1 The above equation can be written in the equivalent form ∫ 1 Umn+1 = GM (tn+1 − tn , xm , y)UMn κM (y) dy 0 ∫ tn+1 ∫ 1 + GM (tn+1 − tn , xm , y) f (UMn κM (y) ) dy ds tn 0 ∫ tn+1 ∫ 1 + GM (tn+1 − tn , xm , y)σ (UMn κM (y) )W (ds, dy), tn 0 where we recall that M−1 GM (t, x, y) = ∑ exp(λ jM t)φ Mj (x)φ j (κM (y)), j=1 and κM (y) = [My] M , φ j (x) = φ j (xl ) for x = xl and φ j (x) = φ j (xl )+(Mx−l)(φ j (xl+1 )− M M φ j (xl )) for x ∈ (xl , xl+1 ]. In order to exhibit a more convenient mild form of the numer- 13 ical solution Umn , we iterate the integral equation above to obtain ∫ 1 Umn+1 = GM (tn+1 , xm , y)u0 (κM (y)) dy 0 n ∫ tr+1 ∫ 1 +∑ GM (tn+1 − tr , xm , y) f (UMr κM (y) ) dy ds r=0 tr 0 n ∫ tr+1 ∫ 1 +∑ GM (tn+1 − tr , xm , y)σ (UMr κM (y) )W (ds, dy), r=0 tr 0 for all m = 1, . . . , M − 1 and n = 0, 1, . . . , N. This implies that ∫ 1 Umn+1 = GM (tn+1 , xm , y)u0 (κM (y)) dy 0 ∫ tn+1 ∫ 1 ( κ T (s)/∆t ) + GM (tn+1 − κNT (s), xm , y) f UMNκ (y) dy ds 0 0 M ∫ tn+1 ∫ 1 ( κ T (s)/∆t ) + GM (tn+1 − κNT (s), xm , y)σ UMNκ (y) W (ds, dy), (16) 0 0 M where we have used the notation κNT (s) := T κN ( Ts ). Set uM,N (tn , xm ) := Umn . Then, equation (16) yields ∫ 1 uM,N (tn , xm ) = GM (tn , xm , y)u0 (κM (y)) dy 0 ∫ tn ∫ 1 + GM (tn − κNT (s), xm , y) f (uM,N (κNT (s), κM (y))) dy ds 0 0 ∫ tn ∫ 1 + GM (tn − κNT (s), xm , y)σ (uM,N (κNT (s), κM (y)))W (ds, dy). (17) 0 0 At this point, we will introduce the weak form associated to the full discretization scheme, and in particular to equation (17). This will allow us to define a continuous version of the scheme, which will be denoted by uM,N (t, x), with (t, x) ∈ [0, T ] × [0, 1]. More precisely, let {v(t, x), (t, x) ∈ [0, T ] × [0, 1]} be the unique Ft -adapted continuous random field satisfying the following: for all Φ ∈ C∞ (R2 ) with Φ(t, 0) = Φ(t, 1) = 0 for all t, it holds ∫ 1 ∫ 1 v(t, κM (y))Φ(t, y)dy = u0 (κM (y))Φ(t, y) dy 0 0 ∫ t∫ 1 ( ) ∂Φ + v(s, κM (y)) ∆M Φ(s, y) + (s, y) dy ds 0 0 ∂s ∫ t∫ 1 + f (v(κNT (s), κM (y)))Φ(s, y) dy ds 0 0 ∫ t∫ 1 + σ (v(κNT (s), κM (y)))Φ(s, y)W (ds, dy), P-a.s., 0 0 (18) 14 for all t ∈ [0, T ]. Here, ∆M denotes the discrete Laplacian, which is defined by, recalling that ∆x = M1 , ∆M Φ(s, y) := (∆x)−2 {Φ(s, y + ∆x) − 2Φ(s, y) + Φ(s, y − ∆x)} . Let us prove that, on the time-space grid points, the random field v fulfills equation (17). That is, we have the following result. Lemma 2.2. With the above notations at hand, we have that, for all m = 1, . . . , M − 1 and n = 0, 1, . . . , N, ∫ 1 v(tn , xm ) = GM (tn , xm , y)u0 (κM (y)) dy 0 ∫ tn ∫ 1 + GM (tn − κNT (s), xm , y) f (v(κNT (s), κM (y))) dy ds 0 0 ∫ tn ∫ 1 + GM (tn − κNT (s), xm , y)σ (v(κNT (s), κM (y)))W (ds, dy). (19) 0 0 Proof. We will follow some of the arguments developed in the proof of [51, Thm. 3.2]. Indeed, for any ϕ ∈ C∞ (R) and any (t, y) ∈ [0, T ] × [0, 1], we define ∫ 1 GtM (ϕ , y) := GM (t, z, y)ϕ (z) dz. 0 Since the Green function GM solves the discretized homogeneous heat equation with Dirichlet boundary conditions, that is, we have GM (t, x, 0) = GM (t, x, 1) = 0 and, for any fixed x ∈ (0, 1), ∂ M G (t, x, y) − ∆M GM (t, x, y) = 0, ∂t we can infer that ∫ 1( ∫ t ) GtM (ϕ , y) = GM (0, z, y) + ∆M GM (s, z, y)ds ϕ (z) dz 0 0 ∫ 1 ∫ t∫ 1 = GM (0, z, y)ϕ (z) dz + ∆M GM (s, z, y)ϕ (z) dz ds. 0 0 0 ∫ 1 ∂ M Hence G (ϕ , y) = ∆M GM (s, z, y)ϕ (z) dz. On the other hand, since ∂t t 0 ∫ 1 ∆M GtM (ϕ , y) = ∆M GM (t, z, y)ϕ (z) dz, 0 we deduce that ∂ M G (ϕ , y) − ∆M GtM (ϕ , y) = 0, (20) ∂t t with (t, y) ∈ [0, T ] × [0, 1]. 15 At this point, we take Φ(s, y) = Gt− M κNT (s) (ϕ , y), with t ∈ [0, T ] and ϕ ∈ C∞ (R), and plug this Φ in (18). Thus, by (20) we get that ∫ 1 ∫ 1 v(t, κM (y))Gt− M κ T (t) (ϕ , y) dy = u0 (κM (y))GtM (ϕ , y) dy 0 N 0 ∫ t∫ 1 + f (v(κNT (s), κM (y)))Gt− M κ T (s) (ϕ , y) dy ds 0 0 N ∫ t∫ 1 + σ (v(κNT (s), κM (y)))Gt− M κ T (s) (ϕ , y)W (ds, dy). 0 0 N Let (ϕε )ε ≥0 be an approximation of the Dirac delta δx , for some x ∈ (0, 1) (e.g. ϕε could be taken to be Gaussian kernels), so that we have ∫ 1 ∫ 1 v(t, κM (y))Gt− M (ϕ , y) dy = κ T (t) ε u0 (κM (y))GtM (ϕε , y) dy 0 N 0 ∫ t∫ 1 + f (v(κNT (s), κM (y)))Gt− M (ϕ , y) dy ds κ T (s) ε 0 0 N ∫ t∫ 1 + σ (v(κNT (s), κM (y)))Gt− M (ϕ , y)W (ds, dy). κ T (s) ε 0 0 N Then, as it is done in the proof of [51, Thm. 3.2], take ε → 0 in the latter equation, so we will end up with ∫ 1 GM (t − κNT (t), x, y)v(t, κM (y)) dy 0 ∫ 1 = GM (t, x, y)u0 (κM (y)) dy 0 ∫ t∫ 1 + GM (t − κNT (s), x, y) f (v(κNT (s), κM (y))) dy ds 0 0 ∫ t∫ 1 + GM (t − κNT (s), x, y)σ (v(κNT (s), κM (y)))W (ds, dy). (21) 0 0 Note that this equation, which is valid for any (t, x) ∈ [0, T ] × [0, 1], is very similar to the one we would like to get, that is (19). In fact, taking t = tn and x = xm in (21) for some n ∈ {0, . . . , N} and m ∈ {1, . . . , M − 1}, respectively, we have, using the explicit expression of GM , ∫ ∫ ( ) 1 1 M−1 0 GM (0, xm , y)v(tn , κM (y)) dy = 0 ∑ φ j (xm )φ j (κM (y)) v(tn , κM (y)) dy j=1 M−1 ∫ 1 = ∑ φ j (xm ) 0 φ j (κM (y))v(tn , κM (y)) dy j=1 M−1 M−1 1 = ∑ v(tn , xk ) M ∑ φ j (xm )φ j (xk ) k=1 j=1 = v(tn , xm ), where in the last step we have applied (11). This concludes the lemma’s proof. 