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Regularity of multiplicative processes on infinite-dimensional Lie groups

Anita Behme, Markus Riedle, Shend Thaqi

TL;DR

The paper addresses regularity questions for Banach-Lie group–valued stochastic processes with independent increments, seeking conditions under which they admit càdlàg or continuous modifications and controlled local displacement. The authors exploit local exponential/logarithm maps and the Baker–Campbell–Hausdorff framework to reduce analysis to the Banach-space setting and develop oscillation-based criteria for modification construction. A concrete infinite-dimensional Heisenberg-group example illustrates how multiplicative processes can be built from additive ones, yielding a multiplicative Lévy process when the underlying components are Lévy. Overall, the work extends finite-dimensional Lévy-type regularity theory to Banach-Lie groups, enabling stochastic-differential-geometric modeling on spaces with infinite-dimensional symmetries.

Abstract

This article studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit càdlàg modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results.

Regularity of multiplicative processes on infinite-dimensional Lie groups

TL;DR

The paper addresses regularity questions for Banach-Lie group–valued stochastic processes with independent increments, seeking conditions under which they admit càdlàg or continuous modifications and controlled local displacement. The authors exploit local exponential/logarithm maps and the Baker–Campbell–Hausdorff framework to reduce analysis to the Banach-space setting and develop oscillation-based criteria for modification construction. A concrete infinite-dimensional Heisenberg-group example illustrates how multiplicative processes can be built from additive ones, yielding a multiplicative Lévy process when the underlying components are Lévy. Overall, the work extends finite-dimensional Lévy-type regularity theory to Banach-Lie groups, enabling stochastic-differential-geometric modeling on spaces with infinite-dimensional symmetries.

Abstract

This article studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit càdlàg modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results.

Paper Structure

This paper contains 9 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic processes on Lie groups
  4. Regularity
  5. A modification with càdlàg paths
  6. Markov property of multiplicative processes
  7. Jumps of a multiplicative process
  8. Results for Multiplicative Processes with Bounded Jumps
  9. Multiplicative Processes on the Heisenberg Group

Key Result

Proposition 2.2

Suppose $\mathcal{G}$ is a Banach-Lie group. Then the tangent space $\mathfrak{g} := T_e \mathcal{G}$ at the identity, is a Banach-Lie algebra. Furthermore, there exists a smooth map denoted $\operatorname{exp} : \mathfrak{g} \to \mathcal{G}$ called exponential such that $\exp(0 ) = e$ and Moreover, there are open sets $U_{ \mathfrak{g} } \subseteq \mathfrak{g}, \, U_{ \mathcal{G}} \subseteq \m

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 14 more