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

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-comparison results for Ornstein--Uhlenbeck processes from bounded drift operators to generators of $C_0$-semigroups on a separable Hilbert space. The argument separates the problem into a Gaussian channel and a jump--drift channel. The Gaussian channel is governed by an inverse-covariance Hilbert--Schmidt perturbation condition. The jump--drift channel is handled by a directional Cameron--Martin condition, formulated conditionally on the jump path; no unconditional Novikov estimate is needed for this step. We prove that these hypotheses give absolute continuity of path laws on the Skorohod space, and equivalence whenever the Gaussian channels are equivalent. For purely jump noise we prove a rigidity phenomenon: absolute continuity forces the two solutions to coincide. For analytic semigroups and compound Poisson jumps we give verifiable sufficient conditions in terms of fractional smoothing and the boundedness of $Q^{-1/2}(\widetilde A-A)A^{-\beta}$. Diagonal examples show the sharp role of the Cameron--Martin and inverse-covariance requirements.