Directional Cameron–Martin Hypothesis

• Directional Cameron–Martin Hypothesis is a principle governing absolute continuity between laws of perturbed stochastic processes, emphasizing the geometry of CM space and integrability conditions.
• It employs Hilbert–Schmidt conditions and a Novikov-type integrability requirement to control discrepancies in both Gaussian and jump–drift channels of Ornstein–Uhlenbeck processes.
• The hypothesis underpins practical applications in filtering, simulation, and control of SPDEs by rigorously characterizing when law equivalence is preserved under operator and noise perturbations.

The Directional Cameron–Martin Hypothesis is a principle governing absolute continuity and equivalence of probability laws for stochastic processes under perturbations in specific directions of function space, particularly the Cameron–Martin (CM) space. This hypothesis is fundamental in the analysis of Ornstein–Uhlenbeck (OU) processes driven by Lévy noise, where one seeks to characterize when changes in the drift generator or noise structure preserve the law of the solution to a stochastic partial differential equation (SPDE) on the path space, such as the Skorokhod space of càdlàg functions. The hypothesis intertwines the structure of the semigroup generator, the stochastic integrability conditions, and the geometry of associated Hilbert spaces, and exhibits nuanced behavior in the presence of jumps, Gaussian noise, and operator perturbations.

1. Formulation of the Directional Cameron–Martin Hypothesis

The directional CM hypothesis is imposed on the jump–drift channel of an OU process driven by Lévy noise $Z$ and Gaussian noise $W$. Consider two generators $A$ and $\widetilde{A}$ of $C_0$-semigroups $S$ and $\widetilde{S}$, with a measurable drift term $b$, and a covariance operator $Q$. The discrepancy arising from a generator perturbation is defined by: $$\Delta_{Jb}^{A \to \widetilde{A}}(t) = \int_0^t [S(t-s) - \widetilde{S}(t-s)] \, dZ_s + \int_0^t [S(t-s) - \widetilde{S}(t-s)] b \, ds$$ The hypothesis postulates the existence of a predictable process $U^{A \to \widetilde{A}}$ in $L^2(\Omega \times [0,T]; H)$, independent of the Gaussian noise, such that: $$\Delta_{Jb}^{A \to \widetilde{A}}(t) = \int_0^t \widetilde{S}(t-s) Q^{1/2} U^{A \to \widetilde{A}}_s \, ds$$ almost surely, and

$$\mathbb{E} \exp \left( \frac{1}{2} \int_0^T \|U^{A \to \widetilde{A}}_s\|^2 \, ds \right) < \infty$$

This Novikov-type condition ensures the applicability of the Girsanov theorem for the Gaussian channel, allowing transfer of absolute continuity via a shift lying in the CM space of the perturbed OU process.

2. Gaussian Channel and Hilbert–Schmidt Perturbation

The Gaussian component of OU dynamics is modeled by a $Q$-Wiener process. Law equivalence of OU processes under generator perturbations requires controlling the difference between the Gaussian convolutions: $$Y^A(W)(t) = \int_0^t S(t-s) \, dW_s, \quad Y^{\widetilde{A}}(W)(t) = \int_0^t \widetilde{S}(t-s) \, dW_s$$ A Hilbert–Schmidt perturbation condition is imposed: $$\int_0^T \|K S(t) Q^{1/2}\|^2_{HS} \, dt < \infty, \quad K = \widetilde{A} - A$$ This criterion, combined with results from Brzeźniak–van Neerven and Peszat, guarantees mutual absolute continuity of the path laws in the Skorokhod space $D_{H,T}$ for the two OU processes.

3. Jump–Drift Channel and Cameron–Martin Representation

For the jump–drift channel, the hypothesis is that the discrepancies between the original and perturbed processes are representable in the CM space of the perturbed process. The necessary condition is that: $\Delta_{Jb}^{A \to \widetilde{A}}(t)$ lies in the range of the operator: $$K_{\widetilde{A}} : u \mapsto \int_0^t \widetilde{S}(t-s) Q^{1/2} u(s) \, ds$$ If this is violated, or the Novikov integrability fails, law equivalence on path space is lost.

