Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

TL;DR

This paper presents a novel recursive multilevel Picard method that effectively addresses the curse of dimensionality in high-dimensional Allen-Cahn PDEs, enabling more efficient numerical solutions for complex reaction-diffusion equations.

Contribution

The authors develop and analyze truncated full-history recursive multilevel Picard schemes to overcome computational challenges in high-dimensional nonlinear PDEs like Allen-Cahn equations.

Findings

Successfully reduces computational complexity in high dimensions

Demonstrates convergence of the proposed schemes

Applicable to reaction-diffusion PDEs with Lipschitz nonlinearities

Abstract

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen-Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

Equations

$$(\frac{\partial }{\partial t}{u}_d)(t,x)=({\Delta }_x{u}_d)(t,x)+f(u_d(t,x)),$$

$$\begin{split}&U^{d,\theta}_{n,M}(t,x)=\sum_{k=1}^{n-1}\frac{t}{M^{n-k}}\Bigg{[}\sum_{m=1}^{M^{n-k}}\bigg{(}\mathbf{f}_{M}\Big{(}U^{d,(\theta,k,m)}_{k,M}\big{(}R^{(\theta,k,m)}_{t},X^{d,(\theta,k,m)}_{R^{(\theta,k,m)}_{t},t,x}\big{)}\Big{)}\\ &-\mathbf{f}_{M}\Big{(}U^{d,(\theta,-k,m)}_{k-1,M}\big{(}R^{(\theta,k,m)}_{t},X^{d,(\theta,k,m)}_{R^{(\theta,k,m)}_{t},t,x}\big{)}\Big{)}\bigg{)}\Bigg{]}+\frac{1}{M^{n}}\!\left[\sum_{m=1}^{M^{n}}\left(u_{d}(0,X^{d,(\theta,0,-m)}_{0,t,x})+t\,f(0)\right)\right],\end{split}$$

$$\sup_{x\in\mathbb{R}^{d}}\left(\mathbb{E}\!\left[\big{|}U^{d,0}_{\mathfrak{N}_{\varepsilon},\mathfrak{N}_{\varepsilon}}(T,x)-u_{d}(T,x)\big{|}^{2}\right]\right)^{\!\nicefrac{{1}}{{2}}}\leq\varepsilon.$$

$$(\frac{\partial }{\partial t}v)(t,x)\le ({\Delta }_{x}v)(t,x),$$

$$\underset{a\in \mathbb{R}}{\inf }\underset{(t,x)\in [0,T]\times {\mathbb{R}}^d}{\sup }(e^{a\Vert x{\Vert }^2}v(t,x))<\infty .$$

$$\underset{(t,x)\in [0,T]\times {\mathbb{R}}^d}{\sup }v(t,x)=\underset{x\in {\mathbb{R}}^d}{\sup }v(0,x).$$

$$\underset{x\in {\mathbb{R}}^d}{\sup }v(0,x)<\infty ,$$

$${\Phi }_{\epsilon }(t,x)=[4\pi (T+\epsilon -t){]}^{-d/2}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t)}\right),$$

$$w_{\epsilon ,M}(t,x)=v(t,x)-M{\Phi }_{\epsilon }(t,x)-\epsilon t.$$

$$({\nabla }_x{\Phi }_{\epsilon })(t,x)={\Phi }_{\epsilon }(t,x)\left[\frac{x}{2(T+\epsilon -t)}\right].$$

$$\begin{array}{rl}(\frac{{\partial }^2}{\partial {x_k}^2}{\Phi }_{\epsilon })(t,x)& =(\frac{\partial }{\partial x_k}{\Phi }_{\epsilon })(t,x)\left[\frac{x_k}{2(T+\epsilon -t)}\right]+{\Phi }_{\epsilon }(t,x)\left[\frac{1}{2(T+\epsilon -t)}\right]\\ & ={\Phi }_{\epsilon }(t,x){\left[\frac{x_k}{2(T+\epsilon -t)}\right]}^2+{\Phi }_{\epsilon }(t,x)\left[\frac{1}{2(T+\epsilon -t)}\right]\\ & ={\Phi }_{\epsilon }(t,x)\left(\frac{|x_k{|}^2}{4(T+\epsilon -t{)}^2}+\frac{1}{2(T+\epsilon -t)}\right).\end{array}$$

$$\begin{array}{rl}({\Delta }_x{\Phi }_{\epsilon })(t,x)& =\sum _{k=1}^d(\frac{{\partial }^2}{\partial {x_k}^2}{\Phi }_{\epsilon })(t,x)\\ & =\sum _{k=1}^d\left[{\Phi }_{\epsilon }(t,x)\left(\frac{|x_k{|}^2}{4(T+\epsilon -t{)}^2}+\frac{1}{2(T+\epsilon -t)}\right)\right]\\ & ={\Phi }_{\epsilon }(t,x)\left(\frac{{\Vert x\Vert }^2}{4(T+\epsilon -t{)}^2}+\frac{d}{2(T+\epsilon -t)}\right).\end{array}$$

$$\begin{array}{rl}& (\frac{\partial }{\partial t}{\Phi }_{\epsilon })(t,x)=-\frac{d}{2}[4\pi (T+\epsilon -t){]}^{-d/2-1}[-4\pi ]\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t)}\right)\\ & +[4\pi (T+\epsilon -t){]}^{-d/2}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t)}\right)\left[\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t{)}^2}\right]\\ & =[4\pi (T+\epsilon -t){]}^{-d/2}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t)}\right)\left[-\frac{d}{2}\left(\frac{-4\pi }{4\pi (T+\epsilon -t)}\right)+\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t{)}^2}\right]\\ & =\left(\frac{d}{2(T+\epsilon -t)}+\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t{)}^2}\right){\Phi }_{\epsilon }(t,x).\end{array}$$

$$(\frac{\partial }{\partial t}{\Phi }_{\epsilon })(t,x)=({\Delta }_x{\Phi }_{\epsilon })(t,x).$$

$$\begin{array}{rl}(\frac{\partial }{\partial t}{w}_{\epsilon ,M})(t,x)& =(\frac{\partial }{\partial t}v)(t,x)-M(\frac{\partial }{\partial t}{\Phi }_{\epsilon })(t,x)-\epsilon \\ & =(\frac{\partial }{\partial t}v)(t,x)-M({\Delta }_x{\Phi }_{\epsilon })(t,x)-\epsilon \\ & \le ({\Delta }_{x}v)(t,x)-M({\Delta }_x{\Phi }_{\epsilon })(t,x)-\epsilon =({\Delta }_x{w}_{\epsilon ,M})(t,x)-\epsilon .\end{array}$$

$$v(t,x)\le C{e}^{a\Vert x{\Vert }^2}.$$

$$\begin{array}{rl}{w}_{\epsilon ,M}(t,x)& \le v(t,x)-M{\Phi }_{\epsilon }(t,x)\\ & =v(t,x)-\frac{M}{[4\pi (T+\epsilon -t){]}^{d/2}}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon -t)}\right)\\ & \le v(t,x)-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon )}\right)\\ & \le C{e}^{a\Vert x{\Vert }^2}-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(\frac{\Vert x{\Vert }^2}{4(T+\epsilon )}\right)\\ & =e^{a\Vert x{\Vert }^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(\Vert x{\Vert }^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right].\end{array}$$

$$\min \left\{0,\left[\underset{s\in [0,T]}{\inf }v(s,0)\right]-\epsilon T-\frac{M}{(4\pi \epsilon {)}^{d/2}}\right\}\in \mathbb{R}.$$

$$\begin{array}{rl}& e^{a|R_{\epsilon ,M}{|}^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(|R_{\epsilon ,M}{|}^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right]\\ & <\min \left\{0,\left[\underset{s\in [0,T]}{\inf }v(s,0)\right]-\epsilon T-\frac{M}{(4\pi \epsilon {)}^{d/2}}\right\}.\end{array}$$

$$\begin{array}{rl}& e^{a|R_{\epsilon ,M}{|}^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(|R_{\epsilon ,M}{|}^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right]\\ & <\left[\underset{s\in [0,T]}{\inf }v(s,0)\right]-\epsilon T-\frac{M}{(4\pi \epsilon {)}^{d/2}}\le \left[\underset{s\in [0,T]}{\inf }v(s,0)\right]-\epsilon t-\frac{M}{(4\pi \epsilon {)}^{d/2}}\\ & \le \left[\underset{s\in [0,T]}{\inf }v(s,0)\right]-\epsilon t-\frac{M}{[4\pi (T+\epsilon -t){]}^{d/2}}\le v(t,0)-\epsilon t-\frac{M}{[4\pi (T+\epsilon -t){]}^{d/2}}\\ & =v(t,0)-\epsilon t-M{\Phi }_{\epsilon }(t,0)=w_{\epsilon ,M}(t,0)\le \underset{\begin{array}{c}(s,x)\in [0,T]\times {\mathbb{R}}^d,\\ \Vert x\Vert \le R_{\epsilon ,M}\end{array}}{\sup }{w}_{\epsilon ,M}(s,x).\end{array}$$

$$\begin{array}{rl}{w}_{\epsilon ,M}(t,x)& \le e^{a\Vert x{\Vert }^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(\Vert x{\Vert }^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right]\\ & \le e^{a\Vert x{\Vert }^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(|R_{\epsilon ,M}{|}^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right]\\ & \le e^{a|R_{\epsilon ,M}{|}^2}\left[C-\frac{M}{[4\pi (T+\epsilon ){]}^{d/2}}\exp \left(|R_{\epsilon ,M}{|}^2\left[\frac{1}{4(T+\epsilon )}-a\right]\right)\right]\\ & \le \underset{\begin{array}{c}(s,y)\in [0,T]\times {\mathbb{R}}^d,\\ \Vert y\Vert \le R_{\epsilon ,M}\end{array}}{\sup }{w}_{\epsilon ,M}(s,y).\end{array}$$

$$\underset{(t,x)\in [0,T]\times {\mathbb{R}}^d}{\sup }{w}_{\epsilon ,M}(t,x)=\underset{\begin{array}{c}(t,x)\in [0,T]\times {\mathbb{R}}^d,\\ \Vert x\Vert \le R_{\epsilon ,M}\end{array}}{\sup }{w}_{\epsilon ,M}(t,x).$$

