On a Cameron--Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

TL;DR

This paper extends classical Cameron--Martin quasi-invariance results to subordinate Brownian motion, establishing an integration by parts formula, a gradient operator on path space, and canonical Dirichlet forms, thus broadening the theoretical framework for stochastic analysis.

Contribution

It introduces a Cameron--Martin type quasi-invariance theorem for subordinate Brownian motion and develops associated analytical tools, extending classical results to a more general setting.

Findings

Established a quasi-invariance theorem for subordinate Brownian motion.

Constructed a gradient operator on the path space.

Derived canonical Dirichlet forms for the process.

Abstract

We present a Cameron--Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and we obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.