Abstract not found

Introduction We study continuity of boundary problems with varying domains. To explain this in more detail, let us consider our standard example: Denote by HGn the Dirichlet Laplacian on the open set G n ae IR d . The basic question which we adress is, whether we have convergence HGn \Gamma! HG ; if the sets G n converge to G in an appropriate sense. Two notions of convergence for the operators appear suitable: Generalized convergence in the strong resolvent sense (srs) and in the norm resolvent sense (nrs) (the "generalized" refers to the fact that the HGn act in different Hilbert spaces; we will frequently omit it). We shall introduce these concepts in some detail below but first we briefly describe the content of the following sections. In Section 1 we a

Studied in this paper is the transformation of an arbitrary symmetric Markov process X by multiplicative functionals which are the exponential of continuous additive functionals of X having zero quadratic variations. We characterize the transformed semigroups by their associated quadratic forms. This is done by rst identifying the symmetric Markov process under Girsanov transform, which may be of independent interest, and then applying Feynman-Kac transform to the Girsanov transformed process. Stochastic analysis for discontinuous martingales is used in our approach.

Introduction Let H = \Gamma\Delta + V be a Schrodinger operator on R d , where V = V + \Gamma V \Gamma is a potential with negative part in the Kato class and positive part in L 1 loc . For any open set G we write HG for the corresponding selfadjoint operator in L 2 (G) which is defined in the following way: denote by h the form associated with H and by h G the closure of hjC 1 c (G); the associated selfadjoint operator is HG . If V 2 L 2 loc then HG is just the Friedrichs extension of HjC 1 c<F27.07

The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

. We discuss the Littlewood-Paley inequality for a di usion process associated with a Dirichlet form of gradient type. We assume that the logarithmic Sobolev inequality holds and the negative part of # 2 is exponentially integrable. Under these and some additional conditions, we showed that ##u# p # C# # 1 - Lu# q for 1 <p<q. We also discuss the Brownian motion on a Riemannian manifold with boundary. 1. Introduction In this paper, we discuss the Littlewood-Paley inequality.T ypical e ample is the Brownian motion on the Euclidean space and it leads to the following inequality: for any p>1 there exist a positive constant C such that C -1 ##u# p ## # -#u# p # C##u# p . (1.1) # -#, the square root of the minus Laplacian, is called the Cauchy operator. (1.1) is equivalent to the L p -boundedness of the Riesz transformation. T#tr kind of inequality also holds for the Ornstein-Uhlenbeck process on an abstract Wiener space, which was proved by P. A. Meyer [11] in a probabili...

Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.

Let X be an m-symmetric right process on Luzin space E and (E, F) be its associated quasiregular Dirichlet form. Let µ be a signed smooth measure of X and A µ be the continuous additive functional (CAF in abbreviation) of X with signed Revuz measure µ. It defines a symmetric Feynman-Kac semigroup Ttf(x): = Ex [exp(A µ t)f(Xt)] for t> 0 and f ≥ 0. (1) Define a symmetric quadratic form (E µ, D(E µ)) by D(E µ): = � u ∈ F: u ∈ L 2 (E, |µ|) �, E µ � (u, v) = E(u, v) − u(x)v(x)µ(dx) for u, v ∈ D(C). E We say that the form (E µ, D(E µ)) is bounded from below if there is some α0 ≥ 0 such that E µ α0 (u, u): = Eµ (u, u) + α0(u, u) ≥ 0 for every u ∈ D(E µ). For a non-negative smooth measure ν, we say it is in the Kato class of X if lim t→0 sup [A x∈E ν t] = 0. It is known (see, e.g., [1, Proposition 2.1(i)] and [6, Theorem 3.1]) that if ν is in the Kato class of X, then for every ε> 0, there is some constant Aε> 0 such that u(x) 2 � ν(dx) ≤ ε E(u, u) + Aε u(x) 2 m(dx) for every u ∈ F.

Introduction We study semigroup differences of the form e \Gammat(H+¯) \Gamma e \Gammat(H+¯+) ; where H is the generator of a regular Dirichlet form and ¯ and are suitable measures. In the first section the appropriate classes of measures are introduced. Moreover, we provide a list of examples which can be treated in the Dirichlet form framework. In the second section the above semigroup is investigated in terms of Hilbert-- Schmidt and trace class properties. It turns out that if the set on which "lives", the so--called quasi--support, has finite capacity then the above semigroup difference is Hilbert--Schmidt. A more restrictive condition on the quasi--support of is exhibited which implies that this differen

Abstract. The problem of reconstructing the drift of a diffusion in R d, d≥2, from the transition probability density observed outside a domain is considered. The solution of this problem also solves a new inverse problem for a class of parabolic partial differential equations. This work considerably extends [2] in terms of generality, both concerning assumptions on the drift coefficient, and allowing for non-constant diffusion coefficient. Sufficient conditions for solvability of this type of inverse problem for d = 1 are also given. 1.