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Small mass limit of expected signature for physical Brownian motion

Siran Li, Hao Ni, Qianyu Zhu

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 Lévy'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 Itô 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 $p=P_0$.

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 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 Lévy'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 for the generalised SDE converges to a nontrivial tensor as , at each degree in the tensor algebra and on each time interval , through a delicate convergence analysis based on the graded PDE system for the expected signature of Itô diffusion processes. Moreover, explicit solutions exhibiting intriguing combinatorial patterns are obtained when the coefficient matrix in our SDE is diagonalisable. In the case of physical Brownian motion, corresponds to the momentum of the particle (viewed as a rough path), and 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 .
Paper Structure (34 sections, 19 theorems, 177 equations)

This paper contains 34 sections, 19 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Physical Brownian motion
  3. From physical to mathematical Brownian motions: non-convergence of signature
  4. Expected signature
  5. Main result and key novelty
  6. Organisation
  7. Preliminaries
  8. Itô diffusion process
  9. Expected signature
  10. PDE for expected signature and a Feynman--Kac type result
  11. Expected signature of physical Brownian motion via the graded PDE system
  12. PDE for physical Brownian motion
  13. The case of one space dimension
  14. Higher dimensional case foreshadowed
  15. Small mass limit in arbitrary dimensions
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\Phi_{n}^{(m)}: \mathbb{R}^d\times\mathbb{R}_+ \to \left(\mathbb{R}^d\right)^{\otimes n}$ be the degree-$n$ expected signature of the momentum path $\left\{P_t\right\}$. Suppose $\|{\mathscr{M}}\| \leq \Lambda$ and that there is $K>0$ such that for each $\delta>0$, $\max\left\{\left\|e^{-{\math where ${\rm Error}_n^{(m)}(p,t)$ is a polynomial in $p$ of degree no more than $\max\{n-2,0\}$, and

Theorems & Definitions (44)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • Remark 3.1
  • ...and 34 more