Small mass limit of expected signature for physical Brownian motion

Abstract

Physical Brownian motion describes the dynamics of a Brownian particle experiencing frictional force. It was investigated in the classical work [L. S. Ornstein and G. E. Uhlenbeck, Phys. Rev. 36 (1930)] as a physically meaningful approach to realising the standard ``mathematical'' Brownian motion, via sending the mass $m \to 0^+$ and performing natural scaling. The analysis was extended to a Brownian particle in an external magnetic field in [P. Friz, P. Gassiat, and T. Lyons, Trans. Amer. Math. Soc. 367 (2015)], discovering the new phenomenon that the area process associated to the physical process converges -- but not to L\'{e}vy's stochastic area. In this paper, we carry out the singular limit analysis of a generalised stochastic differential equation (SDE) model encompassing the physical Brownian motion as a special case. We show that the expected signature of the solution $\{P_t\}_{t \geq 0}$ for the generalised SDE converges to a nontrivial tensor as $m \to 0^+$, at each degree in the tensor algebra and on each time interval $[0,T]$, through a delicate convergence analysis based on the graded PDE system for the expected signature of It\^{o} diffusion processes. Moreover, explicit solutions exhibiting intriguing combinatorial patterns are obtained when the coefficient matrix $\mathscr{M}$ in our SDE is diagonalisable. In the case of physical Brownian motion, $\{P_t\}_{t \geq 0}$ corresponds to the momentum of the particle (viewed as a rough path), and $\mathscr{M}$ is the stress tensor. Our work appears among the very first endeavours to study the singular limit of expected signature of diffusion processes, especially for nonzero initial datum $p=P_0$.