A new class of splitting methods that preserve ergodicity and exponential integrability for stochastic Langevin equation

Abstract

In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the original equation, which is inspired by the inheritance of the Lyapunov structure for obtaining the ergodicity. We prove that the exponential moment of the numerical solution is bounded, thus validating the exponential integrability of the proposed methods. Further, we show that under moderate verifiable conditions, the methods have the first-order convergence in both strong and weak senses, and we present several concrete splitting schemes based on the methods. The splitting strategy of methods can be readily extended to construct conformal symplectic methods and high-order methods that preserve both the ergodicity and the exponential integrability, as demonstrated in numerical experiments. Our numerical experiments also show that the proposed methods have good performance in the long-time simulation.

Figures (7)

• Figure 1: Evolution of $H_0(\tilde{P}(t),\tilde{Q}(t))$.
• Figure 2: Convergence order in $\log$-$\log$ scale.
• Figure 3: Long-time error.
• Figure 4: (a,b,c) The empirical distribution at different time $t=0,2,256$, respectively, and (d) the reference distribution $\rho(p,q)=\frac{\sqrt{\upsilon}}{\sigma\sqrt{\pi}\int_{\mathbb{R}}e^{-\upsilon q^4/(2\sigma^2)}dq}e^{-\frac{2\upsilon}{\sigma^2}(\frac{p^2}{2}+\frac{q^4}{4})}$, see PavGri2014.
• Figure 5: (a) Mean square displacement and (b) $MSD(\infty)-MSD(t)$.

Theorems & Definitions (12)

• Lemma 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• Theorem 3.1
• Theorem 3.2
• proof
• proof : Proof of Theorem \ref{['thm:1']}