$$\underset{(t,x)\in [0,T]\times {\mathbb{R}}^d}{\sup }{w}_{\epsilon ,M}(t,x)=w_{\epsilon ,M}(t_{\epsilon ,M},x_{\epsilon ,M}).$$

$$(\tfrac{\partial}{\partial t}w_{\varepsilon,M})(t_{\varepsilon,M},x_{\varepsilon,M})=0\qquad\text{and}\qquad\big{(}(\tfrac{\partial^{2}}{\partial{x}^{2}}w_{\varepsilon,M})(t_{\varepsilon,M},x_{\varepsilon,M})\big{)}(v,v)\leq 0.$$

$$(\frac{\partial }{\partial t}{w}_{\epsilon ,M})(t_{\epsilon ,M},x_{\epsilon ,M})\ge 0$$

$$({\Delta }_x{w}_{\epsilon ,M})(t_{\epsilon ,M},x_{\epsilon ,M})=\sum _{k=1}^d(\frac{{\partial }^2}{\partial {x_k}^2}{w}_{\epsilon ,M})(t_{\epsilon ,M},x_{\epsilon ,M})\le 0.$$

$$0\le (\frac{\partial }{\partial t}{w}_{\epsilon ,M})(t_{\epsilon ,M},x_{\epsilon ,M})\le ({\Delta }_x{w}_{\epsilon ,M})(t_{\epsilon ,M},x_{\epsilon ,M})-\epsilon \le -\epsilon <0.$$

$$\underset{(t,x)\in [0,T]\times {\mathbb{R}}^d}{\sup }{w}_{\epsilon ,M}(t,x)=w_{\epsilon ,M}(t_{\epsilon ,M},x_{\epsilon ,M})=w_{\epsilon ,M}(0,x_{\epsilon ,M})\le \underset{x\in {\mathbb{R}}^d}{\sup }{w}_{\epsilon ,M}(0,x).$$

$$\begin{array}{rl}v(t,x)& =w_{\epsilon ,M}(t,x)+M{\Phi }_{\epsilon }(t,x)+\epsilon t\le \left[\underset{y\in {\mathbb{R}}^d}{\sup }{w}_{\epsilon ,M}(0,y)\right]+M{\Phi }_{\epsilon }(t,x)+\epsilon t\\ & \le \left[\underset{y\in {\mathbb{R}}^d}{\sup }v(0,y)\right]+M{\Phi }_{\epsilon }(t,x)+\epsilon t.\end{array}$$

$$\begin{array}{rl}v(t,x)& \le \underset{M\searrow 0}{\mathrm{liminf}}\left(\left[\underset{y\in {\mathbb{R}}^d}{\sup }v(0,y)\right]+M{\Phi }_{\epsilon }(t,x)+\epsilon t\right)=\left[\underset{y\in {\mathbb{R}}^d}{\sup }v(0,y)\right]+\epsilon t.\end{array}$$

Full text

1. 1 Introduction
2. 2 A priori bounds for reaction-diffusion equations with coercive nonlinearity
3. 3 Truncated full-history recursive multilevel Picard (MLP) approximations
4. 4 Computational cost analysis for truncated MLP approximations
5. 5 MLP approximations for Allen–Cahn type partial differential equations

1 Introduction

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. Linear parabolic PDEs of second order can be solved approximately without the curse of dimensionality by means of Monte Carlo averages. In the last few years, several probabilistic approximation methods, which seem in certain situations to be capable of efficiently approximating high-dimensional nonlinear PDEs, have been proposed. For instance, the articles [6, 18, 20, 21] propose and study approximation methods based on stochastic representations of solutions of PDEs by means of branching diffusion processes (cf., for example, [32, 35, 37] for theoretical relations and cf., for example, [36] for a related method), the articles [1, 2, 3, 4, 5, 8, 9, 12, 13, 14, 15, 16, 17, 19, 22, 27, 29, 30, 31, 33, 34] propose and study approximation methods based on the reformulation of PDEs as stochastic learning problems involving deep artificial neural networks, and the articles [10, 11, 23, 24, 25, 26] propose and study full-history recursive multilevel Picard (MLP) approximation methods. In particular, the articles [24, 25] prove that MLP approximation schemes do indeed overcome the curse of dimensionality in the numerical approximation of semilinear parabolic PDEs. More formally, Theorem 3.8 in [24] shows that MLP approximation schemes are able to approximate the solutions of semilinear parabolic PDEs with a root mean square error of size $\varepsilon\in(0,\infty)$ and a computational effort which grows at most polynomially both in the dimension as well as in the reciprocal $\nicefrac{{1}}{{\varepsilon}}$ of the desired approximation accuracy. However, the articles [24, 25] are only applicable in the case where the nonlinearity is globally Lipschitz continuous and, to the best of our knowledge, there exists no result in the scientific literature which shows for every $T\in(0,\infty)$ that the solution of a semilinear parabolic PDE with a non-globally Lipschitz continuous nonlinearity can be efficiently approximated at time $T$ without the curse of dimensionality.

In this work we overcome this difficulty by introducing a truncated variant of the MLP approximation schemes introduced in [10, 24] and by proving that this truncated MLP approximation scheme succeeds in approximately solving reaction-diffusion type PDEs with a locally Lipschitz continuous coercive nonlinearity (such as Allen–Cahn type PDEs) without the curse of dimensionality. More specifically, Theorem 4.5 in Section 3 below, which is the main result of this article, proves under suitable assumptions that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that the proposed truncated MLP approximations can achieve a root mean square error of size at most $\varepsilon$ with a computational effort of order $d\varepsilon^{-(2+\delta)}$. To illustrate the findings of this article in more detail, we now present in Theorem 1.1 below a special case of Theorem 4.5.

Theorem 1.1**.**

Let $\delta,\kappa,T\in(0,\infty)$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $f\in C^{1}(\mathbb{R},\mathbb{R})$, $(\mathbf{f}_{d})_{d\in\mathbb{N}}\subseteq C(\mathbb{R},\mathbb{R})$, $(u_{d})_{d\in\mathbb{N}}\subseteq C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, assume that $f^{\prime}$ is at most polynomially growing, assume for every $d\in\mathbb{N}$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$, $v\in\mathbb{R}$ that $vf(v)\leq\kappa(1+v^{2})$, $|u_{d}(0,x)|\leq\kappa$, $u_{d}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, $\inf_{c\in\mathbb{R}}(\sup_{s\in[0,T]}\sup_{y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}}(e^{c(|y_{1}|^{2}+\ldots+|y_{d}|^{2})}|u_{d}(s,y)|))<\infty$, $\mathbf{f}_{d}(v)=f(\min\{\ln(1+\ln(d)),\max\{-\ln(1+\ln(d)),v\}\})$, and

[TABLE]

let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{d,\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $d\in\mathbb{N}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{d,\theta})_{(d,\theta)\in\mathbb{N}\times\Theta}$ are independent, let $R^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $d\in\mathbb{N}$, $s\in[0,T]$, $t\in[s,T]$, $x\in\mathbb{R}^{d}$, $\theta\in\Theta$ let $X^{d,\theta}_{s,t,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{d,\theta}_{s,t,x}=x+\sqrt{2}(W^{d,\theta}_{t}-W^{d,\theta}_{s})$, let $U^{d,\theta}_{n,M}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{d,\theta}_{0,M}(t,x)=0$ and

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and for every $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$ let $\mathfrak{C}_{d,n,M}\in\mathbb{N}_{0}$ be the number of realizations of scalar standard normal random variables which are used to compute one realization of $U^{d,0}_{n,M}(T,0)\colon\Omega\to\mathbb{R}$ (cf. 5.2 for a precise definition). Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $c\in\mathbb{R}$ such that for every $d\in\mathbb{N}$, $\varepsilon\in(0,1]$ it holds that $\mathfrak{C}_{d,\mathfrak{N}_{\varepsilon},\mathfrak{N}_{\varepsilon}}\leq cd\varepsilon^{-(2+\delta)}$ and

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Theorem 1.1 above is an immediate consequence of 5.2 in Section 5 below. 5.2 follows from 5.1 which, in turn, is deduced from Theorem 4.5, the main result of this article. Theorem 1.1 establishes under suitable assumptions that for every $\delta\in(0,\infty)$ there exists $c\in(0,\infty)$ such that for every $d\in\mathbb{N}$ the solution $u_{d}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ of the reaction-diffusion type partial differential equation in (1) can be approximated by the MLP approximation scheme in (2) with a root mean square error of size $\varepsilon\in(0,\infty)$ while the computational effort is bounded by $cd\varepsilon^{-(2+\delta)}$. The numbers $\mathfrak{C}_{d,n,M}$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, in Theorem 1.1 model the computational effort. The nonlinearity $f\colon\mathbb{R}\to\mathbb{R}$ in Theorem 1.1 is required to be locally Lipschitz continuous (which follows from the hypothesis in Theorem 1.1 that $f^{\prime}$ is continuous) and to satisfy a coercivity type condition in the sense that there exists $\kappa\in\mathbb{R}$ such that for all $v\in\mathbb{R}$ it holds that $vf(v)\leq\kappa(1+v^{2})$. This coercivity type condition together with the growth assumption on the solutions $u_{d}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$, $d\in\mathbb{N}$, allows us to deduce in Section 2 that the solutions $u_{d}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$, $d\in\mathbb{N}$, are uniformly bounded. In particular, 2.4 in Section 2 yields that there exists $\mathfrak{M}\in\mathbb{N}$ such that for every $M\in[\mathfrak{M},\infty)\cap\mathbb{N}$, $d\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $(\tfrac{\partial}{\partial t}u_{d})(t,x)=(\Delta_{x}u_{d})(t,x)+\mathbf{f}_{M}(u_{d}(t,x))$. The fact that for every $d,M\in\mathbb{N}$ it holds that $(\tfrac{\partial}{\partial t}u_{d})(t,x)=(\Delta_{x}u_{d})(t,x)+\mathbf{f}_{M}(u_{d}(t,x))$, $(t,x)\in[0,T]\times\mathbb{R}^{d}$, is a parabolic PDE with a globally Lipschitz continuous nonlinearity then permits us to bring the machinery from [24] into play. This will finally allow us to prove Theorem 1.1 (see Sections 2 and 3 for details). We note that although Theorem 1.1 uses the assumption that the nonlinearity $f\colon\mathbb{R}\to\mathbb{R}$ satisfies the coercivity type condition that there exists $\kappa\in\mathbb{R}$ such that for all $v\in\mathbb{R}$ it holds that $vf(v)\leq\kappa(1+v^{2})$, explicit knowledge of the coercivity constant $\kappa$ is not required for the implementation of the MLP approximation scheme.