16 As a consequence of Lemma 2.2, comparing equations (17) and (19) we deduce that uM,N (tn , xm ) = v(tn , xm ) for all m = 1, . . . , M − 1 and n = 0, 1, . . . , N. Thus, we can define a continuous version of uM,N as follows: for any (t, x) ∈ [0, T ] × [0, 1], set ∫ 1 uM,N (t, x) := GM (t − κNT (t), x, y)v(t, κM (y)) dy. 0 Observe that, by (21), the random field {uM,N (t, x), (t, x) ∈ [0, T ] × [0, 1]} satisfies ∫ 1 uM,N (t, x) := GM (t, x, y)u0 (κM (y)) dy 0 ∫ t∫ 1 + GM (t − κNT (s), x, y) f (uM,N (κNT (s), κM (y))) dy ds (22) 0 0 ∫ t∫ 1 + GM (t − κNT (s), x, y)σ (uM,N (κNT (s), κM (y)))W (ds, dy). 0 0 The above mild form of the fully discrete approximation will be used in the proof of the main result of the paper (see Theorem 2.2). Remark 2.2. It can be easily proved that, if tn is any discrete time and x ∈ (xm , xm+1 ), then uM,N (tn , x) turns out to be the linear interpolation between uM,N (tn , xm ) and uM,N (tn , xm+1 ). This is consistent with the definition of the space discrete approximation uM (t, x) when- ever x ∈ (xm , xm+1 ) (see (5)). 2.2.1 Some properties of uM,N This section is devoted to provide three results establishing properties of the full ap- proximation uM,N which will be needed in the sequel. First, we note that the full approximation (22) is bounded. Indeed, the proof of the following proposition is very similar to that of Proposition 2.1 above and is therefore omitted. Proposition 2.3. Assume that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0, and that the func- tions f and σ satisfy the condition (LG). Then, for every p ≥ 1, there exists a constant C such that sup sup E[|uM,N (t, x)|2p ] ≤ C. M,N≥1 (t,x)∈[0,T ]×[0,1] Next, we define the following quantities: ∫ 1 wM,N (t, x) := uM,N (t, x) − GM (t, x, y)u0 (κM (y)) dy 0 and ∫ 1 wM (t, x) := uM (t, x) − GM (t, x, y)u0 (κM (y)) dy, 0 where we recall that uM stands for the spatial discretization introduced in Section 2.1. Then, we have the following result. 17 Proposition 2.4. Assume that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0, and that f and σ satisfy condition (LG). Then, for every p ≥ 1, t, r ∈ [0, T ] and x, z ∈ [0, 1], we have ( )2p E[|wM (t, x) − wM (r, z)|2p ] ≤ C |t − r|1/4 + |x − z|1/2 (23) ( )2p E[|wM,N (t, x) − wM,N (r, z)|2p ] ≤ C |t − r|1/4 + |x − z|1/2 , (24) where the constant C does not depend on M neither on N. Proof. Inequality (23) is proved in [20, Prop. 3.7]. Let us now show inequality (24). By definition, we have ∫ t∫ 1 wM,N (t, x) = GM (t − κNT (s), x, y) f (uM,N (κNT (s), κM (y))) dy ds 0 0 ∫ t∫ 1 + GM (t − κNT (s), x, y)σ (uM,N (κNT (s), κM (y)))W (ds, dy) 0 0 M,N =: F (t, x) + H M,N (t, x), and hence wM,N (t, x) − wM,N (r, z) = F M,N (t, x) − F M,N (r, z) + H M,N (t, x) − H M,N (r, z). Therefore ( E[|wM,N (t, x) − wM,N (r, z)|2p ] ≤ C E[|F M,N (t, x) − F M,N (r, z)|2p ] ) + E[|H M,N (t, x) − H M,N (r, z)|2p ] . We will next prove that ( )2p E[|H M,N (t, x) − H M,N (r, z)|2p ] ≤ C |t − r|1/4 + |x − z|1/2 . The estimate for F M,N follows in a similar way. We have ( |H M,N (t, x) − H M,N (r, z)|2p ≤ C |H M,N (t, x) − H M,N (r, x)|2p ) + |H M,N (r, x) − H M,N (r, z)|2p and define A2p := E[|H M,N (t, x) − H M,N (r, x)|2p ] B2p := E[|H M,N (r, x) − H M,N (r, z)|2p ]. Then A2p ≤ C(A2p 2p 1 + A2 ), where, for r ≤ t without loss of generality, [ ∫ r∫ 1 A2p 1 = E (GM (t − κNT (s), x, y) − GM (r − κNT (s), x, y)) 0 0 ] 2p × σ (uM,N (κNT (s), κM (y)))W (ds, dy) [∫ ∫ ] t 1 2p A2p 2 =E GM (t − κNT (s), x, y)σ (uM,N (κNT (s), κM (y)))W (ds, dy) . r 0 18 Using Burkholder-Davies-Gundy’s inequality, Lemma 2.1, assumption (LG) on σ , Minkowski’s inequality and Proposition 2.3, we have the estimates ( [∫ ∫ ])1/p r 1 2p A21 = E (GM (t − κNT (s), x, y) − GM (r − κNT (s), x, y))σ (uM,N (κNT (s), κM (y))W (ds, dy) 0 0 ( [(∫ r ∫ 1 ) p ])1/p ≤C E |G (t − κN (s), x, y) − G (r − κN (s), x, y)| |σ (u (κN (s), κM (y)))| dy ds M T M T 2 M,N T 2 0 0 ∫ r∫ 1 = C||| |GM (t − κNT (s), x, y) − GM (r − κNT (s), x, y)|2 |σ (uM,N (κNT (s), κM (y)))|2 dy ds||| p 0 0 ∫ r∫ 1 ≤C |GM (t − κNT (s), x, y) − GM (r − κNT (s), x, y)|2 |||σ (uM,N (κNT (s), κM (y)))|||22p dy ds 0 0 ∫ r∫ 1 ≤C |GM (t − κNT (s), x, y) − GM (r − κNT (s), x, y)|2 dy ds 0 0 ≤ C(t − r)1/2 , ( [ ])1/(2p) where we set |||·|||2p = E | · |2p . Using similar arguments we have ( [∫ ∫ ])1/p t 1 2p A22 = E GM (t − κNT (s), x, y)σ (uM,N (κNT (s), κM (y))W (ds, dy) r 0 ( [(∫ t ∫ 1 ) p ])1/p ≤C E |GM (t − κNT (s), x, y)|2 |σ (uM,N (κNT (s), κM (y)))|2 dy ds r 0 ∫ t∫ 1 ≤C |GM (t − κNT (s), x, y)|2 |||σ (uM,N (κNT (s), κM (y)))|||22p dy ds r 0 ∫ t 1 ≤C ds (t − κNT (s))1/2 r ∫ t 1 ≤C ds r (t − s)1/2 ≤ C(t − r)1/2 . Thus, we obtain E[|H M,N (t, x) − H M,N (r, x)|2p ] ≤ C|t − r| p/2 , and we remark that this estimate is uniform with respect to x ∈ [0, 1]. It remains to estimate the term B. We have [∫ ∫ ] r 1 2p B2p := E (GM (r − κNT (s), x, y) − GM (r − κNT (s), z, y))σ (uM,N (κNT (s), κM (y)))W (ds, dy) , 0 0 19 and estimating B as we did for A1 and A2 , we obtain ∫ r∫ 1 B2 ≤ C |GM (r − κNT (s), x, y) − GM (r − κNT (s), z, y)|2 dy ds 0 0 ∫ r M−1 ≤C 0 ∑ exp(−2 j2 π 2 cMj (r − κNT (s)))|φ Mj (x) − φ Mj (z)|2 ds j=1 ∫ r M−1 ≤C 0 ∑ exp(−2 j2 π 2 cMj (r − s))|φ Mj (x) − φ Mj (z)|2 ds. j=1 At this point, we note that the latter term also appears in the proof of [20, Lem. 3.6], so we can estimate it in the same way and obtain E[|H M,N (r, x) − H M,N (r, z)|2p ] ≤ C|x − z| p , with a constant C independent of r. Collecting the estimates obtained so far we obtain the bound ( )2p E[|H M,N (t, x) − H M,N (r, z)|2p ] ≤ C |t − r|1/4 + |x − z|1/2 , which finally leads to (24). Finally, we shall also need the following regularity result for the full approximation. Proposition 2.5. Assume that f and σ satisfy condition (LG). 