4. Rigidity under Pure Jump Noise

A rigidity phenomenon is observed when the Gaussian component is absent ($W \equiv 0$, $b=0$). Theorem 3.4 states that, if $L(X^A) \ll L(X^{\widetilde{A}})$ on path space, necessarily $X^A(t) = X^{\widetilde{A}}(t)$ almost surely for all $t \in [0,T]$. Thus, for pure jump-driven OU dynamics, any nontrivial mismatch in the dynamics cannot be compensated without destroying absolute continuity.

5. Sectorial Elliptic Generators: Verifiable Criteria

When specializing to analytic semigroups with sectorial operators (as in SPDEs with elliptic generators), fractional boundedness is required: $$K A^{-\beta} \in L(H)\quad \text{for some }\beta \in (0,1/2)$$ Duhamel's formula applies,

$$\widetilde{S}(t) - S(t) = \int_0^t \widetilde{S}(t-s) K S(s) ds$$

and analytic smoothing bounds provide effective estimates: $\|A^\beta S(t)\| \leq C_\beta t^{-\beta}$ For compound Poisson noise, the jump law must satisfy an exponential moment bound: $$\int_H \exp\left(c \|Q^{1/2} x\|^2 \right) \nu(dx) < \infty$$ Sectorial resolvent bounds offer explicit pathways for verifying (CM) and Hilbert–Schmidt conditions.

Condition	Mathematical Formulation	Channel
Hilbert–Schmidt	$\int_0^T \|K S(t) Q^{1/2}\|^2_{HS} dt < \infty$	Gaussian
Fractional Boundedness	$K A^{-\beta} \in L(H)$, $\|A^\beta S(t)\| \leq C_\beta t^{-\beta}$	Both
Exponential Moment	$\int_H \exp(c \|Q^{1/2} x\|^2 ) \nu(dx) < \infty$	Jump–Drift
Resolvent Bound	$$\sup_{\lambda \in \Sigma_\theta} \|K (\lambda + A)^{-1}\| | \lambda|^\beta < \infty$$	Both

6. Counterexamples and Sensitivity

Section 6 provides explicit diagonal counterexamples (often in $H = \ell^2(\mathbb{N})$) exposing several failure modes:

• The L$^2$ norm of the CM representation may blow up even if the Hilbert–Schmidt condition holds, e.g.,

$$\|U\|_{L^2}^2 = \sum_n \frac{(\widetilde{a}_n - a_n)^2}{2 a_n q_n}(1 - e^{-2 a_n (T-s)}) |\xi_n|^2 = \infty$$

• Asymmetry can occur: (CM) holds from $A$ to $\widetilde{A}$ but not in reverse, resulting in one-sided absolute continuity.
• Violations of exponential integrability, as with jump laws lacking finite moments, e.g., $\mathbb{E} \exp(c \xi^2) = \infty$ for Student $t$ jumps.
• Breakdown of naive factorizations in the CM argument where required operator norms diverge.

These demonstrate that L$^2$ representability alone is insufficient; Novikov's condition and symmetry are essential for full equivalence. The sensitivity of (CM) to both analytic and probabilistic properties of the underlying dynamics is sharp.

7. Implications and Scope

The Directional Cameron–Martin Hypothesis is pivotal in filtering, simulation, and control for SPDEs with Lévy noise, as it underpins when law equivalence is preserved under operator or noise perturbations. It is sharp in the presence of jumps: unless precise analytic and probabilistic criteria are met, equivalence fails or is only one-sided. The hypothesis generalizes the classical CM theorem to fully non-Gaussian contexts and exposes a deep correspondence between the geometric structure of the CM space, moment conditions on the jump law, and analytic smoothing properties of the generator. The interplay between these ingredients must be checked both analytically (via resolvent/smoothing estimates) and probabilistically (via exponential integrability) to ensure robustness of law equivalence under perturbations.