The remainder of this article is organized as follows. In Section 2 we present elementary a priori bounds for classical solutions of reaction-diffusion type PDEs with coercive nonlinearities. In Section 3 we introduce truncated MLP approximation schemes and we provide upper bounds for the root mean square distance between the truncated MLP approximations and the exact solution of the PDE under consideration. In Section 4 we combine the error estimates from Section 3 with estimates for the computational effort for truncated MLP approximations to show under suitable assumptions that for every $\delta\in(0,\infty)$ a root mean square error of size $\varepsilon\in(0,1]$ can be achieved by truncated MLP approximations with a computational effort of order $d\varepsilon^{-(2+\delta)}$. In Section 5 we specialize our findings to Allen–Cahn type PDEs.

2 A priori bounds for reaction-diffusion equations with coercive nonlinearity

For convenience of the reader, we recall the following well-known maximum principle for subsolutions of the heat equation (cf., e.g., John [28, Pages 216–217 in Section 1 in Chapter 7]).

Lemma 2.1**.**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $v\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}),$ assume that $v|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R}),$ assume for every $t\in(0,T],$ $x\in\mathbb{R}^{d}$ that

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let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be the d-dimensional Euclidean norm, and assume that

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Then it holds that

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Proof of Lemma 2.1.

Throughout this proof assume w.l.o.g. that

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let $\Phi_{\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R},$ $\varepsilon\in(0,\infty),$ be the functions which satisfy for every $\varepsilon\in(0,\infty),$ $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

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and let $w_{\varepsilon,M}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R},$ $\varepsilon,M\in(0,\infty),$ be the functions which satisfy for every $\varepsilon,M\in(0,\infty),$ $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

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Observe that for every $\varepsilon\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

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This implies that for every $\varepsilon\in(0,\infty)$, $t\in[0,T]$, $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ it holds that

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Therefore, we obtain that for every $\varepsilon\in(0,\infty)$, $t\in[0,T]$, $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ it holds that

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Moreover, observe that for every $\varepsilon\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

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Combining this with (12) ensures that for every $\varepsilon\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

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This, (4), and (9) imply that for every $\varepsilon,M\in(0,\infty)$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$ it holds that

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In addition, observe that (5) ensures that there exist $C\in[0,\infty)$ and $a\in(0,\infty)$ such that for every $t\in[0,T],$ $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

To prove (6) we distinguish between the case $T<\frac{1}{4a}$ and the case $T\geq\frac{1}{4a}$. We first prove (6) in the case $T<\frac{1}{4a}$. Observe that (8), (9) and (16) imply that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty),$ $t\in[0,T],$ $x\in\mathbb{R}^{d}$ it holds that

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Furthermore, observe that the hypothesis that $v\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$ and the fact that the interval $[0,T]$ is compact ensure that $\inf_{s\in[0,T]}v(s,0)\in\mathbb{R}.$ Hence, we obtain that for every $\varepsilon,M\in(0,\infty)$ it holds that

[TABLE]

This and the fact that for every $\varepsilon\in(0,\frac{1}{4a}-T)$ it holds that $a<\frac{1}{4(T+\varepsilon)}$ imply that there exists a function $R=(R_{\varepsilon,M})_{(\varepsilon,M)\in(0,\infty)^{2}}\colon(0,\infty)^{2}\to(0,\infty)$ such that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that

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Combining this with (8) and (9) proves that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty),$ $t\in[0,T]$ it holds that

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This, (17), and (19) ensure that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty),$ $t\in[0,T],$ $x\in\mathbb{R}^{d}$ with $\left\|x\right\|>R_{\varepsilon,M}$ it holds that

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Therefore, we obtain that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that

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The fact that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that the function $w_{\varepsilon,M}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ is continuous hence demonstrates that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ there exists $(t_{\varepsilon,M},x_{\varepsilon,M})\in[0,T]\times\mathbb{R}^{d}$ such that it holds that

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The fact that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that $w_{\varepsilon,M}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$ therefore ensures that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty),$ $v\in\mathbb{R}^{d}$ with $t_{\varepsilon,M}>0$ it holds that

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Hence, we obtain that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ with $t_{\varepsilon,M}>0$ it holds that

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and

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This and (15) imply that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ with $t_{\varepsilon,M}>0$ it holds that

[TABLE]

Hence, we obtain for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ that $t_{\varepsilon,M}=0$. Combining this with (23) proves that for every $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that

[TABLE]

This and (9) imply that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $\varepsilon\in(0,\frac{1}{4a}-T)$, $M\in(0,\infty)$ it holds that

[TABLE]

Therefore, we obtain that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $\varepsilon\in(0,\frac{1}{4a}-T)$ it holds that

[TABLE]

Hence, we obtain that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

This establishes (6) in the case $T<\frac{1}{4a}$. We now prove (6) in the case $T\geq\frac{1}{4a}$. For this let $k\in\mathbb{N}$ and $\mathcal{T}\in(0,\frac{1}{8a}]$ be the real numbers which satisfy that

[TABLE]

let $\tau_{l}\in\mathbb{R},$ $l\in\left\{0,1,\dots,k+1\right\}$, be the real numbers which satisfy for all $l\in\{0,1,\dots,k\}$ that

[TABLE]

and let $\mathfrak{v}_{l}\colon[0,\tau_{l+1}-\tau_{l}]\times\mathbb{R}^{d}\to\mathbb{R},$ $l\in\left\{0,1,\dots,k\right\}$, be the functions which satisfy for all $l\in\left\{0,1,\dots,k\right\},$ $t\in[0,\tau_{l+1}-\tau_{l}],$ $x\in\mathbb{R}^{d}$ that

[TABLE]

Next we claim that for every $l\in\{0,1,\dots,k+1\}$ it holds that

[TABLE]

We now prove (35) by induction on $l\in\{0,1,\dots,k+1\}.$ Observe that the fact that

[TABLE]

establishes (35) in the base case $l=0.$ For the induction step $\{0,1,\dots,k\}\ni l\to l+1\in\{1,2,\dots,k+1\}$ assume that there exists $l\in\left\{0,1,\dots,k\right\}$ such that

[TABLE]

In addition, note that (4), (16), and (34) ensure that for every $t\in(0,\tau_{l+1}-\tau_{l}],$ $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

and

[TABLE]

This, (37), and (6) in the case $T<\frac{1}{4a}$ show that

[TABLE]

Therefore, we obtain that

[TABLE]

Induction hence proves (35). Furthermore, note that (35) and the fact that $T=\tau_{k+1}$ imply that

[TABLE]

This establishes (6) in the case $T\geq\frac{1}{4a}$. The proof of Lemma 2.1 is thus completed. ∎

Corollary 2.2**.**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $v\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}),$ assume that $v|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R}),$ assume for every $t\in(0,T],$ $x\in\mathbb{R}^{d}$ that

[TABLE]

let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm, and assume that

[TABLE]

Then it holds that

[TABLE]

Proof of Corollary 2.2.

Throughout this proof let ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\colon\mathbb{R}^{d}\to[0,\infty)$ be the $d$-dimensional Euclidean norm and let $c\in(0,\infty)$ be the real number which satisfies that

[TABLE]

Note that (44) ensures that there exists $a\in(-\infty,0]$ such that

[TABLE]

In addition, observe that (46) implies that for all $x\in\mathbb{R}^{d}\setminus\{0\}$ it holds that

[TABLE]

Combining this with (47) demonstrates that for every $t\in[0,T],$ $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

This ensures that

[TABLE]

Hence, we obtain that

[TABLE]

Combining this with (43) enables us to apply Lemma 2.1 to obtain that

[TABLE]

The proof of Corollary 2.2 is thus completed. ∎

Theorem 2.3**.**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $c\in\mathbb{R}$, let $f\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a function, assume for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $y\in\mathbb{R}$ that

[TABLE]

let $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}),$ assume that $u|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm, assume that

[TABLE]

and assume for every $t\in(0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Then it holds for every $t\in[0,T]$ that

[TABLE]

Proof of Theorem 2.3.

Throughout this proof let $v\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ be the function which satisfies for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Note that (54) ensures that there exists $a\in(-\infty,0]$ such that

[TABLE]

Moreover, observe that (53) demonstrates that $c\geq 0.$ Combining this with (57) and (58) implies that

[TABLE]

Hence, we obtain that

[TABLE]

Next observe that (57), the hypothesis that $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}),$ and the hypothesis that $u|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$ ensure that

[TABLE]

Furthermore, note that (57) demonstrates that for every $t\in(0,T],$ $x\in\mathbb{R}^{d}$ it holds that $v(0,x)=1+|u(0,x)|^{2}$ and

[TABLE]

This, (55), and (53) imply that for every $t\in(0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

The fact that for every twice differentiable function $w\colon\mathbb{R}^{d}\to\mathbb{R}$ and every $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ it holds that

[TABLE]

therefore implies that for every $t\in(0,T]$, $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ it holds that

[TABLE]

Combining this with (60) and (61) enables us to apply Corollary 2.2 to obtain that

[TABLE]

Therefore, we obtain that for every $t\in[0,T]$ it holds that

[TABLE]

The proof of Theorem 2.3 is thus completed. ∎

Corollary 2.4**.**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $c\in\mathbb{R}$, let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm, let $f\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a function which satisfies for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $y\in\mathbb{R}$ that $yf(t,x,y)\leq c(1+y^{2})$, and let $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$ satisfy for every $t\in[0,T)$, $x\in\mathbb{R}^{d}$ that $u|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$, $\inf_{a\in\mathbb{R}}\sup_{(s,y)\in[0,T]\times\mathbb{R}^{d}}(e^{a\left\|y\right\|^{2}}|u(s,y)|)<\infty$, and

[TABLE]

Then it holds for every $t\in[0,T]$ that

[TABLE]

Proof of 2.4.

