Pathwise Uniform Convergence Rates for Stochastic Evolution Equations

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Katharina Klioba, M. Sc.

Collaborators (External): Jan van Neerven, Mark Veraar

Description

We consider stochastic PDEs driven by an additive or multiplicative Gaussian noise of the form \[\begin{align*} \begin{cases} \mathrm{d} u &=(A u + F(t,u))\,\mathrm{d} t + G(t,u) \,\mathrm{d} W\,\,\,\,\text{ on } [0,T],\\ u(0) &= u_0 \in L^p(\Omega;X) \end{cases} \end{align*}\] on a Hilbert space \(X\). Here, \(A\) is the generator of a contractive \(C_0\)-semigroup \((S(t))_{t\geq 0}\), \(W\) is a cylindrical Brownian motion, \(F\) and \(G\) are globally Lipschitz and of linear growth, \(p \in [2,\infty)\), and \(u_0\) is the initial data.

Our aim is to obtain strong convergence rates for a temporal discretisation scheme of the form \(U_0 = u_0\), \[\begin{equation*} U_j = R_k U_{j-1} + k R_k F(t_{j-1},U_{j-1})+ R_k G(t_{j-1},U_{j-1}) \Delta W^{j},~j=1,\ldots,N_k \end{equation*}\]with time step \(k>0\), Wiener increments \(\Delta W^j\), and a contractive time discretisation scheme \(R:[0,\infty) \to \mathcal{L}(X)\) approximating \(S\) to order \(\alpha \in (0,\frac{1}{2}]\) on a subspace \(Y\subseteq X\). Among others, this setting covers the splitting scheme, the implicit Euler, and the Crank-Nicholson method.

Assuming additional structure of \(F\) and \(G\) as well as \(Y\), we obtained the following bound for the pathwise uniform strong error \[\begin{align*} \left(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|u(t_j) - U_j\|_X^p \right)^{1/p} \le C(1+\|u_0\|_{L^p(\Omega;Y)}) \left(\log\left(\frac{T}{k}\right)\right)k^{\alpha}. \end{align*}\]In particular, this implies that the convergence rate of the uniform strong error is given by the order of the scheme up to a logarithmic correction factor. This factor can be avoided for the splitting scheme.

Our work extends [1] to the nonlinear case and has applications for, among others, the stochastic Schrödinger [2], Maxwell [3] and wave equations [4].

References

[1] J. van Neerven and M. Veraar, Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes, Stochastics and Partial Differential Equations: Analysis and computations, (2021).

[2] R. Anton and D. Cohen, Exponential integrators for stochastic Schrodinger equations driven by Itô noise, Journal of Computational Mathematics, 36 (2018), pp. 276–309.

[3] D. Cohen, J. Cui, J. Hong, L. Sun, Exponential integrators for stochastic Maxwell’s equations driven by Itô noise, Journal of Computational Physics, (2020).

[4] X. Wang, An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation, Journal of Scientific Computing, 64 (2015), pp. 234–263.