Capacities, Removable Sets and Lp-Uniqueness on Wiener Spaces

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.


Introduction
The present article deals with capacities associated with Ornstein-Uhlenbeck operators on abstract Wiener spaces (B, μ, H ), [8, 11, 24, 32, 35-37, 53, 58], and applications to L puniqueness 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 L p -uniqueness problems for operators L endowed with a suitable algebra A of functions, the special case p = 2 is the problem of essential self-adjointness. For the 'globally defined' operator L on the entire space L p -uniqueness is well understood, see for instance [18] and the references cited there. If the globally defined operator is L p -unique one can ask whether the removal of a small set (or, in other words, the introduction of a small boundary) destroys this uniqueness or not. A loss of uniqueness means that extensions to generators of C 0 -semigroups, [45], with different boundary conditions exist. The answer to this question depends on the size of the removed set. The most classical example may be the essential self-adjointness problem for the Laplacian on R n , endowed with the algebra C ∞ c (R n \ {0}) of smooth compactly supported functions on R n with the origin {0} removed. It is well known that this operator is essential self-adjoint in L 2 (R n ) if and only if n ≥ 4, [59, p.114] and [47,Theorem X.11,p.161]. Generalizations of this example to manifolds have been provided in [12] and [38], more general examples on Euclidean spaces can be found in [5] and [27], further generalizations to manifolds and metric measure spaces will be discussed in [28]. For the Laplacian on R n one main observation is that, if a compact set of zero measure is removed from R n , the essential self-adjointness of ( , C ∞ c (R n \ )) in L 2 (R n ) implies that dim H ≤ n − 4, where dim H denotes the Hausdorff dimension. See [5,Theorems 10.3 and 10.5] or [27,Theorem 2]. This necessary 'codimension four' condition can be rephrased by saying that we must have H n−d ( ) = 0 for all d < 4, where H n−d denotes the Hausdorff measure of dimension n − d.
Having in mind coefficient regularity or boundary value problems for operators in infinite dimensional spaces, see e.g. [10,13,14,25,26], one may wonder whether a similar 'codimension four' condition can be observed in infinite dimensional situations. For the case of Ornstein-Uhlenbeck operators on abstract Wiener spaces an affirmative answer to this question follows from the present results in the special case p = 2.
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, 35-37, 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, [21]. Uniqueness problems connect easier to another, slightly different definition of capacities, where the functions taken into account in the definition are recruited from the initial algebra A and, roughly speaking, are required to be equal to one on the set in question, see Definitions 3.2 and 3.3. This type of definition connects them to an algebraic ideal property which is helpful to investigate extensions of operators initially defined on ideals of A. For Euclidean Sobolev spaces these two types of capacities are known to be equivalent, see for instance [2,Section 2.7]. The proofs of these equivalences go back to Maz'ya, Khavin, Adams, Hedberg, Polking and others, [1][2][3][41][42][43][44], and rely on bounds in Sobolev norms for certain nonlinear composition operators acting on the cone of nonnegative Sobolev functions, see e.g. [1,Theorem 3], or the cone of potentials of nonnegative functions, see e.g. [1,Theorem 2] or [2,Theorem 3.3.3]. Apart from the first order case r = 1 this is nontrivial, because in finite dimensions Sobolev spaces are not stable under such compositions, see for instance [2,Theorem 3.3.2]. Apart from the case p = 2, where one can also use an integration by parts argument, [1,Theorem 3], the desired bounds are shown using suitable Gagliardo-Nirenberg inequalities, [3,42], or suitable multiplicative estimates of Riesz or Bessel potential operators involving Hardy-Littelwood maximal functions and the L pboundedness of the latter, [ [8,Remark 5.2.1 (i)], but one still needs to establish quantitative bounds. We establish Sobolev norm bounds for nonlinear composition operators acting on potentials of nonnegative functions, Theorem 3.6. To obtain it, we use the L p -boundedness of the maximal function in the sense of Rota and Stein for the Ornstein-Uhlenbeck semigroup, [53,Theorem 3.3], this provides a similar multiplicative estimate as in the finite dimensional case, see Lemma 4.2. From the Sobolev norm estimate for compositions we can then 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 L p -uniqueness results for the Ornstein-Uhlenbeck operator and, under a sufficient condition that ensures they generate C 0 -semigroups, also for its integer powers, see Theorem 5.2. In particular, if ⊂ B is a given closed set of zero Gaussian measure, then the Ornstein-Uhlenbeck operator, endowed with the algebra of cylindrical functions vanishing in a neighborhood of (or the algebra of Watanabe test functions vanishing q.s. on a neighborhood of ) is L p -unique if and only if the (2, p)-capacity of is zero, see Theorem 5.2. Combined with results from [21] on Gaussian Hausdorff measures, we then observe that the L p -uniqueness of this Ornstein-Uhlenbeck operator 'after the removal of ' implies that the Gaussian Hausdorff measure d ( ) of codimension d of must be zero for all d < 2p, see Corollary 6.2. In particular, if the operator is essentially self-adjoint on L 2 (B, μ), then d ( ) must be zero for all d < 4, what is an analog of the necessary 'codimension four' condition known from the Euclidean case. In Sections 8 and 9 we partial rework the arguments to obtain results on essential self-adjointness of Ornstein-Uhlenbeck operators with linear drift as studied for instance in [4,7,51,52], the prominent example being the Hamiltonian of the time zero Gaussian free field in Euclidean quantum field theory, [46,48,49,54], Example 9.3. Again the characterization of essential self-adjointness, Theorem 9.1, is obtained from a capacitary equivalence, Theorem 8.7, based on a truncation result for potentials, Theorem 8.9.
In the next section we recall standard items from the analysis on abstract Wiener spaces. In Section 3 we define Sobolev capacities and prove their equivalence, based on the norm bound on nonlinear compositions, which is proved in Section 4. Section 5 contains the mentioned L p -uniqueness results. The connection to Gaussian Hausdorff measures is briefly discussed in Section 6, followed by some remarks on related Kakutani theorems for multiparameter processes in Section 7. Capacities, truncations and essential-selfadjointness of operators with linear drift are discussed in Sections 8 and 9.

