Law equivalence for Ornstein--Uhlenbeck dynamics driven by L\'evy noise

TL;DR

This paper investigates when changes in the drift of Ornstein-Uhlenbeck processes driven by Lévy noise preserve the law of the solution, extending results to unbounded operators and providing explicit conditions.

Contribution

It generalizes law-equivalence results for OU processes to unbounded generators on Hilbert spaces, including explicit criteria and counterexamples.

Findings

Law equivalence characterized by Gaussian and jump components

Non-degenerate Gaussian noise implies absolute continuity or equivalence

Pure jump noise leads to process coincidence under absolute continuity

Abstract

For stochastic partial differential equations driven by L\'evy noise, understanding when changes in the drift operator preserve the law of the solution is fundamental to filtering, control, and simulation. We extend law-equivalence results for Ornstein--Uhlenbeck (OU) processes from bounded drift operators to generators of $C_0$-semigroups (indeed analytic semigroups) on a separable Hilbert space. Our analysis separates the problem into two channels: a Gaussian component governed by a Hilbert--Schmidt perturbation condition, and a jump-drift component requiring a directional Cameron--Martin hypothesis. We establish that when the Gaussian noise is non-degenerate, these conditions characterise absolute continuity and equivalence of path laws on the Skorohod space. For purely jump noise, we prove a rigidity phenomenon: absolute continuity forces the processes to coincide. Specialising to sectorial elliptic generators with compound Poisson jumps, we provide explicit, verifiable conditions in terms of resolvent estimates and exponential moments. We also construct explicit counterexamples showing that the Cameron--Martin condition can fail, sometimes asymmetrically, yielding only one-sided absolute continuity.