1. If u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0, then for any s,t ∈ [0, T ] and x ∈ [0, 1], p ≥ 1 and 12 < α < 25 , we have E[|uM,N (t, x) − uM,N (s, x)|2p ] ≤ Cs−α p |t − s|τ p , where τ = 21 ∧ (α − 21 ) and with a constant C independent of M, N and x. 2. If u0 ∈ H β ([0, 1]), with u0 (0) = u0 (1) = 0, for some β > 12 , then for any s,t ∈ [0, T ] and x, z ∈ [0, 1], and any p ≥ 1, we have ( ) E[|uM,N (t, x) − uM,N (s, z)|2p ] ≤ C |t − s|τ p + |x − z|2τ p , where τ = 21 ∧ (β − 12 ) and with a constant C independent of M and N. Proof. The proof can be built on the proof of Proposition 2.2, so we will only sketch the main steps. To start with, part 1 can be proved by following the same arguments used in the proof of part 1 of Proposition 2.2 and it is based on three estimates. First, one applies that ∫ 1 |GM (t, x, y) − GM (s, x, y)|2 dy ≤ Cs−α |t − s|α − 2 , 1 0 20 which corresponds to part (iii) in Lemma 2.1. Secondly, we have ∫ t∫ 1 1 |GM (t − κNT (r), x, y)|2 dy dr ≤ C|t − s| 2 , s 0 which can be verified by using (ii) of Lemma 2.1. Finally, it holds that ∫ s∫ 1 1 |GM (t − κNT (r), x, y) − GM (s − κNT (r), x, y)|2 dy dr ≤ C|t − s| 2 . 0 0 The latter estimate can be checked by doing some simple modifications in the proof of part (i) in Lemma 2.1. As far as part 2 is concerned, the time increments can be analyzed following the same steps as those used in the proof of part 2 in Proposition 2.2. We will sketch the proof for the spatial increments. More precisely, taking into account equation (22), in order to control the term E[|uM,N (t, x) − uM,N (t, z)|2p ] first we need to estimate the expression ∫ 1 2p (GM (t, x, y) − GM (t, z, y))u0 (κM (y)) dy . 0 Using the same techniques as in the proof of part 2 in Proposition 2.2, the above term can be bounded by p M−1 ∑ −2β 2 ∥u0 ∥2p Hβ j φM j (x) − φ j (z) M , j=1 where we recall that β > 12 . Next, it can be easily proved that φ M j (x) − φ j (z) ≤ M C(1 ∧ j(z − x)), where the constant C does not depend on M and we have assumed, without loosing generality, that x < z. Hence, ∫ ( )p 1 2p ∞ 0 (GM (t, x, y) − GM (t, z, y))u0 (κM (y)) dy ≤C ∑ j−2β (1 ∧ j2 (z − x)2 ) . j=1 The latter series can be estimated, up to some constant, by (z − x)(2β −1)p . As far as the spatial increments of the remaining two terms in equation (22) is concerned, applying Burkholder-Davies-Gundy and Minkowski’s inequalities, as well as the linear growth on f and σ and Proposition 2.3, the analysis reduces to control the term (∫ t ∫ 1 )p |GM (t − κNT (s), x, y) − GM (t − κNT (s), z, y)|2 dy ds . 0 0 The same arguments as above yield that this term can be bounded by (∫ )p ( )p ∞ 2λ jM (t−s) ( ) ( ) t M−1 ∑e 0 j=1 1 ∧ j (z − x) ds ≤ C ∑ j 1 ∧ j (z − x) 2 2 −2 2 2 j=1 ≤ C(z − x) p . This concludes the proof. 21 Remark 2.3. Whenever u0 ∈ H β ([0, 1]) for some β > 12 , the above result implies, thanks to Kolmogorov’s continuity criterion, that the random field uM,N has a version with Hölder-continuous sample paths. 2.2.2 Main result We are now ready to formulate and prove the main result of this section. Recall that uM is the space discrete approximation given by (8) and uM,N is the full discretization given by (22). Theorem 2.2. Assume that f and σ satisfy the conditions (L) and (LG). 1. If u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0, then for any p ≥ 1, 0 < µ < 1 4 and t ∈ [0, T ], there exists a constant C = C(p, µ ,t) such that ( )1 ≤ C(∆t)µ . 2p sup E[|uM,N (t, x) − uM (t, x)|2p ] x∈[0,1] 2. If u0 ∈ H β ([0, 1]) for some β > 12 , with u0 (0) = u0 (1) = 0, then for any p ≥ 1, we have ( )1 ≤ C(∆t)ν , 2p sup sup E[|uM,N (t, x) − uM (t, x)|2p ] t∈[0,T ] x∈[0,1] where ν = 14 ∧ ( β2 − 14 ). ( [ ])1/(2p) Proof. We have, using the notation |||·|||2p = E | · |2p , |||uM,N (t, x) − uM (t, x)|||2p ∫ t ∫ 1( ≤ ||| GM (t − κNT (s), x, y) f (uM,N (κNT (s), κM (y))) 0 0 ) −GM (t − s, x, y) f (uM (s, κM (y))) dy ds|||2p ∫ t ∫ 1( + ||| GM (t − κNT (s), x, y)σ (uM,N (κNT (s), κM (y))) 0 0 ) −GM (t − s, x, y)σ (uM (s, κM (y))) W (ds, dy)|||2p =: A + B. We show in detail the estimates for B. It will then be clear that similar estimates can be made for A. First we note that B2 ≤ C(B21 + B22 ), where ∫ t∫ 1 B21 = ||| (GM (t − κNT (s), x, y) − GM (t − s, x, y))σ (uM,N (κNT (s), κM (y)))W (ds, dy)|||22p 0 0 22 and ∫ t∫ 1 B22 = ||| GM (t − s, x, y)(σ (uM,N (κNT (s), κM (y))) − σ (uM (s, κM (y))))W (ds, dy)|||22p . 