Throughout this proof let $U\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ and $F\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be the functions which satisfy for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $y\in\mathbb{R}$ that $U(t,x)=u(T-t,\frac{x}{\sqrt{2}})$ and $F(t,x,y)=f(T-t,\frac{x}{\sqrt{2}},y)$. Observe that the assumption that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $y\in\mathbb{R}$ it holds that $vf(t,x,y)\leq c(1+y^{2})$ implies for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $y\in\mathbb{R}$ that

[TABLE]

Moreover, observe that the hypothesis that $\inf_{a\in\mathbb{R}}\sup_{(s,y)\in[0,T]\times\mathbb{R}^{d}}(e^{a\left\|y\right\|^{2}}|u(s,y)|)<\infty$ ensures that there exists $\alpha\in\mathbb{R}$ which satisfies that

[TABLE]

This implies that

[TABLE]

Hence, we obtain that

[TABLE]

In addition, note that the hypothesis that $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, the hypothesis that $u|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$, the chain rule, and (68) ensure that for every $t\in(0,T]$, $x\in\mathbb{R}^{d}$ it holds that $U\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, that $U|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, and that

[TABLE]

Combining this, (70), and (73) with Theorem 2.3 (with $f=F$, $u=U$ in the notation of Theorem 2.3) demonstrates for every $t\in[0,T]$ that

[TABLE]

This completes the proof of 2.4. ∎

3 Truncated full-history recursive multilevel Picard (MLP) approximations

In this section we present and analyze a (truncated) MLP approximation scheme for reaction-diffusion type PDEs with coercive nonlinearity (see 3.1 below for details). The error analysis relies on results in [24, Section 3] (cf. also 3.4 below) in combination with a Feynman–Kac representation (cf. Lemma 3.3) and the a priori estimates in Section 2 above.

Setting 3.1** (Setting and algorithm).**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $f\in C([0,T]\times\mathbb{R}^{d}\times\mathbb{R},\mathbb{R})$, $g\in C(\mathbb{R}^{d},\mathbb{R})$, let ${\bf f}_{r}\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$, $r\in(0,\infty)$, satisfy for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $u\in\mathbb{R}$ that

[TABLE]

let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{\theta})_{\theta\in\Theta}$ are independent, let $R^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t+(T-t)\mathcal{R}^{\theta}$, for every $\theta\in\Theta$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ let $X^{\theta}_{t,s,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{\theta}_{t,s,x}=x+W^{\theta}_{s}-W^{\theta}_{t}$, and let $U^{\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in(0,\infty)$, satisfy for every $\theta\in\Theta$, $n,M\in\mathbb{N}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

The next result, Lemma 3.2 below, is an adaptation of [24, Theorem 3.5] to 3.1.

Lemma 3.2** (Convergence rate for stochastic fixed point equations).**

Assume 3.1, let $\rho\in(0,\infty)$, let $L\colon(0,\infty)\to[0,\infty)$ satisfy for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $v,w\in[-r,r]$ that

[TABLE]

let $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$ satisfy for every $r\in[\rho,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

[TABLE]

Then it holds for every $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Proof of Lemma 3.2.

Throughout this proof let $P_{r}\colon\mathbb{R}\to\mathbb{R}$, $r\in(0,\infty)$, be the functions which satisfy for every $v\in\mathbb{R}$ that $P_{r}(v)=\min\{r,\max\{-r,v\}\}$ and assume w.l.o.g. that there exists a standard Brownian motion $\mathbf{W}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$ which satisfies that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$, $(W^{\theta})_{\theta\in\Theta}$, and $\mathbf{W}$ are independent. Observe that for every $r\in(0,\infty)$ it holds that $P_{r}\colon\mathbb{R}\to\mathbb{R}$ is the projection onto the closed convex interval $[-r,r]$. Therefore, we obtain for every $r\in(0,\infty)$, $v,w\in\mathbb{R}$ that

[TABLE]

(cf., e.g., Brézis [7, Proposition 5.3]). This, (76), and (78) imply for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $v,w\in\mathbb{R}$ that

[TABLE]

This and [24, Theorem 3.5] (with $d=d$, $T=T$, $L=L(r)$, $\xi=x$, $F=(C([0,T]\times\mathbb{R}^{d},\mathbb{R})\ni v\mapsto([0,T]\times\mathbb{R}^{d}\ni(t,x)\mapsto{\bf f}_{r}(t,x,v(t,x))\in\mathbb{R})\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}))$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $g=g$, $u=u$, $\Theta=\Theta$, $W^{\theta}=W^{\theta}$, $\mathfrak{r}^{\theta}=\mathcal{R}^{\theta}$, $\mathcal{R}^{\theta}=R^{\theta}$, $U^{\theta}_{n,M}=U^{\theta}_{n,M,r}$ for $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$ in the notation of [24, Theorem 3.5]) ensure for every $n,M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Moreover, note that (83) and [24, Lemma 3.4] (with $d=d$, $T=T$, $L=L(r)$, $\xi=x$, $F=(C([0,T]\times\mathbb{R}^{d},\mathbb{R})\ni v\mapsto([0,T]\times\mathbb{R}^{d}\ni(t,x)\mapsto{\bf f}_{r}(t,x,v(t,x))\in\mathbb{R})\in C([0,T]\times\mathbb{R}^{d},\mathbb{R}))$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $g=g$, $u=u$, $\Theta=\Theta$, $W^{\theta}=W^{\theta}$, $\mathfrak{r}^{\theta}=\mathcal{R}^{\theta}$, $\mathcal{R}^{\theta}=R^{\theta}$, $U^{\theta}_{n,M}=U^{\theta}_{n,M,r}$ for $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$ in the notation of [24, Lemma 3.4]) yield that for every $M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Combining this with (84) establishes (81). The proof of Lemma 3.2 is thus completed. ∎

Lemma 3.3** (Feyman–Kac formula).**

Let $d\in\mathbb{N}$, $T\in(0,\infty)$, $u,h\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $W\colon[0,T]\times\Omega\to\mathbb{R}^{d}$ be a standard Brownian motion, for every $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ let $X_{t,s,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X_{t,s,x}=x+W_{s}-W_{t}$, and assume for every $t\in[0,T)$, $x\in\mathbb{R}^{d}$ that $\sup_{s\in[0,T],y\in\mathbb{R}^{d}}|u(s,y)|<\infty$, $\mathbb{E}[\int_{t}^{T}|h(s,X_{t,s,x})|\,ds]<\infty$, $u|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$, and

[TABLE]

Then it holds for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Proof of Lemma 3.3.

Throughout this proof let $\langle\cdot,\cdot\rangle\colon\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ be the Euclidean scalar product on $\mathbb{R}^{d}$, let $\|\cdot\|\colon\mathbb{R}^{d}\to[0,\infty)$ be the Euclidean norm on $\mathbb{R}^{d}$, and for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ with $t<T-\nicefrac{{1}}{{r}}$ let the function $\tau^{t,x}_{r}\colon\Omega\to[t,T-\nicefrac{{1}}{{r}}]$ satisfy that $\tau^{t,x}_{r}=\inf(\{s\in[t,T]\colon\|X_{t,s,x}-x\|>r\}\cup\{T-\nicefrac{{1}}{{r}}\})$. Observe that Itô’s formula and the hypothesis that $u|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$ ensure that for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ with $t<T-\nicefrac{{1}}{{r}}$ it holds $\mathbb{P}$-a.s. that

[TABLE]

This implies for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ with $t<T-\nicefrac{{1}}{{r}}$ that

[TABLE]

Combining the fact that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds $\mathbb{P}$-a.s. that $\limsup_{r\to\infty}|\tau^{t,x}_{r}-T|=0$ and the hypothesis that $u\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ is a bounded continuous function with Lebesgue’s dominated convergence theorem hence implies that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

In addition, note that the fact that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds $\mathbb{P}$-a.s. that $\limsup_{r\to\infty}|\tau^{t,x,\mathfrak{t}}_{r}-\mathfrak{t}|=0$, the hypothesis that $h\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ is a continuous function, the hypothesis that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $\int_{t}^{T}\mathbb{E}[|h(s,X_{t,s,x})|]\,ds<\infty$, and Lebesgue’s dominated convergence theorem ensure for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

This, (89), and (90) imply for every $t\in[0,T)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

This establishes (87). The proof of Lemma 3.3 is thus completed. ∎

Proposition 3.4** (Convergence rate for Allen–Cahn PDEs).**

Assume Setting 3.1, let $\rho\in(0,\infty)$, $c\in[0,\infty)$, let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm, let $L\colon(0,\infty)\to[0,\infty)$ satisfy for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $v,w\in[-r,r]$ that

[TABLE]

let $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$ satisfy that $\inf_{a\in\mathbb{R}}[\sup_{t\in[0,T]}\sup_{x\in\mathbb{R}^{d}}(e^{a\left\|x\right\|^{2}}|u(t,x)|)]<\infty$ and $u|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$, and assume for every $t\in[0,T)$, $x\in\mathbb{R}^{d}$, $v\in\mathbb{R}$ that $\rho\geq e^{cT}(1+|g(x)|^{2})^{\nicefrac{{1}}{{2}}}$, $vf(t,x,v)\leq c(1+v^{2})$, $\int_{t}^{T}\mathbb{E}[|f(s,X^{0}_{t,s,x},0)|]\,ds<\infty$, $u(T,x)=g(x)$, and

[TABLE]

Then it holds for every $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Proof of 3.4.