Preliminaries
Following the presentation in [53], we provide some basic definitions and facts. Let (B, μ, H ) be an abstract Wiener space. That is, B is a real separable Banach space, H is a real separable Hilbert space which is embedded densely and continuously on B, and μ is a Gaussian measure on B with The space of all such cylindrical functions on B we denote by FC ∞ b . Clearly FC ∞ b is an algebra under pointwise multiplication and stable under the composition with functions (2) is infinitely many times H -differentiable at any x ∈ B, and for any k ≥ 1 we have where ∂ j denotes the j -th partial differentiation in the Euclidean sense. The space FC ∞ b is dense in L p (B, μ) for any 1 ≤ p < +∞, see e.g. [7, Lemma 2.1].
We write H 0 := R, H 1 := H and generalizing this, denote by H k the space of k-linear maps a : H k → R such that where (e i ) ∞ i=1 is an orthonormal basis in H . The value of this norm does not depend on the choice of this basis. See [9, p.3]. Clearly every such k-linear map a can also be seen as a linear map a : H ⊗k → R, where H ⊗k denotes the k-fold tensor product of H , with this interpretation we have a(e j 1 ⊗ ... ⊗ e j k ) = a(e j 1 , . . . , e j k ) and by Eq. (4) the operator a is a Hilbert-Schmidt operator. For later use we record the following fact. Choose an element y = y 1 ⊗ ... ⊗ y k ∈ H ⊗k such that |y| H ⊗k = 1 and |a| H k ≤ 2|ay|. Without loss of generality we may assume that |y j | H = 1, 1 ≤ j ≤ k. Choosing an orthonormal basis (b i ) n i=1 in the subspace span {y 1 , ..., y k } of H we observe n ≤ k and we obtain the desired result.
We recall the definition of Sobolev spaces on B. For any 1 ≤ p < +∞ and k ≥ 0 let L p (B, μ, H k ) denote the L p -space of functions from B into H k . For any 1 ≤ p < +∞ and integer r ≥ 0 set  (B, μ).
In contrast to Sobolev spaces over finite dimensional spaces, [2, Theorem 3.3.2], also the Sobolev classes W r,p (B, μ), r ≥ 2, are known to be stable under compositions u → T (u) = T •u with functions T ∈ C ∞ b (R), as follows from the evaluation of an integration by parts identity together with the chain rule, applied to cylindrical functions. See [8, Remark 5.2.1 (i)] or [9,Proposition 8.7.5]. In particular, the space W ∞ is stable under compositions with functions from C ∞ b (R). Also, it is an algebra with respect to pointwise multiplication, [37,Corollary 5.8].
Given a bounded (or nonnegative) Borel function f : B → R and t > 0 set The function P t f is again bounded (resp. nonnegative) Borel on B and the operators P t form a semigroup, i.e. that for any s, t > 0 we have P t+s = P t P s .  (p) , D(L (p) )) of (P (p) t ) t>0 is called the Ornstein-Uhlenbeck operator on L p (B, μ), [53, Section 2.1.4]. We will always write P t and L instead of P (p) t and L (p) , the meaning will be clear from the context. Given r > 0 and a bounded (or nonnegative) Borel function f : B → R, set where denotes the Euler Gamma function. The function V r f is again bounded (resp. nonnegative) Borel, and for any 1 ≤ p < ∞ the operators V r form a strongly continuous contraction semigroup (V r ) r>0 on L p (B, μ), see [8,Corollary 5.3.3] or [53,Proposition 4.7], symmetric for p = 2. In any of these spaces the operators V r are the powers (I − L) −r/2 of order r/2 of the respective 1-resolvent operators (I − L) −1 . Meyer's equivalence, [9,Theorem 8.5.2], [53,Theorem 4.4], states that for any integer r ≥ 1 and any 1 < p < +∞ and any u ∈ W r,p (B, μ) we have with constants c 1 > 0 and c 2 > 0 depending only on r and p. By the continuity of the V r and the density of cylindrical functions we observe W r,p (B, μ) = V r (L p (B, μ)) in the sense of equivalently normed spaces. Note that for p = 2 the middle terms in Eq. (8)

Proposition 2.2 For any
Proof From the preceding lines it is immediate that V r (W ∞ ) ⊂ W ∞ . To see the remaining statement suppose f ∈ FC ∞ b with f = F (l 1 , ..., l n ), l i ∈ B * , F ∈ C ∞ b (R n ), and by applying Gram-Schmidt we may assume {l 1 , ..., l n } is an orthonormal system in H . The Ornstein-Uhlenbeck semigroup (T (n) t ) t>0 on L 2 (R n ), defined by . Given x ∈ B and writing ξ = ( x, l 1 , ..., x, l n ), we have , and using Eq. (7) and dominated convergence it follows that V r f ∈ FC ∞ b .
Although different in nature both FC ∞ b and W ∞ can serve as natural replacements in infinite dimensions for algebras of smooth differentiable functions in Euclidean spaces or on manifolds.