0 0 By Burkholder-Davies-Gundy and Minkowski’s inequalities, we have ( [∫ ∫ ])1/p t 1 2p B21 = E (GM (t − κNT (s), x, y) − GM (t − s, x, y))σ (uM,N (κNT (s), κM (y)))W (ds, dy) 0 0 ( [(∫ t ∫ 1 ) p ])1/p ≤C E |G (t − κN (s), x, y) − G (t − s, x, y)| |σ (u (κN (s), κM (y)))| dy ds M T M 2 M,N T 2 0 0 ∫ t∫ 1 = C||| |GM (t − κNT (s), x, y) − GM (t − s, x, y)|2 |σ (uM,N (κNT (s), κM (y)))|2 dy ds||| p 0 0 ∫ t∫ 1 ≤C |GM (t − κNT (s), x, y) − GM (t − s, x, y)|2 |||σ (uM,N (κNT (s), κM (y)))|||22p dy ds. 0 0 By assumption (LG) and Proposition 2.3, we obtain B21 ≤ sup |||σ (uM,N (s, y))|||22p (s,y)∈[0,T ]×[0,1] ∫ t∫ 1 × |GM (t − κNT (s), x, y) − GM (t − s, x, y)|2 dy ds 0 0 ≤ C(∆t) 1/2 . Here we have also used that ∫ t∫ 1 sup |GM (t − κNT (s), x, y) − GM (t − s, x, y)|2 dy ds ≤ C(∆t)1/2 , x∈[0,1] 0 0 where the constant C does not depend on M. This is only a slight variation of (10) in Lemma 2.1. The proof is very similar and is therefore omitted. Concerning the term B2 , using analogous arguments we have ∫ t∫ 1 B22 ≤C |GM (t − s, x, y)|2 dy 0 0 × sup |||σ (uM,N (κNT (s), y)) − σ (uM (s, y))|||22p ds. y∈[0,1] 23 By the Lipschitz assumption on σ and (ii) in Lemma 2.1, we get ∫ t∫ 1 B22 ≤ C |GM (t − s, x, y)|2 dy sup |||uM,N (κNT (s), x) − uM (s, x)|||22p ds 0 0 x∈[0,1] ∫ t ( 1 ≤C √ sup |||uM,N (κNT (s), x) − uM,N (s, x)|||22p 0 t −s x∈[0,1] ) + sup |||u M,N (s, x) − u M (s, x)|||22p ds x∈[0,1] ∫ t 1 ≤C √ sup |||uM,N (κNT (s), x) − uM,N (s, x)|||22p ds 0 t − s x∈[0,1] ∫ t 1 +C √ sup |||uM,N (s, x) − uM (s, x)|||22p ds. (25) 0 t − s x∈[0,1] At this point, We need to distinguish between the two different cases of the initial value u0 . If we assume u0 ∈ C([0, 1]), then we apply Proposition 2.5 to the first term in (25), so we get ∫ t ∫ t 1 sup |||uM,N (κNT (s), x) − uM,N (s, x)|||22p ds ≤ C(∆t)τ (t − s)− 2 s−α ds 1 √ 0 t − s x∈[0,1] 0 ( 1 ) 1 −α = C(∆t)τ B 1 − α , t2 , 2 where B denotes the Beta function. In order to obtain the last equality, we need to restrict the range on α to ( 21 , 1) (part 1 in Proposition 2.5 was valid for any α ∈ ( 12 , 52 )). In this case, notice that we have τ = 12 ∧ (α − 12 ) = α − 12 . Plugging the above estimate 1 in (25) and taking into account that we obtained the bound B21 ≤ C(∆t) 2 , we have thus proved that ∫ t 1 B2 ≤ C(t)(∆t)α − 2 +C 1 √ sup |||uM,N (s, x) − uM (s, x)|||22p ds. 0 t − s x∈[0,1] As commented at the beginning of the proof, the analysis of the term A2 can be per- formed in a similar way, in such a way that the same type of estimate can be obtained. Summing up, we have that ∫ t 1 z(t) ≤ C(t)(∆t)α − 2 +C 1 √ z(s) ds, 0 t −s where z(s) := supx∈[0,1] |||uM,N (s, x) − uM (s, x)|||22p . Then, applying a version of Gron- wall’s Lemma (see for instance [43, Chap. 1]) we conclude this part of the proof. If we instead assume u0 ∈ H β ([0, 1]) for some β > 21 , then we apply part 2 of Proposition 2.5 to the first term in (25), obtaining ∫ t 1 √ sup |||uM,N (κNT (s), x) − uM,N (s, x)|||22p ds ≤ C(∆t)τ , 0 t − s x∈[0,1] 24 where τ = 21 ∧ (β − 21 ). Hence, in this case we get that ∫ t 1 z(t) ≤ C(∆t)τ +C √ z(s) ds, 0 t −s and we conclude applying again a version of Gronwall’s Lemma, see for instance [20, Lem. 3.4]. Combining Theorems 2.1 and 2.2, we arrive at the following error estimate for the full discretization. Theorem 2.3. Let f and σ satisfy conditions (L) and (LG). 1. Assume that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0. Then, for every p ≥ 1, t ∈ (0, T ], 0 < α1 < 14 and 0 < α2 < 14 , there are constants Ci = Ci (t), i = 1, 2, such that ( )1 sup E[|uM,N (t, x) − u(t, x)|2p ] 2p ≤ C1 (∆x)α1 +C2 (∆t)α2 . x∈[0,1] 2. Assume that u0 ∈ H β ([0, 1]) with u0 (0) = u0 (1) = 0, for some β > 12 . Then, for every p ≥ 1, t ∈ (0, T ], 0 < α1 < 14 , there are constants C1 = C1 (t) and C2 such that ( )1 sup E[|uM,N (t, x) − u(t, x)|2p ] 2p ≤ C1 (∆x)α1 +C2 (∆t)τ , x∈[0,1] where τ = 41 ∧ ( β2 − 14 ). Remark 2.4. For ease of presentation, we stated the above results for functions f and σ depending only on u. Observe that the above results remain true in the case of functions f and σ depending on (t, x, u) if one replaces the condition (L) by the following one ( ) | f (t, x, u) − f (s, y, v)| + |σ (t, x, u) − σ (s, y, v)| ≤ C |t − s|1/4 + |x − y|1/2 + |u − v| (H) for all s,t ∈ [0, T ], x, y ∈ [0, 1], u, v ∈ R. In this case, the fully discrete solution reads ∫ 1 uM,N (t, x) = GM (t, x, y)u0 (κM (y)) dy 0 ∫ t∫ 1 + GM (t − κNT (s), x, y) f (κNT (s), κM (y), uM,N (κNT (s), κM (y))) dy ds 0 0 ∫ t∫ 1 + GM (t − κNT (s), x, y)σ (κNT (s), κM (y), uM,N (κNT (s), κM (y)))W (ds, dy), 0 0 [My] where we recall that κM = M and κNT (s) = T κN ( Ts ). 25 10 0 10 -1 Error 10 -2 10 -3 Error SEXP Error SEM Error CNM Slope 1/2 10 -4 10 -5 10 0 Figure 1: Temporal rates of convergence for the exponential integrator (SEXP), the semi-implicit Euler-Maruyama scheme (SEM), and the Crank-Nicolson-Maruyama scheme (CNM). The reference line has slope 1/2 (dashed line). 