In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = \Delta H is the Gross-Laplacian, (E; H; fl) is an abstract Wiener space and B = \Gammaid E +v where v takes values in the Cameron-Martin space H . Using Gross' logarithmic Sobolev-inequality in an essential way we show that ¯ is always absolutely continuous w.r.t. the Gaussian measure fl and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (fl). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L ¯ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove ...

Abstract not found

Abstract not found

: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. On leave from School of Math., Georgia Institute of Technology, Atlanta, GA 30332 Work supported by U.S. National Science Foundation grant no. PHY90--19433--A01. 1 I. INTRODUCTION Observables pertaining to the configuration of a quantum system with n degrees of freedom are operators Q 1 ; Q 2 ; : : : ; Q n which, depending on the system, may or may not commute. Our main concern is with the case in which the configuration variables are amplitudes of certain field modes. For boson fields, these configuration observables do commute, and the state space H can be taken as the space of all complex square integrable functions on their joint spectrum. This is t...

We consider Schrodinger semigroups e.- IH, H =-A+V on Iw ” with V---cIxl- ’ as 1x1--rco, O<c<[(l/2)(n-2)] * with H>O. We determine the exact power law divergence of I~e-‘Hi~p,p and of some IIe-‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x)- 1.x-‘.:Ta 1991 Academic Press, Inc. 1.

In this paper, we generalize Haagerup’s inequality [H] (on convolution norm in the free group) to a very general context of R-diagonal elements in a tracial von Neumann algebra; moreover, we show that in this “holomorphic” setting, the inequality is greatly improved from its originial form. We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for R-diagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem, generalizing and improving the one in [Bi2].

Summary. We point out a simple argument relying on hypercontractivity to describe tail inequalities on the distribution of the largest eigenvalues of random matrices at the rate given by the Tracy–Widom distribution. The result is illustrated on the known examples of the Gaussian and Laguerre unitary ensembles. The argument may be applied to describe the generic tail behavior of eigenfunction measures of hypercontractive operators.

ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic ” algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a q-Segal-Bargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.

We prove that the laws of diffusion processes M on E associated with Dirichlet forms of type E(u; v) = R E hA(z)ru(z); rv(z)i H¯(dz), where H , E are separable Hilbert spaces, are the weak limits of laws of finite dimensional diffusions. These are associated with the image Dirichlet forms obtained from E under projections from E onto finite dimensional subspaces in H. As a by-product we obtain Hoelder continuity of the sample paths as well as a new existence proof for the infinite dimensional diffusion M.

Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the L p -setting is discussed. As a direct application essential self-- adjointness and strong uniqueness in L p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R 2 . AMS Subject Classification Primary: 47 B 25, 81 S 20 Secondary: 31 C 25, 60 H 15, 81 Q 10 Key words and phrases: Dirichlet operators, essential self--adjointness, C 0 -- semigroups, generators, stochastic quantization, Markov uniqueness, a--priori estimates Running head: Strong uniqueness for Dirichlet operators 1 Introduction The theory of Dirichlet forms is a rapidly developing field of modern analysis which has intimate relationships with potential theory, probability theory, diffe...