Capacities and their Equivalence
We define two types of capacities related to W r,p (B, μ)-spaces and verify their equivalence. The following definition is standard, see for instance [20,23,33,52]. We give two further definitions of (r, p)-capacities in which we insist on a strict equality on the set to be tested. In finite dimensional spaces such capacities were introduced in [39], see also [2, Definition 2.7.1], [40] and [43,Chapter 13]. The first definition we give is based on cylindrical functions.
and for an arbitrary set A ⊂ B, The capacities cap have a more 'algebraic' flavor and are well suited to study operator extensions, see Section 5.
One can give a similar definition based on the space W ∞ . To do so, we recall some potential theoretic notions. If a property holds outside a set E ⊂ B with Cap r,p (E) = 0 then we say it holds (r, p)-quasi everywhere (q.e.). We follow [37, Chapter IV, Section 1.2] and call a set E ⊂ B slim if Cap r,p (E) = 0 for all 1 < p < +∞ and all integer r > 0, and if a property holds outside a slim set, we say it holds quasi surely (q.s.). A function u : B → R is said to be (r, p)-quasi continuous if for any ε > 0 we can find an open set U ε ⊂ B such that Cap r,p (U ) < ε and the restriction u| U c ε of u to U c ε is continuous. Every function u ∈ W r,p (B, μ) admits a (r, p)-quasi-continuous version u, unique in the sense that two different quasi continuous versions can differ only on a set of zero (r, p)-capacity. Since continuous functions are dense in W r,p (B, μ) this follows by standard arguments, see for instance [11, Chapter I, Section 8.2]. Now one can follow [37, Chapter IV, Section 2.4] to see that for any u ∈ W ∞ there exists a function u : B → R such that u = u μ-a.e. and for all r and p the function u is (r, p)-quasi continuous. It is referred to as the quasi-sure redefinition of u and it is unique in the sense that the difference of two quasi-sure redefinitions of u is zero (r, p)-quasi everywhere for all r and p, [37].
where u denotes the quasi-sure redefinition of u with respect to the capacities from Definition 3.1, and for an arbitrary set A ⊂ B, For some applications capacities based on the algebra W ∞ may be more suitable that those based on cylindrical functions.

Remark 3.4 In [34, Example 3.13]
Kusuoka introduced capacities based on functions u ∈ W ∞ , but requiring that u ≥ 1 μ-a.e. on U (similarly as in Definition 3.1 above) in place of the condition that u = 1 on U q.s.
The following equivalence can be observed.
and One ingredient of our proof of Theorem 3.5 is a bound in W r,p (B, μ)-norm for compositions with suitable smooth truncation functions. For the spaces W 1,p (B, μ) such a bound is clear from the chain rule for D respectively from general Dirichlet form theory, see [11]. Norm estimates in W r,p (B, μ) for compositions T • u of elements u ∈ W r,p (B, μ) with suitable smooth functions T : R → R can be obtained via the chain rule. For instance, in the special case r = 2 the chain and product rules and the definition of the generator L imply for any u ∈ W 2,p (B, μ). By Eq. (8) it would now suffice to show a suitable bound for LT (u) in L p , and the summand more difficult to handle is the one involving the first derivatives Du. In the finite dimensional Euclidean case an L p -estimate for it follows immediately from a simple integration by parts argument, [1, Theorem 3], or by a use of a suitable Gagliardo-Nirenberg inequality, [3,42]. Integration by parts for Gaussian measures comes with an additional 'boundary' term involving the direction h ∈ H of differentiation that spoils the original trick, and the classical proof of the Gagliardo-Nirenberg inequality involves dimension dependent constants. A simple alternative approach, suitable for any integer r > 0, is to prove truncation results for potentials in a similar way as in [ Theorem 3.6 Assume 1 < p < +∞ and let r > 0 be an integer. Let T ∈ C ∞ (R + ) and suppose that T satisfies Then for every nonnegative f ∈ L p (B, μ) the function T •V r f is an element of W r,p (B, μ), and there is a constant c T > 0 depending only on p, r and L such that for every nonnegative Remark 3.7 To prove Theorem 3.5 the function T can be chosen much more specifically than in Theorem 3.6. However, Eq. (12) is the classical hypothesis introduced by Maz'ya, [40] (see also [2, Theorem 3.3.3]), and since the statement may be of independent interest, we prove Theorem 3.6 in this general form.
Another useful tool in our proof of Theorem 3.5 is the following 'intermediate' Due to Proposition 2.2 the right hand side in Eq. (14) Approximating f by bounded nonnegative functions in L p (B, μ), taking their cylindrical approximations, which are nonnegative as well, and smoothing by convolution in finite dimensional spaces, we can approximate f in L p (B, μ) by a sequence of nonnegative func- for all i = 1, 2, ... For any k = 1, 2, ... let now Now choose l such that Cap r,p (A l ) < ε 3 and then j large enough so that The topological support of μ is B, see for instance [8, Theorem 3.6.1, Definition 3.6.2 and the remark following it]. Since u n j is continuous by Proposition 2.2 we therefore have u n j ≥ 1 everywhere on V . Now, since Cap r,p is clearly subadditive and monotone, Using Theorem 3.6 and Lemma 3.8 we can now verify Theorem 3.5.
Proof We show Eq. (9). It suffices to consider open sets U .
with c 2 as in Eq. (8), so that it suffices to show with a suitable constant c > 0 depending only on r and p. Let and we arrive at Eq. (9) with c 3 := 1/c p T and c 4 := c (10) is an easy consequence.