2.2.3 Numerical experiments: strong convergence We now numerically illustrate the results from Theorem 2.2. To do so, we first dis- cretize the problem (1), with u0 (x) = cos(π (x − 1/2)), f (u) = u/2, σ (u) = 1 − u with centered finite differences using the mesh ∆x = 2−9 . The time discretizations are done using the semi-implicit Euler-Maruyama scheme (see e.g. [21]), the Crank- Nicolson-Maruyama scheme (see e.g. [52]) and the explicit exponential integrator (15) with step sizes ∆t ranging from 2−1 to 2−16 . The loglog plots of the errors sup(t,x)∈[0,0.5]×[0,1] E[|uM,N (t, x) − uM (t, x)|2 ] are shown is Figure 1, where convergence of order 1/2 for the exponential integrator is observed. The reference solution is com- puted with the exponential integrator using ∆xref = 2−9 and ∆tref = 2−16 . The expected values are approximated by computing averages over Ms = 500 samples. Next, we compare the computational costs of the explicit stochastic exponential method (15), the semi-implicit Euler-Maruyama scheme, and the Crank-Nicolson- Maruyama scheme for the numerical integration of problem (1) with the same parame- ters as in the previous numerical experiments. We run the numerical methods over the time interval [0, 1]. We discretize the spatial domain [0, 1] with a mesh ∆x = 2−6 . We run 100 samples for each numerical method. For each method and each sample, we run several time steps and compare the error at final time with a reference solution provided for the same sample with the same method for the very small time step ∆tref = 2−15 . Figure 2 shows the total computational time for all the samples, for each method and each time step, as a function of the averaged final error we obtain. We observe that the computational cost of the Crank-Nicolson-Maruyama scheme 26 4 SEXP SEM Total comp. time in sec. (log10-scale) 3.5 CNM 3 2.5 2 1.5 1 0.5 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 Aver. final error over 100 samples (log10-scale) Figure 2: Computational time as a function of the averaged final error for the following numerical methods: the stochastic exponential scheme (15) (SEXP), the semi-implicit Euler-Maruyama (SEM), and the Crank-Nicholson-Maruyama scheme (CNM). is slightly higher than the cost of the semi-implicit Euler-Maruyama scheme which is a little bit higher than the one for the explicit scheme (15). 2.3 Full discretization: almost sure convergence In this subsection we prove almost sure convergence of the fully discrete approxima- tion uM,N (22) to the exact solution u of the stochastic heat equation (1) with globally Lipschitz continuous coefficients. The main result is the following. Theorem 2.4. Assume that the functions f and σ satisfy the conditions (LG) and (L), and that u0 ∈ C([0, 1]) with u0 (0) = u0 (1) = 0. Then, the full approximation uM,N (t, x) converges to u(t, x) almost surely, as M, N → ∞, uniformly in t ∈ [0, T ] and x ∈ [0, 1]. Proof. In [20, Thm. 3.1], it was shown that uM (t, x) converges to u(t, x) almost surely uniformly in (t, x) as M → ∞. It is therefore enough to show that uM,N (t, x) converges to uM (t, x) almost surely, as N → ∞, uniformly in (t, x) and M ∈ N. To achieve this, it suffices to prove that wM,N (t, x) converges to wM (t, x) almost surely in (t, x) as N → ∞. This is because the terms involving u0 in the approximations uM given by (8) and uM,N given by (22) are the same. We first observe that |wM,N (t, x) − wM (t, x)|2p ≤ C(A1 + A2 + A3 ), 27 where N N ∑∑ 2p A1 = wM,N (tn , xi ) − wM (tn , xi ) n=0 i=0 2p A2 = sup sup sup sup wM,N (t, x) − wM,N (tn , xi ) n=0,...,N i=0,...,N |x−xi |≤1/N |t−tn |≤∆t 2p A3 = sup sup sup sup wM (t, x) − wM (tn , xi ) n=0,...,N i=0,...,N |x−xi |≤1/N |t−tn |≤∆t and we recall that xi and tn are the discrete points in space and time, respectively, given by xi = Ni for i = 0, 1 . . . , N and tn = n∆t for n = 0, 1, . . . , N. By Theorem 2.2 we obtain ( )2µ p−2 1 E[A1 ] ≤ C , N for all 0 < µ < 41 . Also, by Proposition 2.4 we have ( )2pδ 1 E[A2 + A3 ] ≤ C N for δ ∈ (0, 1/4). Using that ( )2µ p−2 ( )2pδ ( )2p min(δ ,µ )−2 1 1 1 + ≤2 N N N we thus get [ ] ( )2p min(δ ,µ )−2 1 E sup sup |w M,N (t, x) − w (t, x)| M 2p ≤C , M≥1 (t,x)∈[0,T ]×[0,1] N where the constant C does not depend on M neither on N. Hence, using Markov’s inequality we obtain that ( ( )2 ) ( )2p min(δ ,µ )−4 1 1 P sup sup |w (t, x) − w (t, x)| > M,N M 2p ≤C M≥1 (t,x)∈[0,T ]×[0,1] N N for all integers N ≥ 1. It thus follows that ( ( )2 ) ∞ 1 ∑P sup sup M≥1 (t,x)∈[0,T ]×[0,1] |w M,N (t, x) − w (t, x)| M 2p > N <∞ N=1 for p large enough. By the Borel-Cantelli lemma we now know that for sufficiently large p we have 1 sup sup |wM,N (t, x) − wM (t, x)|2p ≤ , M≥1 (t,x)∈[0,T ]×[0,1] N2 with probability one. Taking the limit N → ∞ concludes the proof. 28 0.2 SEXP 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 Figure 3: Almost sure convergence of the exponential integrator (SEXP). The reference solution is displayed in red. 2.3.1 Numerical experiments: almost sure convergence We now numerically illustrate Theorem 2.4. To do so, we first discretize the stochastic heat equation (1), with u0 (x) = cos(π (x − 1/2)), f (u) = 1 − u, σ (u) = sin(u) with centered finite differences using the mesh ∆x = 2−9 . The time discretization is done using the explicit exponential integrator (15) with step sizes ∆t ranging from 2−6 to 2−18 (only every second power). Figure 3 displays, for a fixed spatial discretization, profiles of one realization of the numerical solution at the fixed time T = 0.5 as well as a reference solution computed with the exponential integrator using ∆xref = 2−9 and ∆tref = 2−18 . Convergence to this reference solution as the time step goes to zero (from light to dark grey plots) is observed. 