First, observe that the hypothesis that $\sup_{x\in\mathbb{R}^{d}}|g(x)|<\infty$ implies that for every $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Next note that 2.4 (with $d=d$, $T=T$, $c=c$, $\left\|\cdot\right\|=\left\|\cdot\right\|$, $f=f$, $u=u$ in the notation of 2.4) ensures for every $t\in[0,T]$ that

[TABLE]

Combining this with (76) yields for every $r\in[\rho,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

This and (94) demonstrate that for every $r\in[\rho,\infty)$, $t\in[0,T)$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Next observe that the fact that $\sup_{t\in[0,T],x\in\mathbb{R}^{d}}|u(t,x)|\leq\rho$ and (93) ensure that for every $r\in[\rho,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $(\mathbb{E}[|u(s,X^{0}_{0,s,x})|^{2}])^{\nicefrac{{1}}{{2}}}\leq\rho<\infty$ and

[TABLE]

Hence, we obtain that (99) and Lemma 3.3 (with $d=d$, $T=T$, $u=u$, $h=([0,T]\times\mathbb{R}^{d}\ni(t,x)\mapsto{\bf f}_{r}(t,x,u(t,x))\in\mathbb{R})$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $W=W^{0}$, $X_{t,s,x}=X^{0}_{t,s,x}$ for $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ in the notation of Lemma 3.3) demonstrate that for every $r\in[\rho,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Lemma 3.2 (with $\rho=\rho$, $L=L$, $u=u$ in the notation of Lemma 3.2), (96), and (100) hence establish (95). The proof of 3.4 is thus completed. ∎

Proposition 3.5**.**

Let $d\in\mathbb{N}$, $\rho,T\in(0,\infty)$, $c\in[0,\infty)$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $f\in C([0,T]\times\mathbb{R}^{d}\times\mathbb{R},\mathbb{R})$, $({\bf f}_{r})_{r\in(0,\infty)}\subseteq C([0,T]\times\mathbb{R}^{d}\times\mathbb{R},\mathbb{R})$, $u\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, let $\left\|\cdot\right\|\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm on $\mathbb{R}^{d}$, let $L\colon(0,\infty)\to[0,\infty)$ be a function, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{\theta})_{\theta\in\Theta}$ are independent, let $R^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $\theta\in\Theta$, $s\in[0,T]$, $t\in[s,T]$, $x\in\mathbb{R}^{d}$ let $X^{\theta}_{s,t,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{\theta}_{s,t,x}=x+\sqrt{2}(W^{\theta}_{t}-W^{\theta}_{s})$, assume for every $r\in(0,\infty)$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$, $v\in\mathbb{R}$, $w,\mathfrak{w}\in[-r,r]$ that $vf(t,x,v)\leq c(1+v^{2})$, $|f(t,x,w)-f(t,x,\mathfrak{w})|\leq L(r)|w-\mathfrak{w}|$, $\int_{0}^{t}\mathbb{E}[|f(s,X^{0}_{s,t,x},0)|]\,ds<\infty$, ${\bf f}_{r}(t,x,v)=f(t,x,\min\{r,\max\{-r,v\}\})$, $e^{cT}(1+|u(0,x)|^{2})^{\nicefrac{{1}}{{2}}}\leq\rho$, $u|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, $\inf_{a\in\mathbb{R}}[\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}(e^{a\left\|y\right\|^{2}}|u(s,y)|)]<\infty$, and

[TABLE]

and let $U^{\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in(0,\infty)$, satisfy for every $\theta\in\Theta$, $n,M\in\mathbb{N}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

Then it holds for every $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

Proof of 3.5.

Throughout this proof let $v\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ be the function which satisfies for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $v(t,x)=u(T-t,x\sqrt{2})$, let $F\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be the function which satisfies for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ that $F(t,x,w)=f(T-t,x\sqrt{2},w)$, let $\mathbf{F}_{r}\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$, $r\in(0,\infty)$, be the functions which satisfy for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ that $\mathbf{F}_{r}(t,x,w)=F(t,x,\min\{r,\max\{-r,w\}\})$, let $\mathcal{S}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$ that $\mathcal{S}^{\theta}=1-\mathcal{R}^{\theta}$, let $S^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $S^{\theta}_{t}=t+(T-t)\mathcal{S}^{\theta}$, for every $\theta\in\Theta$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ let $Y^{\theta}_{t,s,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy that $Y^{\theta}_{t,s,x}=\frac{1}{\sqrt{2}}X^{\theta}_{T-s,T-t,x\sqrt{2}}=x+W^{\theta}_{T-t}-W^{\theta}_{T-s}=x+(W^{\theta}_{T}-W^{\theta}_{T-s})-(W^{\theta}_{T}-W^{\theta}_{T-t})$, and let $V^{\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in(0,\infty)$ satisfy for every $n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $V^{\theta}_{n,M,r}(t,x)=U^{\theta}_{n,M,r}(T-t,x\sqrt{2})$. Note that (102) hence ensures for every $t\in[0,T)$, $x\in\mathbb{R}^{d}$ that $v\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, $v|_{[0,T)\times\mathbb{R}^{d}}\in C^{1,2}([0,T)\times\mathbb{R}^{d},\mathbb{R})$, $\inf_{a\in\mathbb{R}}[\sup_{(s,y)\in[0,T]\times\mathbb{R}^{d}}(e^{a\left\|y\right\|^{2}}|v(s,y)|)]<\infty$, and

[TABLE]

In addition, note that the hypothesis that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ it holds that $wf(t,x,w)\leq c(1+w^{2})$ guarantees that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ it holds that

[TABLE]

Moreover, observe that it holds for every $t\in[0,T]$, $\theta\in\Theta$ that

[TABLE]

Next observe that the assumption that for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w,\mathfrak{w}\in[-r,r]$ it holds that $|f(t,x,w)-f(t,x,\mathfrak{w})|\leq L(r)|w-\mathfrak{w}|$ implies that for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w,\mathfrak{w}\in[-r,r]$ it holds that

[TABLE]

In addition, note that

[TABLE]

Furthermore, note that for every $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

and

[TABLE]

Moreover, observe that (103) guarantees for every $n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that

[TABLE]

The fact that for every $\theta\in\Theta$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ it holds that $X^{\theta}_{t,s,x\sqrt{2}}=\sqrt{2}Y^{\theta}_{T-s,T-t,x}$ and (107) therefore imply that for every $n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Combining this with (103) and the fact that for every $M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $V^{\theta}_{n,M,r}(t,x)=U^{\theta}_{n,M,r}(T-t,x\sqrt{2})$ yields hat for every $\theta\in\Theta$, $n,M\in\mathbb{N}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $V^{\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

This and the fact that for every $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ it holds that $u(0,x\sqrt{2})=v(T,x)$ and $\mathbf{F}_{r}(t,x,w)=F(t,x,\min\{r,\max\{-r,w\}\})=f(T-t,x\sqrt{2},\min\{r,\max\{-r,w\}\})={\bf f}_{r}(T-t,x\sqrt{2},w)$ demonstrate that for every $\theta\in\Theta$, $n,M\in\mathbb{N}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $V^{\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

This, (105)–(111), and 3.4 (with $d=d$, $T=T$, $\Theta=\Theta$, $f=F$, $g=(\mathbb{R}^{d}\ni x\mapsto v(T,x)=u(0,x\sqrt{2})\in\mathbb{R})$, ${\bf f}_{r}=\mathbf{F}_{r}$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{R}^{\theta}=\mathcal{S}^{\theta}$, $W^{\theta}=([0,T]\times\Omega\ni(t,\omega)\mapsto W^{\theta}_{T}(\omega)-W^{\theta}_{T-t}(\omega)\in\mathbb{R}^{d})$, $X^{\theta}_{t,s,x}=Y^{\theta}_{t,s,x}$, $R^{\theta}=S^{\theta}$, $U^{\theta}_{n,M,r}=V^{\theta}_{n,M,r}$, $\rho=\rho$, $c=c$, $\left\|\cdot\right\|=\left\|\cdot\right\|$, $L=L$, $u=v$ for $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $s\in[t,T]$ $x\in\mathbb{R}^{d}$ in the notation of 3.4) demonstrate that for every $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Combining this with (111) and the fact that for every $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that $u(t,x)=v(T-t,\nicefrac{{x}}{{\sqrt{2}}})$, $U^{\theta}_{n,M,r}(t,x)=V^{\theta}_{n,M,r}(T-t,\nicefrac{{x}}{{\sqrt{2}}})$, and $\sqrt{2}\,Y^{0}_{0,T,\nicefrac{{x}}{{\sqrt{2}}}}=X^{0}_{0,T,x}$ establishes (104). The proof of 3.5 is thus completed. ∎

4 Computational cost analysis for truncated MLP approximations

Our next goal is to estimate the overall complexity of the MLP approximation scheme. This is achieved in Theorem 4.5 below. We first quote an elementary result (see [24, Lemma 3.6]) which provides a bound for the computational cost. Lemma 4.2–Lemma 4.4 are technical statements needed for the proof of Theorem 4.5.

Lemma 4.1** (Computational cost).**

Let $d\in\mathbb{N}$, $(\mathfrak{C}_{n,M})_{n\in\mathbb{N}_{0},M\in\mathbb{N}}\subseteq\mathbb{N}_{0}$ satisfy for every $n,M\in\mathbb{N}$ that $\mathfrak{C}_{0,M}=0$ and

[TABLE]

Then it holds for every $n,M\in\mathbb{N}$ that $\mathfrak{C}_{n,M}\leq d(5M)^{n}$.

Proof of Lemma 4.1.

This is an immediate consequence of [24, Lemma 3.6] (with $d=d$, $RV_{n,M}=\mathfrak{C}_{n,M}$ for $n\in\mathbb{N}_{0}$, $M\in\mathbb{N}$ in the notation of [24, Lemma 3.6]). The proof of Lemma 4.1 is thus completed. ∎

Lemma 4.2**.**

Let $\alpha,\beta,c,\kappa,\rho\in(0,\infty)$, $K\in\mathbb{N}_{0}$, $(\gamma_{n})_{n\in\mathbb{N}}\subseteq[0,\infty)$, $(\epsilon_{n,r})_{n\in\mathbb{N},r\in[\rho,\infty)}\subseteq[0,\infty)$, let $L\colon(0,\infty)\to[0,\infty)$ be a function, assume for every $n\in\mathbb{N}$, $r\in[\rho,\infty)$ that $\gamma_{n}\leq(\alpha n)^{n}$ and $\epsilon_{n,r}\leq ce^{L(r)}\kappa^{n}(1+\beta L(r))^{n}n^{-\nicefrac{{n}}{{2}}}$, and let $\varrho\colon\mathbb{N}\to(0,\infty)$ satisfy that

[TABLE]

Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that $\sup_{n\in[1,\mathfrak{N}_{\varepsilon}+K]\cap\mathbb{N}}\gamma_{n}\leq\mathfrak{c}_{\delta}\varepsilon^{-(2+2\delta)}$ and $\sup_{n\in[\mathfrak{N}_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\varrho_{n}}\leq\varepsilon$.