Smooth Truncations
To verify Theorem 3.6 we begin with the following generalization of [8, formula (5.4.4) in Proposition 5.4.8].
where the functionsĥ i are as in Eq. (1) and Q : R n → R, n ≤ k, is a polynomial of degree k whose coefficients are constants or products of scalar products A straightforward calculation shows that with a polynomial Q as stated.
The next inequality is a counterpart to [2, Proposition 3.1.8]. It provides a pointwise multiplicative estimate for derivatives of potentials in terms of powers of the potential and a suitable maximal function.
Proof Suppose h 1 , ..., h k ∈ H are members of an orthonormal system in H , not necessarily distinct. Then for any δ > 0 we have, by dominated convergence, y), ...,ĥ k (y)) μ(dy)dt with a polynomial Q of degree k as in Proposition 4.1. Now let β > 1 be a real number such that r 2k < β < r k .
Hölder's inequality yields Using the fact that r ≥ 2, the elementary inequality e −t t ≤ 1 − e −2t , t ≥ 0, and the left inequality in Eq. (17), so that another application of Hölder's inequality, now with q, shows that the first factor on the right hand side of Eq. (18) is bounded by According to Proposition 4.1 the coefficients of the polynomial Q are bounded for fixed k, and since its degree does not exceed k, it involves only finitely many distinct products of powers of the functionsĥ i . Together with the fact that eachĥ i is N(0, 1)-distributed, this implies that there is a constant c 1 (k, q, β) > 0, depending on k but not on the particular choice of the elements h 1 , ..., h k , such that B |Q(ĥ 1 (y), ...,ĥ k (y))| βq μ(dy) Taking into account Eq. (17), we therefore obtain To estimate In a similar fashion we can then obtain the estimate where c 2 (k, q, γ ) > 0 is a constant depending on n but not on the particular choice of h 1 , ..., h k . We finally choose suitable δ > 0. The function can attain any value in (0, 1). Since Jensen's inequality implies we have sup t>0 (P t (f q )(x)) 1/q ≥ V r f (x) and can choose δ > 0 such that note that the denominator cannot be zero unless f is zero μ-a.e. Combining with Eqs. (19) and (21) we obtain for some constants c 1 (k, q, β), c 2 (k, q, γ ). For any given r there exist only finitely many numbers k < r and for any such k numbers β and γ as in Eqs. (17) and (20)  Proof If r = 1 then T has a bounded first derivative, and the desired bound is immediate from the definition of the norm · W 1,p (B,μ) , the chain rule for the gradient D and Meyer's equivalence, [53,Theorem 4.4]. In the following we therefore assume r ≥ 2.
We verify that for any k ≤ r the inequality holds with a constant c(k, L, p, r) > 0 depending only on k, L, p and r. If so, then summing up yields with a constant c T > 0 depending on L, p and r, as desired. To see Eq. (24) suppose k ≤ r and that h 1 , ..., h k are members of an orthonormal system (g i ) k i=1 , not necessarily distinct. To simplify notation, we use multiindices with respect to this orthonormal system: Given a multiindex α = (α 1 , ..., α k ) we write D α := ∂ α 1 g 1 · · · ∂ α k g k , where for β = 0, 1, 2, ..., a function u : B → R and an element g ∈ H we define ∂ β g u as the image of u under the application of β differentiations in direction g, Now let α be a multiindex such that D α = ∂ h 1 · · · ∂ h k . Then clearly |α| = k. Moreover, we have by the chain rule, where the interior sum is over all j -tuples (α 1 , ..., α j ) of multiindices α i such that |α i | ≥ 1 for all i and α 1 +α 2 +...+α j = α. The interior sum has k−1 j −1 summands. The C α 1 ,...,α j are real valued coefficients, and since there are only finitely many different C α 1 ,...,α j , there exists a constant C(k) > 0 which for all multiindices α with |α| = k dominates these constants, C α 1 ,...,α j ≤ C(k). In particular, C(k) does not depend on the particular choice of the elements h 1 , ..., h k . More explicit computations can for instance be obtained using [19].
The hypothesis Eq. (12) on T implies with a constant c(k) > 0 depending only on k and with L being as in Eq. (12). Since where 1 < q < +∞ is arbitrary and c(k, q, r) > 0 is a constant depending only on k, q and r. For the case j = 1 we have Taking the supremum over all h 1 , ..., h k ∈ H as above we obtain with a constant c(k, L, q, r) > 0 by Proposition 2.1. Fixing 1 < q < p and using the boundedness of the semigroup maximal function, [53, Theorem 3.3], we see that there is a constant c(p, q) > 0 depending only on p and q such that (sup μ) . On the other hand, by Eq. (8), we have Combining, we arrive at Eq. (24).