3 Convergence analysis for non-globally Lipschitz con- tinuous coefficients In this section, we remove the globally Lipschitz assumption on the coefficients f and σ in equation (1) and we prove convergence in probability of the fully discrete approx- imation uM,N given by (22) to the exact solution u of (1). Throughout the section we will assume that the initial condition u0 belongs to H β for some β > 12 . Furthermore, we shall consider the following hypotheses: (PU) Pathwise uniqueness holds for problem (1): whenever u and v are carried by the same filtered probability space and if both u and v are solutions to problem (1) 29 on the stochastic time interval [0, τ ), then u(t, ·) = v(t, ·) for all t ∈ [0, τ ), almost surely. (C) The coefficient functions f (t, x, u) and σ (t, x, u) are continuous in the variable u. Remark 3.1. For general conditions ensuring pathwise uniqueness in equation (1), we refer the reader to [24, 25]. Nevertheless, note that pathwise uniqueness for parabolic stochastic partial differential equations is an active research topic. Indeed, we mention, for instance, the works [19] (Lipschitz coefficients), [42, 41] (Hölder coefficients), [13, 14] (additive noise), where this question is investigated. These results provide examples of parabolic stochastic partial differential equations where assumption (PU) is fulfilled. In order to prove the main result of the section (cf Theorem 3.1), we will follow a similar approach as in [20] (see also [44]). More precisely, we will first use the re- sults from Section 2 to deduce that the family of laws determined by uM,N are tight in the space of continuous functions. Then, we will apply Skorokhod’s representation theorem and make use of the weak form (18) corresponding to the fully discrete ap- proximation uM,N . Finally, a suitable passage to the limit and assumption (PU) will let us conclude the proof. We will use the above strategy in a successful way thanks the following two auxil- iary results. Lemma 3.1 (Lemma 4.5 in [20]). For all k ≥ 0, let zk = {zk (t, x) : t ≥ 0, x ∈ [0, 1]} be a continuous Ftk -adapted random field and let W k = {W k (t, x) : t ≥ 0, x ∈ [0, 1]} be a Brownian sheet carried by some filtered probability space (Ω, F , (Ftk )t≥0 , P). Assume also that, for every ε > 0 ( ) lim P sup sup (|zk − z0 | + |W k −W 0 |)(t, x) ≥ ε = 0. k→∞ t∈[0,T ] x∈[0,1] Let h = h(t, x, r) be a bounded Borel function of (t, x, r) ∈ R+ × [0, 1] × R, which is continuous in r ∈ R. Then, letting k → ∞, ∫ t∫ 1 ∫ t∫ 1 h(s, x, zk (s, x)) dx ds −→ h(s, x, z0 (s, x)) dx ds, 0 0 0 0 ∫ t∫ 1 ∫ t∫ 1 h(s, x, zk (s, x))W k (ds, dx) −→ h(s, x, z0 (s, x))W 0 (ds, dx), 0 0 0 0 in probability for every t ∈ [0, T ]. Lemma 3.2 (Lemma 4.4 in [20]). Let E be a Polish space equipped with the Borel σ -algebra. A sequence of E-valued random elements (zn )n≥1 converges in probability if and only if, for every pair of subsequences zl := znl and zm := znm , there exists a subsequence vk := (zlk , zmk ) converging weakly to a random element v supported on the diagonal {(x, y) ∈ E × E : x = y}. 30 We are now ready to state and prove the main result of this section. Theorem 3.1. Assume that the coefficients f and σ satisfy condition (LG), and that hy- potheses (PU) and (C) are fulfilled. Then, there exists a random field u = {u(t, x) : t ≥ 0, x ∈ [0, 1]} such that, for every ε > 0, ( ) P sup sup |uMk ,Nk (t, x) − u(t, x)| ≥ ε → 0, t∈[0,T ] x∈[0,1] as k tends to infinity, for all sequences of positive integers (Mk , Nk )k≥1 such that Mk , Nk → ∞, as k → ∞, where we recall that uM,N denotes the fully discrete solution (22). Fur- thermore, the random field u is the unique solution to the stochastic heat equation (1). Proof. We first show that the sequence (uM,N )M,N≥1 defines a tight family of laws in the space C([0, T ] × [0, 1]). To do so, we invoke part 2 in Proposition 2.5 on the regularity of the numerical solution and we apply the tightness criterion on the plane [4, Thm. 2.2], which generalizes a well-known result of Billingsley. Furthermore, Prokhorov’s theorem implies that the sequence of laws (uM,N )M,N≥1 is relatively com- pact in C([0, T ] × [0, 1]). Fix any pair of sequences (Mk , Nk )k≥1 such that Mk , Nk → ∞, as k → ∞. Then, the laws of vk := uMk ,Nk , k ≥ 1, form a tight family in the space C([0, T ] × [0, 1]). Let now (v1j ) j≥1 and (v2ℓ )ℓ≥1 be two subsequences of (vk )k≥1 . By Skorokhod’s Rep- resentation Theorem, there exist subsequences of positive integers ( jr )r≥1 and (ℓr )r≥1 b c c b of the indices j and ℓ, a probability space ( (Ω, F , ()Ft )t≥1 , P), and a sequence of con- br , r ≥ 1, such that tinuous random fields (zr )r≥1 with zr := uer , ur , W 1. zr −→ z := (e b ) a.s. in C([0, T ] × [0, 1], R3 ), where the random field z is u, u, W r→∞ b F defined on (Ω, c, (F ct )t≥1 , P), b W b is a Brownian sheet defined on this basis, and c Ft = σ (z(s, x), (s, x) ∈ [0,t] × [0, 1]) (and conveniently completed). 