Splitting integrators for stochastic Lie--Poisson systems

Charles-Edouard Bréhier (1, 2, 3) , David Cohen (4) , Tobias Jahnke (5)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan
3 UPPA - Université de Pau et des Pays de l'Adour
4 Chalmers - Department of Mathematical Sciences
5 Institut für Angewandte und Numerische Mathematik [Karlsruhe]

We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie-Poisson systems, namely: stochastically perturbed Maxwell-Bloch, rigid body and sine Euler equations.

Domaines: Probabilités [math.PR]; Analyse numérique [math.NA]

Charles-Edouard Bréhier, David Cohen, Tobias Jahnke. Splitting integrators for stochastic Lie--Poisson systems. Mathematics of Computation, 2023, 92 (343), pp.2167-2216. ⟨10.1090/mcom/3829⟩. ⟨hal-03431169⟩