L p -uniqueness of Powers of the Ornstein-Uhlenbeck Operator
We discuss related uniqueness problems for the Ornstein Uhlenbeck operator L and its integer powers. Recall first that a densely defined operator (L, A) on L p (B, μ), 1 ≤ p < +∞ is said to be L p -unique if there is only one C 0 -semigroup on L p (B, μ) whose generator extends From Eq. (8) it follows that for any m = 1, 2, ... and 1 < p < +∞ we have D((−L) m ) = W 2m,p (B, μ). The density of FC ∞ b and W ∞ in the spaces W 2m,p (B, μ) and the completeness of the latter imply that ((−L) m , W 2m,p (B, μ)) is the closure in L p (B, μ) of ((−L) m , FC ∞ b ) and also of ((−L) m , W ∞ ). Since obviously (P t ) t>0 is a C 0 -semigroup, (L, FC ∞ b ) and (L, W ∞ ) are L p -unique in all L p (B, μ), 1 ≤ p < +∞. To discuss the its powers −(−L) m for m ≥ 2 we quote well known facts to provide a sufficient condition for them to generate C 0 -semigroups. Since (P t ) t>0 is a symmetric Markov semigroup on L 2 (B, μ), for any 1 < p < +∞ the operator L = L (p) generates a bounded holomorphic semigroup on L p (B, μ) with angle θ satisfying [17, Theorem 8] shows that (up to a discussion of limit cases) this is a sharp condition for −(−L) m to generate a bounded C 0 -semigroup. For 1 < p < +∞ this also recovers the L p -uniqueness in the case m = 1. For p = 2 condition Eq. (25) is always satisfied.
Alternatively we can conclude the generation of C 0 -semigroups on L 2 (B, μ) directly from the spectral theorem. For later use we fix the following fact. Here we are interested in L p -uniqueness after the removal of a small closed set ⊂ B of zero measure. This is similar to our discussion in [27] and, in a sense, similar to a removable singularities problem, see for instance [42] or [43] or [2, Section 2.7].
Let ⊂ B be a closed set of zero Gaussian measure and N := B \ . We define and it remains to show the converse inclusion.
Given  (N ). Now let j be fixed. For any 1 ≤ k ≤ 2m let h 1 , ..., h k be members of an orthonormal system (g i ) k i=1 , not necessarily distinct. As in the proof of Theorem 3.6 we use multiindex notation with respect to this orthonormal system. Let α be such that D α = ∂ h 1 · · · ∂ h k . Then, by the general Leibniz rule, where for two multiindices α and β we write β ≤ α if β i ≤ α i for all i = 1, ..., k. For any such β we clearly have and taking the supremum over all h 1 , ..., h k as above, The proof for W ∞ is similar.

Comments on Gaussian Hausdorff Measures
For finite dimensional Euclidean spaces the link between Sobolev type capacities and Hausdorff measures is well known and the critical size of a set in order to have (r, p)capacity zero or not is, roughly speaking, determined by its Hausdorff codimension, see e.g. [2,Chapter 5]. For Wiener spaces one can at least provide a partial result of this type.
Hausdorff measures on Wiener spaces of integer codimension had been introduced in [21, Section 1]. We briefly sketch their method but allow non-integer codimensions, this is an effortless generalization and immediate from their arguments.
Given an m-dimensional Euclidean space F and a real number 0 ≤ d ≤ m the spherical Hausdorff measure S d of dimension d can be defined as follows: For any ε > 0 set and finally, S d (A) := sup ε>0 S d ε (A), A ⊂ F . A priori S d is an outer measure, but its σ -algebra of measurable sets contains all Borel sets. For any 0 ≤ d ≤ m we define Combined with Theorem 5.2 this yields a necessary codimension condition which is similar as in the case of Laplacians on Euclidean spaces, [5,27].

Comments on Stochastic Processes
We finally like to briefly point out connections to known Kakutani type theorems for related multiparameter Ornstein-Uhlenbeck processes. The connection between Gaussian capacities, [20], and the hitting behavious of multiparameter processes, [29][30][31], has for instance been investigated in [6,55,56]. We briefly sketch the construction and main result of [56], later generalized in [6]. Let and initial law μ (0) . Let μ (1) denote the law of the process Z (1) , clearly a centered Gaussian measure on (1) (B). Next, let (T (1) t ) t>0 be the Ornstein-Uhlenbeck semigroup on (1) for any bounded Borel function f on (1) (B), and let Z (2) be the Ornstein-Uhlenbeck process taking values in (1) (B) with semigroup ( (1) ) t>0 and initial law μ (1) . Iterating this construction yields, for any integer r ≥ 1, an Ornstein-Uhlenbeck process Z (r) taking values in (r−1) (B). This process may also be viewed as an r-parameter process Z (r) = A more causal connection between uniqueness problems for operators and classical probability should involve certain branching diffusions rather than multiparameter processes, but even for finite dimensional Euclidean spaces the problem is not fully settled and remains a future project.

