Law equivalence for Ornstein--Uhlenbeck dynamics driven by Lévy noise
Tomasz Kania
TL;DR
The paper develops a comprehensive framework for law equivalence and rigidity of infinite-dimensional Ornstein–Uhlenbeck processes driven by Lévy noise, extending results from bounded to unbounded drift generators that are generators of analytic semigroups. It decomposes the problem into Gaussian and jump–drift channels, establishing absolute continuity and equivalence under a Gaussian Hilbert–Schmidt perturbation condition and a directional Cameron–Martin hypothesis, with full equivalence requiring both directions. In the Gaussian-dominated case, the authors provide verifiable criteria via fractional boundedness K A^{−β} and HS bounds; for compound Poisson jumps they translate the Cameron–Martin condition into exponential moment requirements on the jump law and resolvent-based checks. A rigidity result in the pure-jump setting shows that, under absolute continuity, the two processes must coincide, while the paper also presents counterexamples illustrating potential failures of the Cameron–Martin condition and one-sided absolute continuity, clarifying the boundaries of the theory. The results have practical implications for numerical simulation and statistical inference in SPDEs driven by Lévy noise, including importance sampling and model identifiability through law equivalence.
Abstract
For stochastic partial differential equations driven by Lévy 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.