[Paper Review] Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

Beniamin Gołdys, Jan van Neerven|ArXiv.org|Jun 30, 2006

Stochastic processes and financial applications30 references21 citations

TL;DR

This paper establishes explicit conditions for key properties—such as the strong Feller property, spectral gap, and analyticity—of transition semigroups associated with Banach space-valued Ornstein-Uhlenbeck processes. It links these properties to the behavior of the underlying $C_0$-semigroup $\mathbf{S}$ restricted to the Cameron-Martin space $H$, showing that $\mathbf{S}_H$-analyticity is both necessary and sufficient for the semigroup's analyticity in $L^2(E, \mu_\infty)$, with similar characterizations for the strong Feller and spectral gap properties under $\mathbf{S}$-invariance of $H$. The results generalize and refine existing criteria for stochastic PDEs with dissipative drifts.

ABSTRACT

We investigate the transition semigroup of the solution to a stochastic evolution equation $dX(t) = AX(t)dt +dW_H(t)$, $t\ge 0,$ where $A$ is the generator of a $C_0$-semigroup $S$ on a separable real Banach space $E$ and $W_H$ is cylindrical white noise with values in a real Hilbert space $H$ which is continuously embedded in $E$. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of $S$ in $H$. In particular we investigate the interplay between analyticity of the transition semigroup, $S$-invariance of $H$, and analyticity of the restricted semigroup $S_H$.

Motivation & Objective

• To characterize the strong Feller property, spectral gap, and analyticity of transition semigroups for Banach space-valued Ornstein-Uhlenbeck processes.
• To establish necessary and sufficient conditions for these properties in terms of the restricted semigroup $\mathbf{S}_H$ on the Cameron-Martin space $H$.
• To extend existing criteria for stochastic PDEs with dissipative drifts by linking semigroup properties to the invariance and analyticity of $H$ under $\mathbf{S}$.
• To provide a unified framework for analyzing transition semigroups when $E$ is a general separable Banach space, not necessarily a Hilbert space.

Proposed method

• Analyzes the transition semigroup $\mathbf{P} = \{P(t)\}_{t \geq 0}$ defined by $P(t)\phi(x) = \mathbb{E}[\phi(S(t)x + \int_0^t S(t-s)\,dW_H(s))]$, with $W_H$ cylindrical Wiener process on Hilbert space $H \subset E$.
• Investigates the interplay between the $C_0$-semigroup $\mathbf{S}$ on Banach space $E$, its restriction $\mathbf{S}_H$ to $H$, and the reproducing kernel Hilbert spaces $H_t$ associated with the finite-time laws $\mu_t$.
• Uses the Liapunov equation $AX + XA^* = -Q$ to characterize $Q$-symmetry and $\mathbf{S}$-invariance of $H$, linking spectral properties of $A$ to $H$-invariance.
• Applies interpolation theory and the BIP (bounded analyticity in the positive sector) property to identify conditions under which $\mathbf{S}_H$ is analytic.
• Establishes that $\mathbf{P}$ is analytic in $L^2(E, \mu_\infty)$ if and only if $\mathbf{S}_H$ is analytic, under the assumption that $H$ is $\mathbf{S}$-invariant.
• Employs the mixed topology $\tau_{\text{mixed}}$ on $C_b(E)$ to study the $C_0$-semigroup structure of $\mathbf{P}$, avoiding strong continuity in the sup-norm topology.

Experimental results

Research questions

• RQ1 Under what conditions is the transition semigroup of a Banach space-valued Ornstein-Uhlenbeck process strongly Feller?
• RQ2 When does the generator of the Ornstein-Uhlenbeck semigroup possess a spectral gap in $L^2(E, \mu_\infty)$?
• RQ3 What is the precise relationship between the analyticity of the transition semigroup $\mathbf{P}$ and the analyticity of the restricted semigroup $\mathbf{S}_H$ on the Cameron-Martin space $H$?
• RQ4 How does $\mathbf{S}$-invariance of $H$ affect the spectral and regularity properties of the transition semigroup?
• RQ5 What conditions ensure the existence of an invariant measure $\mu_\infty$ and the validity of the semigroup formula $P(t)\phi(x) = \int_E \phi(S(t)x + y)\,d\mu_t(y)$ in general Banach spaces?

Key findings

• The Ornstein-Uhlenbeck semigroup $\mathbf{P}$ is analytic in $L^2(E, \mu_\infty)$ if and only if the restricted semigroup $\mathbf{S}_H$ is analytic on $H$, under the assumption that $H$ is $\mathbf{S}$-invariant.
• The strong Feller property of $\mathbf{P}$ holds if and only if $\mathbf{S}_H$ is analytic and satisfies a certain decay condition on the Hilbert-Schmidt norm of $S(t) \circ i_H$.
• The spectral gap property of the generator of $\mathbf{P}$ in $L^2(E, \mu_\infty)$ is equivalent to the spectral gap of $A$ in the reproducing kernel Hilbert space $H_\infty$ associated with the invariant measure $\mu_\infty$, under $\mathbf{S}$-invariance of $H$.
• For $E = L^p(\Omega)$ with $p \in [2, \infty)$ and $H = H_0^\alpha(\Omega)$, the semigroup $\mathbf{P}$ is analytic in $L^2(E, \mu_\infty)$ whenever $\alpha > \frac{d}{4} - \frac{1}{2}$ and $\alpha$ satisfies appropriate Sobolev embedding conditions.
• The condition $\int_0^\infty \|A^{-\beta/2} S(t)\|_{\mathscr{L}_2(L^2(\Omega))}^2 dt < \infty$ implies Hypothesis (H$\mu_\infty$), which is sufficient for analyticity of $\mathbf{P}$ in $L^2(E, \mu_\infty)$.
• In the case $E = L^2(\Omega)$ and $H = H_0^\beta(\Omega)$ with $\beta > \frac{d}{4} - \frac{1}{2}$, the semigroup $\mathbf{S}_H$ is analytic and contractive on $H$, ensuring analyticity of $\mathbf{P}$.