Capacities and Truncations for Ornstein-Uhlenbeck Operators with Linear Drift
In this section we investigate Ornstein-Uhlenbeck semigroups with linear drift as considered for instance in [4,7,51]. The main example we have in mind is the time zero Gaussian free field, [46,48,49], see Example 9.3. We therefore restrict attention to the special cases r = 1, 2, p = 2 and m = 1 (in the notation of Section 5) and follow [7] and [51].
Let (E, H, μ) be an abstract Wiener space and let (A, D(A)) be a strictly positive selfadjoint operator on H such that the operators e −tA , t > 0, extend to a strongly continuous contraction semigroup on E. We assume that K ⊂ E * ∩ D(A) is a dense subspace of E * , dense in D(A) w.r.t. the graph norm, that A(K) ⊂ K and that e −tA (K) = K, t > 0. (26) Remark 8.1 If H is a real separable Hilbert space and (A, D(A)) a strictly positive selfadjoint operator on H then one can find an inner product norm q, continuous on H and such that the embedding of H into the completion E of H w.r.t. q is Hilbert-Schmidt and the operators e −tA , t > 0, behave as stated. One can also find a space K as above. This is part of the statement of [7, Theorem 3.1]. If in this situation μ * is a standard Gaussian cylindrical measure on H then it induces a Gaussian measure μ on E, [7, Remark 3.2 (ii)].
For the special cases r = 1, 2, p = 2 and m = 1 under consideration it is convenient to use the corresponding semigroup as the starting point for subsequent developments. Given a bounded (or nonnegative) Borel function f : E → R and t > 0 set Since the operators √ 1 − e −2tA are bounded on E, [7,Lemma 3.5], this definition makes sense. The family of operators (P A,t ) t>0 is a symmetric sub-Markovian semigroup on L 2 (E, μ), referred to as the Mehler semigroup corresponding to (E, H, A, μ). Actually, it is a Feller semigroup on E, as shown in [7, Corollary 3.6] and commented in [51, Section 2, p. 732]. If A is the identity operator on H then we recover Eq. (6) form Eq. (27). Let (L A , D(L A )) denote the infinitesimal generator in L 2 (E, μ) of (P A,t ) t>0 , that is, the unique non-positive definite self-adjoint operator on L 2 (E, μ) such that P A,t = e tA , t > 0, see for instance [11]. It is referred to as the Ornstein-Uhlenbeck operator on L 2 (E, μ) with linear drift A. Similarly as in Eq. (7) we define V A,r f := 1 (r/2) ∞ 0 t r/2−1 e −t P A,t f dt for any r > 0 and bounded (or nonnegative) Borel function f : E → R. Being symmetric and Markovian, the semigroup (P A,t ) t>0 also induces (unique) strongly continuous contraction semigroups on the spaces L p (E, μ), 1 ≤ p < +∞ (as mentioned in Section 2), and for simplicity we denote them by the same symbol; likewise for their generators and the contractive operators V A,r = (I − L A ) −r/2 . For any 1 ≤ p < +∞ and r > 0 we define the Sobolev spaces W r,p A (E, μ) := V A,r (L p (E, μ)). Endowed with the norms these spaces are Banach, and for p = 2 Hilbert. We also consider the space Let FC ∞ b,K denote the collection of functions on E of form f = F (l 1 , ..., l n ), where n ≥ 1, F ∈ C ∞ b (R n ) and l 1 , ..., l n ∈ K. Clearly this space is an algebra, and it is dense in L p (E, μ), 1 ≤ p < +∞ (as can be seen using arguments as in [7, Lemma 2.1 and Proposition 5.4]).
With D defined on FC ∞ b,K by formula Eq. (3) with k = 1 we now set D A := √ AD. Then for f = F (l 1 , ..., l n ) ∈ FC ∞ b,K with F ∈ C ∞ b (R n ) and l i ∈ K and x ∈ E we obtain The following was proved in [7, Theorem 5.3, Proposition 5.4 and Theorem 5.5].
For background on Dirichlet form theory see for instance [11] or in the present context, [7]. Here we only point out that E A and L A are uniquely associated by the identity In terms of the Sobolev type spaces defined above, we observe that D(

Remark 8.3
For systematical reasons we mention the following results from [51], although we will not use them explicitely. Consider the norms on FC ∞ b,K defined by where H 2 is as in Eq. (4), and let W 1,2 A (E, μ) and W 2,2 A (E, μ) denote the completions of FC ∞ b,K in these norms, respectively. By the Meyer equivalence proved in [51, Theorems 3.1 and 3.6] the spaces W r,2 A (E, μ) and W r,2 A (E, μ), r = 1, 2, coincide in the sense of equivalently normed spaces. We remark that in [51] not the space FC ∞ b,K was used, but a space of polynomial functions based on E * ∩ ∞ k=1 D(A k ). However, for the cases r = 1, 2 the necessary modifications in the proof are inessential.
To discuss capacities based on the space W ∞ A below, we have to import two implications of the Meyer equivalence in [51]: The first is the fact that the space W ∞ A is an algebra (which can be seen as in the proof of [51,Theorem 4.3]) and the second is the fact that FC ∞ b,K ⊂ W r,p A (E, μ) for all r and p, so that in particular, The following is an analog of Proposition 2.2.