Proof of Lemma 4.2.

Throughout this proof let $\mathfrak{a}_{\delta}\in[0,\infty]$, $\delta\in(0,\infty)$, and $\mathfrak{b}\in[0,\infty)$ satisfy for every $\delta\in(0,\infty)$ that

[TABLE]

and

[TABLE]

First, observe that the fact that for every $t\in(0,\infty)$ it holds that $\ln(t)\leq t-1$ and (118) ensure that

[TABLE]

This and the fact that $\lim_{s\to-\infty}e^{s}=0$ imply that

[TABLE]

Hence, we obtain that there exist $N_{\varepsilon}\in\mathbb{N}$, $\varepsilon\in(0,\infty)$, which satisfy for every $\varepsilon\in(0,\infty)$ that

[TABLE]

Moreover, the assumption that $\liminf_{n\to\infty}\varrho_{n}=\infty$ implies that there exists $\mathfrak{n}\in\mathbb{N}$ which satisfies that $\inf_{n\in[\mathfrak{n},\infty)\cap\mathbb{N}}\varrho_{n}\geq\rho$. Next let $\eta\in(0,\infty)$ satisfy that $\eta<ce^{L(\varrho_{\mathfrak{n}})}\kappa^{\mathfrak{n}}(1+\beta L(\varrho_{\mathfrak{n}}))^{\mathfrak{n}}\mathfrak{n}^{-\nicefrac{{\mathfrak{n}}}{{2}}}$. This implies for every $\varepsilon\in(0,\eta]$ that $N_{\varepsilon}>\mathfrak{n}$. Hence, we obtain that for every $\varepsilon\in(0,\eta]$ it holds that $\inf_{n\in[N_{\varepsilon},\infty)\cap\mathbb{N}}\varrho_{n}\geq\inf_{n\in[\mathfrak{n},\infty)\cap\mathbb{N}}\varrho_{n}\geq\rho$. This, the assumption that for every $n\in\mathbb{N}$, $r\in[\rho,\infty)$ it holds that $\epsilon_{n,r}\leq ce^{L(r)}\kappa^{n}(1+\beta L(r))^{n}n^{-\nicefrac{{n}}{{2}}}$, and (123) ensure that for every $\varepsilon\in(0,\eta]$ it holds that

[TABLE]

Next let $E=\{\varepsilon\in(0,\infty)\colon N_{\varepsilon}>1\}$. Observe that (123) yields for every $\varepsilon\in E$ that

[TABLE]

This and the assumption that for every $n\in\mathbb{N}$ it holds that $\gamma_{n}\leq(\alpha n)^{n}$ imply that for every $\varepsilon\in E$, $\delta\in(0,\infty)$ it holds that

[TABLE]

Next observe that the fact that for every $t\in(0,\infty)$ it holds that $\ln(t)\leq t-1$ and (118) ensure once again that for every $\delta\in(0,\infty)$ it holds that

[TABLE]

This, (118), and (119) imply for every $\delta\in(0,\infty)$ that

[TABLE]

Next observe that the assumption that for every $n\in\mathbb{N}$ it holds that $\gamma_{n}\leq(\alpha n)^{n}$ and (120) ensure that for every $\varepsilon\in(0,\eta]\setminus E$, $\delta\in(0,\infty)$ it holds that

[TABLE]

Combining this with (119), (120), (124), and (126) we obtain that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,\eta]$ it holds that $\sup_{n\in[N_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\varrho_{n}}\leq\varepsilon$ and

[TABLE]

Next let $\mathfrak{N}_{\varepsilon}\in\mathbb{N}_{0}$, $\varepsilon\in(0,1]$, satisfy for every $\varepsilon\in(0,1]$ that

[TABLE]

This and (130) ensure that for every $\delta\in(0,\infty)$, $\varepsilon\in(\eta,1]$ it holds that $\sup_{n\in[\mathfrak{N}_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\rho_{n}}=\sup_{n\in[{N}_{\eta},\infty)\cap\mathbb{N}}\epsilon_{n,\rho_{n}}\leq\eta\leq\varepsilon$ and

[TABLE]

Combining this with (130) and (131) establishes that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that

[TABLE]

The proof of Lemma 4.2 is thus completed. ∎

Lemma 4.3**.**

Let $\alpha\in[1,\infty)$. Then it holds for every $n\in\mathbb{N}$ that $\sum_{m=1}^{n}(\alpha m)^{m}\leq 2(\alpha n)^{n}$.

Proof of Lemma 4.3.

First, note that the claim is clear in the case $n=1$. Next observe that for all $n\in\mathbb{N}\cap[2,\infty)$ it holds that $\alpha n\geq 2$. This implies that for all $n\in\mathbb{N}\cap[2,\infty)$ it holds that

[TABLE]

The proof of Lemma 4.3 is thus completed. ∎

Lemma 4.4**.**

Let $\alpha,\beta,c,\kappa,\rho\in(0,\infty)$, $K\in\mathbb{N}_{0}$, $(\gamma_{n})_{n\in\mathbb{N}}\subseteq[0,\infty)$, $(\epsilon_{n,r})_{n\in\mathbb{N},r\in[\rho,\infty)}\subseteq[0,\infty)$, let $L\colon(0,\infty)\to[0,\infty)$ be a function, assume for every $n\in\mathbb{N}$, $r\in[\rho,\infty)$ that $\gamma_{n}\leq(\alpha n)^{n}$ and $\epsilon_{n,r}\leq ce^{L(r)}\kappa^{n}(1+\beta L(r))^{n}n^{-\nicefrac{{n}}{{2}}}$, and let $\varrho\colon\mathbb{N}\to(0,\infty)$ satisfy $\limsup_{n\to\infty}(\frac{L(\varrho_{n})}{\ln(n)}+\frac{1}{\varrho_{n}})=0$. Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that $\sum_{n=1}^{\mathfrak{N}_{\varepsilon}+K}\gamma_{n}\leq\mathfrak{c}_{\delta}\varepsilon^{-(2+2\delta)}$ and $\sup_{n\in[\mathfrak{N}_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\varrho_{n}}\leq\varepsilon$.

Proof of Lemma 4.4.

First, observe that for every $n\in\mathbb{N}$ it holds that $\gamma_{n}\leq(\max\{\alpha,1\}n)^{n}$. Lemma 4.2 (with $\alpha=\max\{\alpha,1\}$, $\beta=\beta$, $c=c$, $\kappa=\kappa$, $\rho=\rho$, $K=K$, $L=L$, $\varrho_{n}=\varrho_{n}$, $\gamma_{n}=(\max\{\alpha,1\}n)^{n}$, $\epsilon_{n,r}=\epsilon_{n,r}$ for $r\in[\rho,\infty)$, $n\in\mathbb{N}$ in the notation of Lemma 4.2) therefore guarantees that there exist $\mathfrak{N}_{\varepsilon}\in\mathbb{N}$, $\varepsilon\in(0,1]$, and $\mathfrak{c}_{\delta}\in[0,\infty)$, $\delta\in(0,\infty)$, such that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that $\sup_{n\in[1,\mathfrak{N}_{\varepsilon}+K]\cap\mathbb{N}}(\max\{\alpha,1\}n)^{n}\leq\mathfrak{c}_{\delta}\varepsilon^{-(2+2\delta)}$ and $\sup_{n\in[\mathfrak{N}_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\varrho_{n}}\leq\varepsilon$. The fact that for every $n\in\mathbb{N}$ it holds that $\gamma_{n}\leq(\max\{\alpha,1\}n)^{n}$, the fact that for every $N\in\mathbb{N}$ it holds that $\sup_{n\in[1,N]\cap\mathbb{N}}(\max\{\alpha,1\}n)^{n}=(\max\{\alpha,1\}N)^{N}$, and Lemma 4.3 hence imply that for every $\varepsilon\in(0,1]$ it holds that $\sup_{n\in[\mathfrak{N}_{\varepsilon},\infty)\cap\mathbb{N}}\epsilon_{n,\varrho_{n}}\leq\varepsilon$ and

[TABLE]

The proof of Lemma 4.4 is thus completed. ∎

Theorem 4.5**.**

Let $\rho,T\in(0,\infty)$, $c,\gamma,p\in[0,\infty)$, $K\in\mathbb{N}_{0}$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $(f_{d})_{d\in\mathbb{N}},({\bf f}_{d,r})_{d\in\mathbb{N},r\in(0,\infty)}\subseteq C([0,T]\times\mathbb{R}^{d}\times\mathbb{R},\mathbb{R})$, let $L\colon(0,\infty)\to[0,\infty)$ be a function, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{d,\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $d\in\mathbb{N}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{d,\theta})_{(d,\theta)\in\mathbb{N}\times\Theta}$ are independent, let $R^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $d\in\mathbb{N}$, $\theta\in\Theta$, $s\in[0,T]$, $t\in[s,T]$, $x\in\mathbb{R}^{d}$ let $X^{d,\theta}_{s,t,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{d,\theta}_{s,t,x}=x+\sqrt{2}(W^{d,\theta}_{t}-W^{d,\theta}_{s})$, assume for every $d\in\mathbb{N}$, $r\in(0,\infty)$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$, $u,v\in[-r,r]$, $w\in\mathbb{R}$ that $wf_{d}(t,x,w)\leq c(1+w^{2})$, ${\bf f}_{d,r}(t,x,w)=f_{d}(t,x,\min\{r,\max\{-r,w\}\})$, $\mathbb{E}[\int_{0}^{t}|f_{d}(s,X^{d,0}_{s,t,x},0)|\,ds]<\infty$, and $|f_{d}(t,x,u)-f_{d}(t,x,v)|\leq L(r)|u-v|$, let $u_{d}\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, $d\in\mathbb{N}$, satisfy for every $d\in\mathbb{N}$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$ that $e^{cT}(1+|u_{d}(0,x)|^{2})^{\nicefrac{{1}}{{2}}}\leq\rho$, $\inf_{a\in\mathbb{R}}[\sup_{s\in[0,T]}\sup_{y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}}(e^{a(|y_{1}|^{2}+\ldots+|y_{d}|^{2})}|u_{d}(s,y)|)]<\infty$, $u_{d}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, and

[TABLE]

let $U^{d,\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{d,\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

let $\varrho\colon\mathbb{N}\to(0,\infty)$ satisfy $\limsup_{n\to\infty}(\frac{L(\varrho_{n})}{\ln(n)}+\frac{1}{\varrho_{n}})=0$, and let $\mathfrak{C}_{d,n,M}\in\mathbb{N}_{0}$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$ that $\mathfrak{C}_{d,0,M}=0$ and

[TABLE]

Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$, $x\in\mathbb{R}^{d}$ with $(\int_{0}^{T}\mathbb{E}[|f_{d}(s,X^{d,0}_{s,T,x},0)|^{2}]\,ds)^{\nicefrac{{1}}{{2}}}\leq\gamma d^{p}$ it holds that $\sum_{n=1}^{\mathfrak{N}_{(\varepsilon/d^{p})}+K}\mathfrak{C}_{d,n,n}\leq\mathfrak{c}_{\delta}d^{1+p(2+\delta)}\varepsilon^{-(2+\delta)}$ and

[TABLE]

Proof of Theorem 4.5.

