[Paper Review] On the Ornstein-Uhlenbeck operator in convex sets of Banach spaces

Gianluca Cappa|arXiv (Cornell University)|Mar 10, 2015

Numerical methods in inverse problems7 references3 citations

TL;DR

This paper establishes Poincaré and Logarithmic-Sobolev inequalities for the Ornstein-Uhlenbeck operator on open convex subsets of infinite-dimensional separable Banach spaces equipped with a Gaussian measure. Using finite-dimensional approximation and Deuschel-Stroock's method, it proves spectral properties of the operator and derives functional inequalities without relying on Malliavin calculus or Wiener chaos decomposition.

ABSTRACT

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.

Motivation & Objective

• To study the Ornstein-Uhlenbeck operator and its associated semigroup in $L^2(\Omega, \gamma)$ for an open convex subset $\Omega$ of an infinite-dimensional separable Banach space $X$.
• To establish Poincaré and Logarithmic-Sobolev inequalities for the operator in such domains, despite the absence of explicit eigenfunction representations or Wiener chaos decomposition.
• To derive spectral properties of the Ornstein-Uhlenbeck operator using the functional inequalities.
• To provide a simplified proof of these inequalities using analytic tools and finite-dimensional approximation, avoiding heavy reliance on Malliavin calculus.

Proposed method

• Approximates the infinite-dimensional Ornstein-Uhlenbeck operator via finite-dimensional Ornstein-Uhlenbeck operators using cylindrical approximation of $\Omega$.
• Employs the Deuschel-Stroock method to derive the Poincaré and Log-Sobolev inequalities from semigroup properties.
• Uses the integration-by-parts formula for the Cameron-Martin space gradient $\nabla_H$ and properties of the Cameron-Martin space $H$.
• Establishes submarkovian and contractive properties of the semigroup $T^\Omega(t)$, including $[T^\Omega(t)(fg)]^2 \leq T^\Omega(t)(f^2)T^\Omega(t)(g^2)$ and $|\nabla_H T^\Omega(t)f|_H \leq e^{-t} T^\Omega(t)|\nabla_H f|_H$.
• Applies Fatou's lemma and approximation by cylindrical functions in $\mathcal{F}C^1_b(\Omega)$ to extend inequalities to the full Sobolev space $W^{1,2}(\Omega, \gamma)$.
• Uses Fernique's theorem to control growth of $\|x\|_X$ and ensures that multiplication by sublinear functions preserves $L^2(\Omega, \gamma)$ integrability.

Experimental results

Research questions

• RQ1 Can Poincaré and Logarithmic-Sobolev inequalities be established for the Ornstein-Uhlenbeck operator on convex subsets of infinite-dimensional Banach spaces without Wiener chaos decomposition?
• RQ2 What are the spectral properties of the Ornstein-Uhlenbeck operator on such domains, and how can they be deduced from functional inequalities?
• RQ3 Can the semigroup generated by the Ornstein-Uhlenbeck operator be shown to be submarkovian and contractive using finite-dimensional approximation?
• RQ4 How does the finite-dimensional approximation of the domain $\Omega$ allow for the derivation of global inequalities in infinite dimensions?
• RQ5 What is the role of the Cameron-Martin space gradient $\nabla_H$ in ensuring the validity of the inequalities in the infinite-dimensional setting?

Key findings

• The Poincaré inequality holds: $\int_\Omega \left|f - \int_\Omega f\,d\gamma\right|^2 d\gamma \leq \int_\Omega |\nabla_H f|_H^2 d\gamma$ for all $f \in W^{1,2}(\Omega, \gamma)$.
• The Logarithmic-Sobolev inequality holds: $\int_\Omega f^2 \log(f^2)\,d\gamma \leq \int_\Omega |\nabla_H f|_H^2\,d\gamma + \|f\|_{L^2(\Omega,\gamma)}^2 \log(\|f\|_{L^2(\Omega,\gamma)}^2)$ for all $f \in W^{1,2}(\Omega, \gamma)$.
• The semigroup $T^\Omega(t)$ is submarkovian: if $0 \leq f \leq 1$ $\gamma$-a.e., then $0 \leq T^\Omega(t)f \leq 1$ $\gamma$-a.e. for all $t > 0$.
• The semigroup satisfies the pointwise inequality $|\nabla_H T^\Omega(t)f|_H \leq e^{-t} T^\Omega(t)|\nabla_H f|_H$ for all $f \in W^{1,2}(\Omega, \gamma)$ and $t \geq 0$.
• The functional $\Lambda_v: f \mapsto fv$ is continuous on $W^{1,2}(\Omega, \gamma)$ for any measurable $v$ with $|v(x)| \leq k(\|x\|_X + 1)$, due to the Log-Sobolev inequality.
• The limit $\lim_{t \to \infty} T^\Omega(t)\varphi = m_\Omega(\varphi)$ holds in $L^2(\Omega, \gamma)$ for $\varphi \in \mathcal{F}C^1_b(\Omega)$ with $\varphi \geq c^2$, and this is used to derive the Log-Sobolev inequality via integration of the derivative of the entropy.