Regularization and Asymptotic Behaviour of Ornstein-Uhlenbeck Evolution Operators in Infinite Dimension

Technical Summary: Regularization and Asymptotic Behaviour of Ornstein-Uhlenbeck Evolution Operators in Infinite Dimension

Problem Statement
The paper investigates the properties of Ornstein-Uhlenbeck (OU) evolution operators acting on functions defined in a separable Hilbert space $X$. Specifically, the study focuses on the non-autonomous, infinite-dimensional setting where the coefficients of the associated backward Kolmogorov equation are time-dependent. The primary objectives are twofold:

1. To establish general regularization properties of these operators when acting on $L^p$ spaces with respect to a suitable evolution system of measures.
2. To analyze the asymptotic behavior of the operators in the time-periodic case, specifically identifying the optimal convergence rate to the mean with respect to time and providing a criterion for this optimality based solely on the drift term.

While the theory for autonomous OU operators in infinite dimensions is well-established, the non-autonomous case remains less complete. Previous work has addressed finite-dimensional non-autonomous equations and specific functional inequalities (e.g., strong Feller property, hypercontractivity) in infinite dimensions, but a comprehensive analysis of asymptotic decay rates and their optimality in the $L^p$ context was lacking.

Methodology
The analysis is structured around the evolution operator $\{U(t, s)\}$ associated with a family of linear operators $\{A(t)\}$ and a bounded family $\{B(t)\}$. The OU evolution operator $P_{s,t}$ is defined via a Gaussian measure with mean $U(t, s)x$ and covariance $Q(t, s)$.

The methodology proceeds through several key stages:

• Evolution Systems of Measures: The authors introduce the family of centered Gaussian measures $\mu_t = N(0, Q(t, -\infty))$, termed an "evolution system of measures." These generalize the concept of invariant measures to the non-autonomous context, satisfying the invariance property $\int P_{s,t}\phi \, d\mu_s = \int \phi \, d\mu_t$.
• Extension to $L^p$ Spaces: The operator $P_{s,t}$, initially defined on bounded continuous functions $C_b(X)$, is extended to $L^p(X, \mu_t)$. This extension relies on the invariance property and the contraction nature of the operator.
• Regularization via Sobolev Spaces: A core technical component involves proving that $P_{s,t}$ maps $L^p$ functions into first-order Sobolev spaces $W^{1,p}(X, \mu_s)$. This requires handling both scalar and vector-valued functions. The authors define vector-valued Sobolev spaces using cylindrical functions and the closure of the gradient operator in the Hilbert-Schmidt norm. The proof of regularization utilizes the Cameron-Martin theorem and the "controllability condition" (Assumption 8), which ensures the image of the evolution operator is contained within the image of the square root of the covariance operator.
• Spectral Analysis in the Periodic Case: Assuming the coefficients $A(\cdot)$ and $B(\cdot)$ are $T$-periodic, the authors employ the spectral theory of compact operators. They analyze the operator $P_{0,T}$ restricted to the subspace of zero-mean functions. The compactness of $P_{0,T}$ (derived from the regularization results) allows the use of spectral radius formulas to estimate convergence rates.
• Linking Spectra: A critical step involves relating the spectrum of the OU operator $P_{0,T}$ to the spectrum of the adjoint evolution operator $U(T, 0)^*$. The authors demonstrate that eigenvalues of $U(T, 0)^*$ correspond to eigenvalues of $P_{0,T}$, specifically for linear eigenfunctions.

Key Contributions and Results

1. Regularization and Compactness: The paper proves that for $p > 1$, the extended OU operator $P_{s,t}: L^p(X, \mu_t) \to L^p(X, \mu_s)$ is a compact operator. Its image is contained in the Sobolev space $W^{1,p}(X, \mu_s)$, and the gradient satisfies a specific estimate involving the operator $\Lambda(t, s) = Q(t, s)^{-1/2}U(t, s)$. This compactness is pivotal for the subsequent spectral analysis.

2. Exponential Convergence Rate: In the time-periodic case, the paper establishes that the convergence of $P_{s,t}f$ to its mean $\int f d\mu_t$ is exponential. Specifically, for any $\omega$ strictly greater than the decay rate $\omega_0$ of the evolution operator $U(t, s)$ (where $\|U(t, s)\| \leq Me^{\omega_0(t-s)}$), there exists a constant $C_{T,\omega}$ such that:
   $$\| P_{s,t}f - \int_X f d\mu_t \|_{L^p(X, \mu_s)} \leq C_{T,\omega} e^{\omega(t-s)} \|f\|_{L^p(X, \mu_t)}$$
   The spectral radius of $P_{0,T}$ restricted to zero-mean functions is shown to be bounded by $e^{\omega_0 T}$.

3. Optimality Criterion: The paper provides a sufficient condition for the convergence rate to be optimal, meaning the rate cannot be improved beyond $e^{\omega_0(t-s)}$. The condition is that all eigenvalues of $U(T, 0)^*$ with maximum modulus $e^{\omega_0 T}$ must be semisimple. If this condition holds, the estimate holds with $\omega = \omega_0$.

   • Crucially, the authors note that this optimality criterion depends only on the drift term $A(\cdot)$ (via $U(T, 0)^*$) and not on the diffusion term $B(\cdot)$.

4. Concrete Example: Section 5 constructs a concrete example in $X = L^2(\Omega)$ where $A(t)$ is a time-periodic elliptic differential operator and $B(t)$ is a fractional power of $A(t)^{-1}$. The authors verify that this system satisfies all assumptions, including the controllability condition and the non-degeneracy of measures. Furthermore, they prove that for this specific example, the eigenvalue of $U(T, 0)^*$ with maximum modulus is unique and semisimple, thereby confirming that the derived convergence rate is indeed optimal for this class of operators.

Significance
The paper claims to advance the theory of non-autonomous evolution equations in infinite dimensions by:

• Providing a rigorous framework for $L^p$ regularization of OU operators with respect to evolution systems of measures.
• Establishing a direct link between the spectral properties of the underlying deterministic evolution operator $U(t, s)$ and the asymptotic decay of the stochastic OU operator $P_{s,t}$.
• Offering a verifiable criterion (semisimplicity of the maximal eigenvalues of the drift operator) to determine the sharpness of the convergence rate.
• Demonstrating that the optimal convergence speed is governed entirely by the first-order coefficients (drift) of the generator, a result that simplifies the analysis of complex stochastic systems.

The work bridges the gap between the well-understood autonomous case and the more complex non-autonomous scenario, providing tools to analyze the long-term behavior of solutions to infinite-dimensional stochastic differential equations.