Search

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.

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $\Omega$, it is only known that the radius of convergence for the expected signature on sufficiently regular $\Omega$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $\Omega$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,\alpha}$-domain $\Omega \subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains., Comment: 35 pages, 1 figure. The results obtained in V1 have been extended from $d=2$ to $2\leq d \leq 8$