Throughout this proof let $\mathfrak{X}_{d}\subseteq\mathbb{R}^{d}$, $d\in\mathbb{N}$, satisfy for every $d\in\mathbb{N}$ that

[TABLE]

let $\epsilon_{n,r}\in[0,\infty]$, $n\in\mathbb{N}$, $r\in(0,\infty)$, satisfy for every $n\in\mathbb{N}$, $r\in(0,\infty)$ that

[TABLE]

and let $\gamma_{n}\in[0,\infty]$, $n\in\mathbb{N}$, satisfy for every $n\in\mathbb{N}$ that

[TABLE]

Note that Lemma 4.1 demonstrates that for every $d,n,M\in\mathbb{N}$ it holds that $\mathfrak{C}_{d,n,M}\leq d(5M)^{n}$. This implies for every $n\in\mathbb{N}$ that

[TABLE]

Next observe that 3.5 (with $d=d$, $T=T$, $\Theta=\Theta$, $f=f_{d}$, ${\bf f}_{r}={\bf f}_{d,r}$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{R}^{\theta}=\mathcal{R}^{\theta}$, $W^{\theta}=W^{d,\theta}$, $R^{\theta}=R^{\theta}$, $X^{\theta}_{t,s,x}=X^{d,\theta}_{t,s,x}$, $U^{\theta}_{n,M,r}(t,x)=U^{d,\theta}_{n,M,r}(t,x)$, $\rho=\rho$, $c=c$, $\left\|\cdot\right\|=(\mathbb{R}^{d}\ni y=(y_{1},\ldots,y_{d})\mapsto|y_{1}|^{2}+\ldots+|y_{d}|^{2}\in\mathbb{R})$, $L=L$, $u=u_{d}$ for $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ in the notation of 3.5) ensures that it holds for every $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ that

[TABLE]

This implies that for every $n\in\mathbb{N}$, $r\in[\rho,\infty)$ it holds that

[TABLE]

This, (143), and Lemma 4.4 (with $\alpha=5$, $\beta=2$, $c=1+\sup_{d\in\mathbb{N}}\sup_{x\in\mathbb{R}^{d}}|u_{d}(0,x)|+\gamma\sqrt{T}$, $\kappa=\sqrt{e}$, $\rho=\rho$, $K=K$, $L(s)=L(s)T$, $\gamma_{n}=\gamma_{n}$, $\epsilon_{n,r}=\epsilon_{n,r}$, $\varrho_{n}=\varrho_{n}$ for $n\in\mathbb{N}$, $r\in[\rho,\infty)$, $s\in(0,\infty)$ in the notation of Lemma 4.2) guarantee that there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ which satisfy that for every $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that

[TABLE]

This implies that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$, $x\in\mathfrak{X}_{d}$ it holds that

[TABLE]

and

[TABLE]

This establishes (139). The proof of Theorem 4.5 is thus completed. ∎

5 MLP approximations for Allen–Cahn type partial differential equations

In this section we consider sample applications of Theorem 4.5. This provides us with examples of Allen–Cahn PDEs for which the curse of dimensionality can be broken in numerical approximations.

Corollary 5.1**.**

Let $T\in(0,\infty)$, $c,\rho\in[0,\infty)$, $K\in\mathbb{N}_{0}$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $f\in C(\mathbb{R},\mathbb{R})$, $({\bf f}_{r})_{r\in(0,\infty)}\subseteq C(\mathbb{R},\mathbb{R})$, let $L\colon(0,\infty)\to[0,\infty)$ be a function, assume for every $r\in(0,\infty)$, $u,v\in[-r,r]$, $w\in\mathbb{R}$ that $wf(w)\leq c(1+w^{2})$, ${\bf f}_{r}(w)=f(\min\{r,\max\{-r,w\}\})$, and $|f(u)-f(v)|\leq L(r)|u-v|$, let $\varrho\colon\mathbb{N}\to(0,\infty)$ satisfy $\limsup_{n\to\infty}(\frac{L(\varrho_{n})}{\ln(n)}+\frac{1}{\varrho_{n}})=0$, let $u_{d}\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, $d\in\mathbb{N}$, satisfy for every $d\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $e^{cT}(1+|u_{d}(0,x)|^{2})^{\nicefrac{{1}}{{2}}}\leq\rho$, $u_{d}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, $\inf_{a\in\mathbb{R}}[\sup_{s\in[0,T]}\sup_{y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}}(e^{a(|y_{1}|^{2}+\ldots+|y_{d}|^{2})}|u_{d}(s,y)|)]<\infty$, and

[TABLE]

let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{d,\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $d\in\mathbb{N}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{d,\theta})_{(d,\theta)\in\mathbb{N}\times\Theta}$ are independent, let $R^{\theta}_{t}\colon\Omega\to[0,t]$, $\theta\in\Theta$, $t\in[0,T]$, satisfy for every $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $d\in\mathbb{N}$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$, $\theta\in\Theta$ let $X^{d,\theta}_{t,s,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{d,\theta}_{t,s,x}=x+\sqrt{2}(W^{d,\theta}_{s}-W^{d,\theta}_{t})$, let $U^{d,\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $\theta\in\Theta$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{d,\theta}_{0,M,r}(t,x)=0$ and

[TABLE]

and let $\mathfrak{C}_{d,n,M}\in\mathbb{N}_{0}$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$ that $\mathfrak{C}_{d,0,M}=0$ and

[TABLE]

Then

1. (i)

it holds for every $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, $r\in[\rho,\infty)$ that

[TABLE]

and 2. (ii)

there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that $\sum_{n=1}^{\mathfrak{N}_{\varepsilon}+K}\mathfrak{C}_{d,n,n}\leq d\mathfrak{c}_{\delta}\varepsilon^{-(2+\delta)}$ and

[TABLE]

Proof of 5.1.

Throughout this proof let $F_{d}\colon[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$, $d\in\mathbb{N}$, be the functions which satisfy for every $d\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ that

[TABLE]

Observe that the fact that for every $d\in\mathbb{N}$, $x\in\mathbb{R}^{d}$ it holds that $|u_{d}(0,x)|\leq\rho<\infty$ and (154) ensure that for every $d\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $w\in\mathbb{R}$ it holds that ${\bf f}_{r}(w)=F_{d}(t,x,\min\{r,\max\{-r,w\}\})$, $\mathbb{E}[\int_{0}^{t}|F_{d}(s,X^{d,0}_{s,t,x},0)|\,ds]=t|f(0)|<\infty$, $(\mathbb{E}[\int_{0}^{T}|F_{d}(s,X^{d,0}_{s,T,x},0)|^{2}\,ds])^{\nicefrac{{1}}{{2}}}=\sqrt{T}|f(0)|$, and

[TABLE]

This and Theorem 4.5 (with $\rho=\rho$, $T=T$, $c=c$, $\gamma=\sqrt{T}|f(0)|$, $p=0$, $K=K$, $\Theta=\Theta$, $L=L$, $f_{d}=F_{d}$, ${\bf f}_{d,r}=([0,T]\times\mathbb{R}^{d}\times\mathbb{R}\ni(t,x,y)\mapsto{\bf f}_{r}(y)\in\mathbb{R})$, $u_{d}=u_{d}$, $\varrho=\varrho$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{R}^{\theta}=\mathcal{R}^{\theta}$, $W^{d,\theta}=W^{d,\theta}$, $R^{\theta}=R^{\theta}$, $X^{d,\theta}_{t,s,x}=X^{d,\theta}_{t,s,x}$, $U^{d,\theta}_{n,M,r}(t,x)=U^{d,\theta}_{n,M,r}(t,x)$, $\mathfrak{C}_{d,n,M}=\mathfrak{C}_{d,n,M}$ for $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$ in the notation of Theorem 4.5) ensure that

1. (I)

for every $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, $r\in[\rho,\infty)$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

and 2. (II)

there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

This establishes Items (i) and (ii). The proof of 5.1 is thus completed. ∎

Corollary 5.2**.**

Let $T\in(0,\infty)$, $c\in[0,\infty)$, $K\in\mathbb{N}_{0}$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, $f\in C(\mathbb{R},\mathbb{R})$, $(\mathbf{f}_{n})_{n\in\mathbb{N}}\subseteq C(\mathbb{R},\mathbb{R})$, let $\varrho\colon\mathbb{N}\to(0,\infty)$ satisfy $\limsup_{n\to\infty}(\frac{\varrho_{n}}{\ln(\ln(n))})<\infty=\liminf_{n\to\infty}\varrho_{n}$, assume for every $n\in\mathbb{N}$, $u,v\in\mathbb{R}$ that $|f(u)-f(v)|\leq c(1+|u|^{c}+|v|^{c})|u-v|$, $vf(v)\leq c(1+v^{2})$, and $\mathbf{f}_{n}(v)=f(\min\{\varrho_{n},\max\{-\varrho_{n},v\}\})$, let $u_{d}\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, $d\in\mathbb{N}$, satisfy for every $d\in\mathbb{N}$, $t\in(0,T]$, $x\in\mathbb{R}^{d}$ that $\inf_{a\in\mathbb{R}}[\sup_{s\in[0,T]}\sup_{y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}}(e^{a(|y_{1}|^{2}+\ldots+|y_{d}|^{2})}|u_{d}(s,y)|)]<\infty$, $|u_{d}(0,x)|\leq c$, $u_{d}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, and

[TABLE]

let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{d,\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $d\in\mathbb{N}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{d,\theta})_{(d,\theta)\in\mathbb{N}\times\Theta}$ are independent, let $R^{\theta}\colon[0,T]\times\Omega\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $d\in\mathbb{N}$, $t\in[0,T]$, $s\in[t,T]$, $x\in\mathbb{R}^{d}$, $\theta\in\Theta$ let $X^{d,\theta}_{t,s,x}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{d,\theta}_{t,s,x}=x+\sqrt{2}(W^{d,\theta}_{s}-W^{d,\theta}_{t})$, let $U^{d,\theta}_{n,M}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{d,\theta}_{0,M}(t,x)=0$ and

[TABLE]

and let $\mathfrak{C}_{d,n,M}\in\mathbb{N}_{0}$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$ that $\mathfrak{C}_{d,0,M}=0$ and

[TABLE]

Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that

[TABLE]

Proof of 5.2.

