Small mass limit of expected signature for physical Brownian motion

The model of physical Brownian motion describes the dynamics of a Brownian particle experiencing both friction and external magnetic field. It was investigated as a physically meaningful approach to realising the standard ''mathematical'' Brownian motion, via sending the mass $m \to 0^+$ and performing natural scaling. In this paper, we are concerned with the singular limit analysis of a generalised stochastic differential equation (SDE) model, motivated by and encompassing the physical Brownian motion. We show that the expected signature of the solution $\left\{P_t\right\}_{t \geq 0}$ for the generalised SDE converges in the limit $m \to 0^+$ to a nontrivial tensor, at each degree of tensors and on each compact time interval $[0,T]$. The solution $\left\{P_t\right\}_{t \geq 0}$, viewed as a rough path, generalises the momentum of particle in classical physical Brownian motion. This singular limit is identified through a delicate convergence analysis based on the graded PDE system for the expected signature of It\^{o} diffusion processes. Explicit closed-form solutions exhibiting intriguing combinatorial patterns are obtained when the coefficient matrix $\mathbf{M}$ in our SDE, which generalises the stress tensor in physical Brownian motion, is diagonalisable. Our work appears among the very first endeavours to study the singular limit of expected signature of diffusion processes, in addition to the literature on the analysis of the expected signature of stochastic processes.