Proposition 8.4 For any
Proof Again the statement for W ∞ A is immediate. The statement for FC ∞ b,K can be proved similarly as in Proposition 2.2: If f = F (l 1 , ..., l n ) ∈ FC ∞ b,K with F ∈ C ∞ b (R n ) and l 1 , ..., l n ∈ K orthogonal in H , we have, for any x ∈ E,   [7,Theorem 6.7] it was shown that one can always find a Banach space E 1 such that E is continuously and densely embedded into E 1 , the operators e −tA , t > 0, extend to a strongly continuous contraction semigroup on E 1 , and when μ is considered as a measure on E 1 , the capacities Cap A,r,p , 1 < p < +∞, r > 0, associated with the Mehler semigroup corresponding to (E 1 , H, A, μ) are tight. The key items in the proof of this fact were the density of K in E * and Eq. (26). In [4,Corollary 1.5], quoted to prove [7,Theorem 6.7], the space E 1 was constructed as the completion of H w.r.t. the norm x E 1 := e −sA x E for fixed s > 0. Using Eq. (26) one can then conclude that e −sA l E * 1 ≤ l E * , for all l ∈ K, so that even the initial assumptions involving the space K remain valid.
Under Assumption 8.5 one can now define the notions (r, p)-quasi everywhere (q.e.), slim, quasi surely (q.s.) and (r, p)-quasi continuous in the same manner as in Section 3 (see [22], [37,Chapter IV] and (ii) Using the theory in [51] one can study capacities of type cap for general r and p and establish a more general version of Theorem 8.7. However, as this is not needed to discuss our main example and since in the presence of a drift A the corresponding Sobolev spaces of higher order are considerably more complicated to handle, we leave it to the interested reader.
We provide two results from which Theorem 8.7 is obtained by similar arguments as Theorem 3.5. The first is the following Theorem 8.9 which is a partial analog of Theorem 3.6 and of course interesting only for r = 2. Theorem 8.9 Let r = 1, 2 and let T ∈ C 2 (R) be as in Theorem 8.9, so that Eq. (12) holds. Then for any nonnegative f ∈ L 2 (E, μ) the function T • V A,r f is in W r,2 A (E, μ), and there is a constant c T > 0, depending only on r and on L in Eq. (12) such that for any nonnegative f ∈ L 2 (E, μ) we have The second result employed to prove Theorem 8.7 is the following Lemma 8.10, and since it can be shown in the same manner as Lemma 3.8, we omit its proof.
In the sequel we provide a proof for Theorem 8.9. To do so we use a spectral theoretic refinement of the arguments used to show Theorem 3.6. Recall that the self-adjoint operator (A, D(A)) on H is assumed to be strictly positive, hence we can find λ 0 > 0 so that Let ( λ ) λ≥λ 0 denote the family of spectral projectors in H uniquely associated with (A, D(A)), so that Note that by the above bound we have λ 0 (H ) = {0}. Given t > 0 let now ϕ t : (0, +∞) → (0, +∞) be the function defined by obviously nonnegative, continuous and decreasing with lim λ→∞ ϕ t (λ) = 0. Recall from Section 2 that h →ĥ denotes the isometry from H into L 2 (E, μ). Clearly its rangeĤ is a closed subspace of L 2 (E, μ), we denote the orthogonal projection in L 2 (E, μ) ontoĤ by Ĥ , and for the inverse of the bijection· : H →Ĥ we write· :Ĥ → H . We fix some straightforward consequences of the spectral representation and the isometry.
(ii) For any t > 0, any h ∈ H and any g ∈ L 2 (E, μ) we have where the integral over (λ 0 , +∞) on the right hand side is taken w.r.t. the signed measure (iii) For any β > 1, t > 0, h ∈ H and nonnegative g ∈ L 2 (E, μ), we have Proof Statement (i) is clear from the spectral theorem and since ϕ t is decreasing. To see (ii) note that by the isometry and the spectral theorem, Proof To shorten notation we will use the abbreviation g t,x (y) := f (e −tA x + √ 1 − e −2tA y). We first assume that f is bounded. Let h ∈ D( √ A) be such that h D( √ A) = 1, and let δ > 0 be arbitrary. By Proposition 8.12 we see that for μ-a.e. x ∈ E we have Then by Lemma 8.11 and Hölder's inequality, with 1 β + 1 β = 1, and by Hölder, now with 1 q + 1 q = 1, the first factor in Eq. (35) admits the bound .
where c 1 (λ 0 , β, q) > 0 is a constant depending only on λ 0 , β and q. In a similar fashion we can obtain a bound with arbitrary γ > 2, 1 γ + 1 γ = 1 and a constant c 2 (λ 0 , γ, q) > 0 depending only on λ 0 , γ and q. Choosing suitable δ > 0 as in the proof of Lemma 4.2, we obtain the statement for bounded f . For general nonnegative f ∈ L 2 (E, μ) consider f N := f ∧ N , for which we obtain by the positivity of the operators V A,2 and P t and Eq. (34). This allows to conclude the result using standard Dirichlet form theory.
We prove Theorem 8.9.
Proof We provide a proof only for the case r = 2. Let u = V A,2 f . Since T (0) = 0 and |T | is bounded by L, we have By Eq. (28) and the triangle inequality it therefore suffices to obtain a suitable bound for L A T (u) L 2 (E,μ) . For any ε > 0 we also consider u ε := V A,2 (f + ε). By Eq. (34) we have u ε ≥ ε and, because it implies 1 ∈ ker L A , also L A u ε = L A u. By Eq. (29) clearly also D A u ε = D A u. The analog of the chain rule Eq. (11) for the case of linear drift, applied to u ε , yields L A T (u ε ) L 2 (E,μ) ≤ L L A u L 2 (E,μ) + 1 u ε D A u ε , D A u ε H L 2 (E,μ) .
Clearly L A u L 2 (E,μ) ≤ f L 2 (E,μ) . On the other hand, by Lemma 8.13 with some 1 < q < 2 and the boundedness of the maximal function in L 2/q (E, μ), [ Together with the preceding argument this implies that for any 0 < ε < f L 2 (E,μ) we have L A T (u ε ) L 2 (E,μ) ≤ c f L 2 (E,μ) .
Accordingly we can find a sequence (ε k ) k with lim k ε k = 0 and an element w ∈ L 2 (E, μ) such that lim k L A T (u ε k ) = w weakly in L 2 (E, μ) and such that with w N := N k=1 T (u ε k ) ∈ D(L A ) also w = lim N L A w N = lim N N k=1 L A T (u ε k ) strongly in L 2 (E, μ). Since lim ε T (u ε ) − T (u) L 2 (E,μ) ≤ L lim ε u ε − u L 2 (E,μ) = 0, we also have lim N w N = T (u) strongly in L 2 (E, μ). Since (L A , D(L A )) is closed, it follows that T (u) ∈ D(L A ) and L A T (u) = w. We can therefore conclude that and combined with the above this yields T • u W 2,2 A (E,μ) ≤ c T f L 2 (E,μ) , as desired.

