Non-analytic semigroups and H^∞ calculus: fractional boundedness hypotheses for HS-perturbation and CM-like factorisation

Ascertain whether there exist fractional boundedness conditions on the generator perturbation K = \widetilde{A} − A, in settings where A and \widetilde{A} generate non-analytic C_0-semigroups (for example, admitting a bounded H^∞ functional calculus or exhibiting polynomial stability), that both (i) imply the Gaussian Hilbert–Schmidt perturbation integrability condition \int_0^T \|K S(t) Q^{1/2}\|_{HS}^2 dt < \infty for all T > 0, where S is the C_0-semigroup generated by A and Q is the covariance of the non-degenerate Q-Wiener process W, and (ii) enable a Cameron–Martin-like factorisation argument sufficient to establish absolute continuity or equivalence of the path laws of the Ornstein–Uhlenbeck processes driven by the same L\'evy noise.

Background

The main results in the paper rely on analyticity of the semigroups generated by A and \widetilde{A} to obtain smoothing estimates and to validate Duhamel-type representations in the strong operator topology. These properties are used to verify a Hilbert–Schmidt perturbation condition for the Gaussian component and to structure the jump-drift discrepancy within a Cameron–Martin framework. Extending these results beyond analytic semigroups is natural but technically challenging, especially in settings where only a bounded H^∞ functional calculus or polynomial stability is available.

The authors explicitly pose whether suitable fractional boundedness assumptions on the perturbation K can replace analyticity while still ensuring the key Hilbert–Schmidt integrability condition and permitting a Cameron–Martin-like factorisation needed for change-of-measure arguments. This would broaden the applicability of their equivalence and rigidity results to a wider class of generators encountered in SPDEs.