Throughout this proof let $L\colon(0,\infty)\to[0,\infty)$ satisfy for every $r\in(0,\infty)$ that $L(r)=c(1+2r^{c})$, let $F_{r}\colon\mathbb{R}\to\mathbb{R}$, $r\in(0,\infty)$, be the functions which satisfy for every $r\in(0,\infty)$, $v\in\mathbb{R}$ that $F_{r}(v)=f(\min\{r,\max\{-r,v\}\})$, and let $V^{d,\theta}_{n,M,r}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $V^{d,\theta}_{n,M,r}(0,x)=0$ and

[TABLE]

Next observe that the hypothesis that $\limsup_{n\to\infty}(\frac{\varrho_{n}}{\ln(\ln(n))})<\infty$ implies that there exists $\gamma\in(0,\infty)$ which satisfies that for every $n\in[3,\infty)\cap\mathbb{N}$ it holds that $\varrho_{n}\leq\gamma\ln(\ln(n))$. This yields that

[TABLE]

Next let $A\subseteq\mathbb{N}_{0}$ be the set given by

[TABLE]

Note that the fact that for every $d,M\in\mathbb{N}$, $\theta\in\Theta$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$, $r\in(0,\infty)$ it holds that $V^{d,\theta}_{0,M,r}(t,x)=0=U^{d,\theta}_{0,M}(t,x)$ ensures that $1\in A$. Moreover, note that (159), (162), and (164) ensure that for every $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in A$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

Hence, we obtain that for every $n\in A$ it holds that $n+1\in A$. Combining this with the fact that $1\in A$ and induction ensures that $A=\mathbb{N}$. This yields that for every $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ it holds that

[TABLE]

The fact that for all $r\in(0,\infty)$, $w,\mathfrak{w}\in[-r,r]$ it holds that $|f(w)-f(\mathfrak{w})|\leq L(r)|w-\mathfrak{w}|$, (163), and 5.1 (with $\rho=\exp(T\sup_{v\in\mathbb{R}}(\frac{vf(v)}{1+v^{2}}))[1+\sup_{d\in\mathbb{N}}\sup_{x\in\mathbb{R}^{d}}|u_{d}(0,x)|^{2}]^{\nicefrac{{1}}{{2}}}$, $c=\sup_{v\in\mathbb{R}}(\frac{vf(v)}{1+v^{2}})$, $T=T$, $K=K$, $\Theta=\Theta$, $f=f$, ${\bf f}_{r}={\bf f}_{r}$, $L=L$, $\left\|\cdot\right\|_{d}=\left\|\cdot\right\|_{d}$, $\varrho=\varrho$, $u_{d}=u_{d}$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{R}^{\Theta}=\mathcal{R}^{\theta}$, $W^{d,\theta}=W^{d,\theta}$, $X^{d,\theta}_{t,s,x}=X^{d,\theta}_{t,s,x}$, $U^{d,\theta}_{n,M,r}(t,x)=V^{d,\theta}_{n,M,r}(t,x)$, $\mathfrak{C}_{d,n,M}=\mathfrak{C}_{d,n,M}$ for $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ in the notation of 5.1) therefore guarantee that there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that $\sum_{n=1}^{\mathfrak{N}_{\varepsilon}+K}\mathfrak{C}_{d,n,n}\leq\mathfrak{c}_{\delta}d\varepsilon^{-(2+2\delta)}$ and

[TABLE]

The proof of 5.2 is thus completed. ∎

Corollary 5.3**.**

Let $c,T\in(0,\infty)$, $K\in\mathbb{N}_{0}$, $\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}$, for every $d\in\mathbb{N}$ let $\left\|\cdot\right\|_{d}\colon\mathbb{R}^{d}\to[0,\infty)$ be a norm on $\mathbb{R}^{d}$, let $u_{d}\in C([0,T]\times\mathbb{R}^{d},\mathbb{R})$, $d\in\mathbb{N}$, satisfy for every $d\in\mathbb{N}$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $|u_{d}(0,x)|\leq c$, $\inf_{a\in\mathbb{R}}[\sup_{s\in[0,T]}\sup_{y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}}(e^{a(|y_{1}|^{2}+\ldots+|y_{d}|^{2})}|u_{d}(s,y)|)]<\infty$, $u_{d}|_{(0,T]\times\mathbb{R}^{d}}\in C^{1,2}((0,T]\times\mathbb{R}^{d},\mathbb{R})$, and

[TABLE]

let $\varrho\colon\mathbb{N}\to(0,\infty)$ be a function which satisfies that $\limsup_{n\to\infty}(\frac{\varrho_{n}}{\ln(\ln(n))})<\infty=\liminf_{n\to\infty}\varrho_{n}$, let $f_{n}\colon\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, be the functions which satisfy for every $n\in\mathbb{N}$, $v\in\mathbb{R}$ that $f_{n}(v)=(\min\{\varrho_{n},\max\{-\varrho_{n},v\}\})-(\min\{\varrho_{n},\max\{-\varrho_{n},v\}\})^{3}$, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{R}^{\theta}\colon\Omega\to[0,1]$, $\theta\in\Theta$, be independent $\mathcal{U}_{[0,1]}$-distributed random variables, let $W^{d,\theta}\colon[0,T]\times\Omega\to\mathbb{R}^{d}$, $d\in\mathbb{N}$, $\theta\in\Theta$, be independent standard Brownian motions, assume that $(\mathcal{R}^{\theta})_{\theta\in\Theta}$ and $(W^{d,\theta})_{(d,\theta)\in\mathbb{N}\times\Theta}$ are independent, let $R^{\theta}\colon\Omega\times[0,T]\to[0,T]$, $\theta\in\Theta$, satisfy for every $\theta\in\Theta$, $t\in[0,T]$ that $R^{\theta}_{t}=t\mathcal{R}^{\theta}$, for every $d\in\mathbb{N}$, $s\in[0,T]$, $t\in[s,T]$, $x\in\mathbb{R}^{d}$, $\theta\in\Theta$ let $X^{d,\theta}_{s,t}\colon\Omega\to\mathbb{R}^{d}$ satisfy $X^{d,\theta}_{s,t,x}=x+\sqrt{2}\,(W^{d,\theta}_{t}-W^{d,\theta}_{s})$, let $U^{d,\theta}_{n,M}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to\mathbb{R}$, $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$, $\theta\in\Theta$, $r\in(0,\infty)$, $t\in[0,T]$, $x\in\mathbb{R}^{d}$ that $U^{d,\theta}_{0,M}(t,x)=0$ and

[TABLE]

and let $\mathfrak{C}_{d,n,M}\in\mathbb{N}_{0}$, $d,M\in\mathbb{N}$, $n\in\mathbb{N}_{0}$, satisfy for every $d,n,M\in\mathbb{N}$ that $\mathfrak{C}_{d,0,M}=0$ and

[TABLE]

Then there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1)$ it holds that

[TABLE]

Proof of 5.3.

First, observe that for every $r\in(0,\infty)$, $v,w\in[-r,r]$ it holds that

[TABLE]

Moreover, note that for every $v\in\mathbb{R}$ it holds that $v(v-v^{3})=v^{2}-v^{4}\leq 1+v^{2}$. This, the hypothesis that for every $d\in\mathbb{N}$, $x\in\mathbb{R}^{d}$ it holds that $|u_{d}(0,x)|\leq c$, the fact that for every $d\in\mathbb{N}$ it holds that $\left\|\cdot\right\|_{d}$ and the Euclidean norm on $\mathbb{R}^{d}$ are equivalent, (172), and 5.2 (with $T=T$, $c=\max\{c,2\}$, $K=K$, $\Theta=\Theta$, $f=(\mathbb{R}\ni u\mapsto u-u^{3}\in\mathbb{R})$, $\mathbf{f}_{M}=f_{M}$, $\varrho=\varrho$, $u_{d}=u_{d}$, $(\Omega,\mathcal{F},\mathbb{P})=(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{R}^{\theta}=\mathcal{R}^{\theta}$, $W^{d,\theta}=W^{d,\theta}$, $R^{\theta}=R^{\theta}$, $X^{d,\theta}_{s,t,x}=X^{d,\theta}_{s,t,x}$, $U^{d,\theta}_{n,M}(t,x)=U^{d,\theta}_{n,M}(t,x)$, $\mathfrak{C}_{d,n,M}=\mathfrak{C}_{d,n,M}$ for $d,M\in\mathbb{N}$, $\theta\in\Theta$, $n\in\mathbb{N}_{0}$, $r\in(0,\infty)$, $t\in[0,T]$, $s\in[0,t]$, $x\in\mathbb{R}^{d}$ in the notation of 5.1) ensure that there exist $\mathfrak{N}\colon(0,1]\to\mathbb{N}$ and $\mathfrak{c}\colon(0,\infty)\to[0,\infty)$ such that for every $d\in\mathbb{N}$, $\delta\in(0,\infty)$, $\varepsilon\in(0,1]$ it holds that

[TABLE]

The proof of 5.3 is thus completed. ∎

Acknowledgements

This project has been partially supported by the Deutsche Forschungsgemeinschaft (DFG) via research grant HU 1889/6-1. F. H. gratefully acknowledges a Research Travel Grant by the Karlsruhe House of Young Scientists (KHYS) supporting his stay at ETH Zurich.