Essential Self-Adjointness of Ornstein-Uhlenbeck Operators with Linear Drift
In this section we consider the essential self-adjointness of L A in L 2 (E, μ), endowed with subspaces of FC ∞ b,K or W ∞ A after the removal of a small set from E. Let (A, D(A)) and K be as in in the preceding section. In [7,Proposition 5.4] it was shown that L A , endowed with FC ∞ b,K , is essentially self-adjoint in L 2 (E, μ), and that its unique self-adjoint extension is (L A , D(L A )) with D(L A ) = W 2,2 A (E, μ) as discussed above. By Eq. (31) then also L A , endowed with W ∞ A , is essentially self-adjoint with the same unique self-adjoint extension. Similarly as before let now ⊂ E be a closed set of zero Gaussian measure and write N := E \ . Let Theorem 9.1 follows by (a simpler version of) the same arguments as Theorem 5.2.

Remark 9.2
Of course an analogous statement is true for capacities of type Cap A,1,2 and with essential self-adjointness replaced by Markov uniqueness, but this is the special case of a well known standard result in Dirichlet form theory.
Theorem 9.1 can be applied to the time zero Gaussian free field, [54], as discussed in [4, Examples 3.3 (ii)], [7, Examples 5.6. (ii) and 6.6.(i)] and [48,49]. We follow [50,Examples 3.5], referred to as the 'second approach to the free Dirichlet form' in [7, Examples 5.6.(ii)]. Obviously A is stricly positive and we may choose any 0 < λ 0 < m in Eq. (32). Let E and K be as constructed in [7,Theorem 3.1]. Alternatively, let E be the space B α as described in [50,Examples 3.5] and constructed in [48,49] and K = S(R d ). Let E be extended according to [7,Theorem 6.7] and Remark 8.6 above (we keep the same symbol E). The mean zero Gaussian measure μ on E with covariance E l 1 (y)l 2 (y)μ(dy) = l 1 , l 2 H , l 1 , l 2 ∈ E , makes (E, H, μ) into an abstract Wiener space. It is called the time zero Gaussian free field with mass m. This setup satisfies all assumptions made in the beginning of Section 8, and it satisfies Assumption 8.5. The corresponding generator (L A , D(L A )) and Dirichlet form (E A , D(E A )) as in Eq. (30) are called the free Hamiltonian and the free-field Dirichlet form, respectively.
If a closed set ⊂ E of zero Gaussian measure is removed from E and N := E \ , then the free Hamiltonian L A , endowed with FC ∞ b,K (N ) or W ∞ A (N ) as defined above, is essentially self-adjoint on L 2 (N, μ) = L 2 (E, μ) with unique self-adjoint extension (L A , D(L A )) if and only if Cap A,2,2, ( ) = 0. In other words, small 'boundaries' of zero Cap A,2,2, -capacity are not seen when extending the operator, and if a small boundary is not seen, it must have zero Cap A,2,2, -capacity.