Long-Term Analysis of Stochastic Hamiltonian Systems Under Time Discretizations

Abstract

In this paper, we focus our investigation on providing long-term estimates of the Hamiltonian deviation computed along numerical approximations to the solutions of stochastic Hamiltonian systems, of both Itô and Stratonovich types. It is well known that the expected Hamiltonian of an Itô Hamiltonian system with additive noise exhibits a linear drift in time [C. Chen et al., Adv. Comput. Math., 46 (2020), 27], while the Hamiltonian function is conserved along the exact flow of a Stratonovich Hamiltonian system [T. Misawa, Japan J. Indust. Appl. Math., 17 (2000), pp. 119–128; T. Misawa, Math. Probl. Eng., 2010 (2010), 384937]. Here, we focus our attention on providing modified differential equations associated to suitable discretizations for the above problems, by means of weak backward error analysis arguments [K. C. Zygalakis, SIAM J. Sci. Comput., 33 (2011), pp. 102–130]. Then, long-term estimates are provided for both Itô and Stratonovich Hamiltonian systems, revealing the presence of parasitic terms affecting the overall conservation accuracy. Finally, selected numerical experiments are provided to confirm the theoretical analysis.