2. For every r ≥ 1, the finite ( dimensional) distributions of zr coincide with those of the random field ζr := v1jr , v2ℓr ,W , and thus law(zr ) = law(ζr ) for all r ≥ 1. Note that W br is a Brownian sheet defined on (Ω, c, (F b F ctr )t≥1 , P), ctr = b where F σ (zr (s, x), (s, x) ∈ [0,t] × [0, 1]) (and conveniently completed). We now fix (t, x) ∈ [0, T ] × [0, 1]. Since the laws of zr and ζr coincide and the first two components of ζr satisfy the weak form (18), so do the components of zr . Namely, 31 for all Φ ∈ C∞ (R2 ) with Φ(t, 0) = Φ(t, 1) = 0 for all t, it holds ∫ 1 ∫ 1 uer (t, κM (y))Φ(t, y) dy = u0 (κM (y))Φ(t, y) dy 0 0 ∫ t∫ 1 ( ) ∂Φ + uer (s, κM (y)) ∆M Φ(s, y) + (s, y) dy ds 0 0 ∂s ∫ t∫ 1 + ur (κNT (s), κM (y)))Φ(s, y) dy ds f (e 0 0 ∫ t∫ 1 + σ (e ur (κNT (s), κM (y)))Φ(s, y)W (ds, dy), b P-a.s., 0 0 (26) for all t ∈ [0, T ], and also ∫ 1 ∫ 1 ur (t, κM (y))Φ(t, y) dy = u0 (κM (y))Φ(t, y) dy 0 0 ∫ t∫ 1 ( ) ∂Φ + ur (s, κM (y)) ∆M Φ(s, y) + (s, y) dy ds 0 0 ∂s ∫ t∫ 1 + f (ur (κNT (s), κM (y)))Φ(s, y) dy ds 0 0 ∫ t∫ 1 + σ (ur (κNT (s), κM (y)))Φ(s, y)W (ds, dy), b P-a.s., 0 0 (27) for all t ∈ [0, T ]. We recall that ∆M denotes the discrete Laplacian, which is defined by ∆M Φ(s, y) := (∆x)−2 {Φ(s, y + ∆x) − 2Φ(s, y) + Φ(s, y − ∆x)} , where we remind that ∆x = M1 . Taking r → ∞ in the above formulas (26) and (27), and using Lemma 3.1, we show that the random fields ũ and ū are solutions of (2), and hence of equation (1), on the same stochastic basis (Ω, b F c, (Fct )t≥1 , P). b Thus, by the pathwise uniqueness b assumption, we obtain that ue(t, x) = u(t, x) for all (t, x) ∈ [0, T ] × [0, 1] P-a.s. Hence, by Lemma 3.2, we get that {u M k ,Nk }k≥1 converges in probability to u, uniformly on [0, T ] × [0, 1], the solution of the stochastic heat equation (1). 4 Acknowledgement L. Quer-Sardanyons’ research is supported by grants 2014SGR422 and MTM2015- 67802-P. This work was partially supported by the Swedish Research Council (VR) (project nr. 2013 − 4562). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N, Umeå Univer- sity. 32 References [1] E. J. Allen, S. J. Novosel, and Z. Zhang. Finite element and difference approxima- tion of some linear stochastic partial differential equations. Stochastics Stochas- tics Rep., 64(1-2):117–142, 1998. [2] R. Anton and D. Cohen. Exponential integrators for stochastic Schrödinger equa- tions driven by Itô noise. To appear in the special issue on SPDEs of J. Comput. Math, 2017. [3] R. Anton, D. Cohen, S. Larsson, and X. Wang. Full discretization of semilinear stochastic wave equations driven by multiplicative noise. SIAM J. Numer. Anal., 54(2):1093–1119, 2016. [4] X. Bardina, M. Jolis, and L. Quer-Sardanyons. Weak convergence for the stochas- tic heat equation driven by Gaussian white noise. Electron. J. Probab., 15:no. 39, 1267–1295, 2010. [5] A. Barth and A. Lang. Simulation of stochastic partial differential equations using finite element methods. Stochastics, 84(2-3):217–231, 2012. [6] A. Barth and A. Lang. L p and almost sure convergence of a Milstein scheme for stochastic partial differential equations. Stochastic Process. Appl., 123(5):1563– 1587, 2013. [7] S. Becker, A. Jentzen, and P. E. Kloeden. An exponential Wagner-Platen type scheme for SPDEs. SIAM J. Numer. Anal., 54(4):2389–2426, 2016. [8] D. Cohen and G. Dujardin. Exponential integrators for nonlinear schrödinger equations with white noise dispersion. Stochastics and Partial Differential Equa- tions: Analysis and Computations, pages 1–22, 2017. [9] D. Cohen, S. Larsson, and M. Sigg. A trigonometric method for the linear stochastic wave equation. SIAM J. Numer. Anal., 51(1):204–222, 2013. [10] D. Cohen and L. Quer-Sardanyons. A fully discrete approximation of the one- dimensional stochastic wave equation. IMA J. Numer. Anal., 36(1):400–420, 2016. [11] S. Cox and J. van Neerven. Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems. SIAM J. Numer. Anal., 48(2):428– 451, 2010. [12] S. Cox and J. van Neerven. Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise. Numer. Math., 125(2):259–345, 2013. [13] G. Da Prato, F. Flandoli, E. Priola, and M. Röckner. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded mea- surable drift. Ann. Probab., 41(5):3306–3344, 2013. 33 [14] G. Da Prato, F. Flandoli, E. Priola, and M. Röckner. Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theo- ret. Probab., 28(4):1571–1600, 2015. [15] A. M. Davie and J. G. Gaines. Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comp., 70(233):121– 134, 2001. [16] E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012. [17] Q. Du and T. Zhang. Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal., 40(4):1421–1445, 2002. [18] J. G. Gaines. Numerical experiments with S(P)DE’s. In Stochastic partial differ- ential equations (Edinburgh, 1994), volume 216 of London Math. Soc. Lecture Note Ser., pages 55–71. Cambridge Univ. Press, Cambridge, 1995. [19] I. Gyöngy. Existence and uniqueness results for semilinear stochastic partial dif- ferential equations. Stochastic Process. Appl., 73(2):271–299, 1998. [20] I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I. Potential Anal., 9(1):1– 25, 1998. [21] I. Gyöngy. Lattice approximations for stochastic quasi-linear parabolic partial dif- ferential equations driven by space-time white noise II. Potential Anal., 11(1):1– 37, 1999. [22] I. Gyöngy and D. Nualart. Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise. Stochastic Process. Appl., 58(1):57–72, 1995. [23] I. Gyöngy and D. Nualart. Implicit scheme for stochastic parabolic partial differ- ential equations driven by space-time white noise. Potential Anal., 7(4):725–757, 1997. [24] I. Gyöngy and É. Pardoux. On quasi-linear stochastic partial differential equa- tions. Probab. Theory Related Fields, 94(4):413–425, 1993. [25] I. Gyöngy and É. Pardoux. Weak and strong solutions of white-noise driven parabolic spdes. Prepub. Laboratoire de Mathématiques Marseille, 92-22, 1993. [26] E. Hausenblas. Approximation for semilinear stochastic evolution equations. Po- tential Anal., 18(2):141–186, 2003. [27] M. Hochbruck and A. Ostermann. Exponential integrators. Acta Numer., 19:209– 286, 2010. 34 [28] A. Jentzen. Pathwise numerical approximation of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal., 31(4):375–404, 2009. [29] A. Jentzen. Higher order pathwise numerical approximations of SPDEs with additive noise. SIAM J. Numer. Anal., 49(2):642–667, 2011. [30] A. Jentzen, P. Kloeden, and G. Winkel. Efficient simulation of nonlinear parabolic SPDEs with additive noise. Ann. Appl. Probab., 21(3):908–950, 2011. [31] A. Jentzen and P. E. Kloeden. Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465(2102):649–667, 2009. [32] A. Jentzen and P. E. Kloeden. Taylor approximations for stochastic partial differ- ential equations, volume 83 of CBMS-NSF Regional Conference Series in Ap- plied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. [33] A. Jentzen and M. Röckner. A Milstein scheme for SPDEs. Found. Comput. Math., 15(2):313–362, 2015. [34] P. E. Kloeden, G. J. Lord, A. Neuenkirch, and T. Shardlow. The exponential inte- grator scheme for stochastic partial differential equations: pathwise error bounds. J. Comput. Appl. Math., 235(5):1245–1260, 2011. [35] R. Kruse. Strong and weak approximation of semilinear stochastic evolution equations, volume 2093 of Lecture Notes in Mathematics. Springer, Cham, 2014. [36] A. Lang, A. Petersson, and A. Thalhammer. Mean-square stability analysis of ap- proximations of stochastic differential equations in infinite dimensions. to appear in BIT Num. Math, 2017. [37] G. J. Lord, C. E. Powell, and T. Shardlow. An introduction to computational stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge Univer- sity Press, New York, 2014. [38] G. J. Lord and A. Tambue. Stochastic exponential integrators for the finite ele- ment discretization of SPDEs for multiplicative and additive noise. IMA J. Numer. Anal., 33(2):515–543, 2013. [39] A. Millet and P. Morien. On implicit and explicit discretization schemes for parabolic SPDEs in any dimension. Stochastic Process. Appl., 115(7):1073–1106, 2005. [40] T. Müller-Gronbach and K. Ritter. An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise. BIT, 47(2):393– 418, 2007. 35 [41] L. Mytnik and E. Neuman. Pathwise uniqueness for the stochastic heat equation with Hölder continuous drift and noise coefficients. Stochastic Process. Appl., 125(9):3355–3372, 2015. [42] L. Mytnik and E. Perkins. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. Probab. Theory Related Fields, 149(1-2):1–96, 2011. [43] B. G. Pachpatte. Integral and finite difference inequalities and applications, vol- ume 205. Elsevier Science B.V., 2006. [44] R. Pettersson and M. Signahl. Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal., 22(4):375–393, 2005. [45] J. Printems. On the discretization in time of parabolic stochastic partial differen- tial equations. M2AN Math. Model. Numer. Anal., 35(6):1055–1078, 2001. [46] L. Quer-Sardanyons and M. Sanz-Solé. Space semi-discretisations for a stochas- tic wave equation. Potential Anal., 24(4):303–332, 2006. [47] M. Sauer and W. Stannat. Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition. Math. Comp., 84(292):743–766, 2015. [48] T. Shardlow. Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim., 20(1-2):121–145, 1999. [49] C. Ta, D. Wang, and Q. Nie. An integration factor method for stochastic and stiff reaction-diffusion systems. J. Comput. Phys., 295:505–522, 2015. [50] H. Triebel. Theory of function spaces II, volume 84 of Monographs in Mathemat- ics. Birkhäuser Verlag, Basel, 1992. [51] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pages 265–439. Springer, Berlin, 1986. [52] J. B. Walsh. Finite element methods for parabolic stochastic PDE’s. Potential Anal., 23(1):1–43, 2005. [53] X. Wang. Strong convergence rates of the linear implicit euler method for the finite element discretization of spdes with additive noise. IMA Journal of Numer- ical Analysis, 37(2):965, 2017. [54] X. Wang and S. Gan. A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise. Numer. Algorithms, 62(2):193–223, 2013. [55] Y. Yan. Galerkin finite element methods for stochastic parabolic partial differen- tial equations. SIAM J. Numer. Anal., 43(4):1363–1384 (electronic), 2005. 36