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Invariance properties of Brownian motion via Lie's symmetries

Susanna Dehò, Francesco C. De Vecchi, Paola Morando, Stefania Ugolini

TL;DR

The paper develops a Lie symmetry framework for stochastic differential equations to study invariance properties of Brownian motion, unifying classical stochastic symmetries (strong/weak/G-weak) with explicit stochastic transformations that act on both the solution and the driving noise. It constructs a group–Lie algebra structure for stochastic transformations via a principal-bundle embedding, yields infinitesimal determining equations, and demonstrates flow reconstruction to obtain finite symmetries. Applied to Brownian motion, the approach recovers symmetries such as reflection, time-scaling (self-similarity), and rotations, and shows how GBM and Brownian Bridge arise as finite transformations within this framework. A central outcome is an integration-by-parts formula, from which Stein identities and Lévy area-type results emerge as specific instances, illustrating the method’s power to derive classical Gaussian identities from diffusion symmetries and to extend to general diffusions.

Abstract

The invariance properties of Brownian motion are investigated and revisited within a recent Lie symmetry approach to stochastic differential equations. Some notable properties of the process can be recovered by a related integration by parts formula developed in the same research area.

Invariance properties of Brownian motion via Lie's symmetries

TL;DR

The paper develops a Lie symmetry framework for stochastic differential equations to study invariance properties of Brownian motion, unifying classical stochastic symmetries (strong/weak/G-weak) with explicit stochastic transformations that act on both the solution and the driving noise. It constructs a group–Lie algebra structure for stochastic transformations via a principal-bundle embedding, yields infinitesimal determining equations, and demonstrates flow reconstruction to obtain finite symmetries. Applied to Brownian motion, the approach recovers symmetries such as reflection, time-scaling (self-similarity), and rotations, and shows how GBM and Brownian Bridge arise as finite transformations within this framework. A central outcome is an integration-by-parts formula, from which Stein identities and Lévy area-type results emerge as specific instances, illustrating the method’s power to derive classical Gaussian identities from diffusion symmetries and to extend to general diffusions.

Abstract

The invariance properties of Brownian motion are investigated and revisited within a recent Lie symmetry approach to stochastic differential equations. Some notable properties of the process can be recovered by a related integration by parts formula developed in the same research area.
Paper Structure (16 sections, 6 theorems, 114 equations)

This paper contains 16 sections, 6 theorems, 114 equations.

Table of Contents

  1. Finite stochastic transformations
  2. From Brownian motion to Geometric Brownian motion via strong transformation
  3. From Brownian motion to Brownian Bridge via weak transformation
  4. Invariance properties and finite symmetries
  5. Reflection of Brownian motion
  6. Algebraic structure of stochastic transformations
  7. Probabilistic foundation of the algebraic structure
  8. Geometric Brownian motion (again)
  9. Lie algebra structure and infinitesimal transformations
  10. Time invariance of Brownian motion
  11. Infinitesimal symmetries
  12. Symmetries of Brownian Motion
  13. First Family ($V_\alpha$)
  14. Second Family ($V_\beta$)
  15. Lie symmetries and integration by parts formulas
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $(X,W)$ be a (weak) solution to $SDE_{\mu,\sigma}$ and let $T=(\Phi, f, B, h)$ be a stochastic transformation. Then, the transformed process $P_T(X,W)$ is a solution to $E_T(SDE_{\mu,\sigma})$. The transformed process $P_T(X,W):=(P_T(X),P_T(W))$is defined by: where $P_T(W)$ is a Brownian motion with respect to the filtration $\mathcal{F}'_t=\mathcal{F}_{f^{-1}(t)}$ and the probability measure

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2: Double action of the transformation
  • proof
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • ...and 15 more