Abstract

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the Lp-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian measure. To prove the equivalence we show the Wr,p(B,μ)-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We discuss connections to Gaussian Hausdorff measures. Roughly speaking, if Lp-uniqueness holds then the ‘removed’ set Σ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least 2p. For p = 2 we obtain parallel results on truncations, capacities and essential self-adjointness for Ornstein-Uhlenbeck operators with linear drift. These results apply to the time zero Gaussian free field as a prototype example.

Highlights

  • The present article deals with capacities associated with Ornstein-Uhlenbeck operators on abstract Wiener spaces (B, μ, H ), [8, 11, 24, 32, 35,36,37, 53, 58], and applications to Lpuniqueness problems for Ornstein-Uhlenbeck operators and their integer powers, endowed with algebras of functions vanishing in a neighborhood of a small closed set

  • Our original motivation comes from Lp-uniqueness problems for operators L endowed with a suitable algebra A of functions, the special case p = 2 is the problem of essential

  • The basic tools to describe the critical size of a removed set ⊂ B are capacities associated with the Sobolev spaces W r,p(B, μ) for the H -derivative respectively the Ornstein-Uhlenbeck semigroup, [8, 11, 24, 32, 35,36,37, 53, 58]

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Summary

Introduction

The present article deals with capacities associated with Ornstein-Uhlenbeck operators on abstract Wiener spaces (B, μ, H ), [8, 11, 24, 32, 35,36,37, 53, 58], and applications to Lpuniqueness problems for Ornstein-Uhlenbeck operators and their integer powers, endowed with algebras of functions vanishing in a neighborhood of a small closed set. The basic tools to describe the critical size of a removed set ⊂ B are capacities associated with the Sobolev spaces W r,p(B, μ) for the H -derivative respectively the Ornstein-Uhlenbeck semigroup, [8, 11, 24, 32, 35,36,37, 53, 58] Such capacities can be introduced following usual concepts of potential theory, [11, 20, 37, 52, 53, 55, 56, 58], see Definition 3.1 below, and they are known to be connected to Gaussian Hausdorff measures,. From the Sobolev norm estimate for compositions we can deduce the desired equivalence of capacities, Theorem 3.5, where A is chosen to be the set of smooth cylindrical functions or the space of Watanabe test functions Applications of this equivalence provide Lp-uniqueness results for the Ornstein-Uhlenbeck operator and, under a sufficient condition that ensures they generate C0-semigroups, for its integer powers, see Theorem 5.2. Capacities, truncations and essential-selfadjointness of operators with linear drift are discussed in Sections 8 and 9

Preliminaries
Capacities and their Equivalence
Smooth Truncations
Lp-uniqueness of Powers of the Ornstein-Uhlenbeck Operator
Comments on Gaussian Hausdorff Measures
Comments on Stochastic Processes
Capacities and Truncations for Ornstein-Uhlenbeck Operators with Linear Drift
Essential Self-Adjointness of Ornstein-Uhlenbeck Operators with Linear Drift
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