pre-Dirichlet Forms Research Articles

AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials. Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝd\{0}.

On a Relation between Intrinsic and Extrinsic Dirichlet Forms for Interacting Particle Systems

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre–Dirichlet form ℰΓ of the Poisson measure πσ in terms of the extrinsic one ℰP. More precisely, replacing πσ by a Gibbs measure μ on the configuration space ΓX we derive a relation between the intrinsic prend–Dirichlet form ℰΓμ of the measure μ and the extrinsic one ℰP. As a consequence we prove the closability of ℰΓμ on L2(ΓX, μ) under very general assumptions on the interaction potential of the Gibbs measures μ.

Stochastic Quantization of the Two-Dimensional Polymer Measure

We prove that there exists a diffusion process whose invariant measure is the two-dimensional polymer measure nu(g). The diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. We prove the closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer measure nu(g)) but requires the quasi-invariance of nu(g) along a basis of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives (have versions which) form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diffusion process is ergodic under time translations.

Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness

We study the Dirichlet forms and the associated Dirichlet operators for Gibbs measures on infinite particle configuration space. The Dirichlet forms are defined to be “gradient-type” forms by introducing a measurable field of rigged Hilbert spaces on the configuration space. Under mild conditions on the interaction including singular potentials, we show that the pre-Dirichlet operator is symmetric and that the closure of the pre-Dirichlet form satisfies the Markovian property. When the interaction is three times differentiable and decreasing sub-exponentially, we show that the Dirichlet operator is essentially self-adjoint on a domain consisting of bounded smooth local functions.

Integration by Parts and Quasi-Invariance for Heat Kernel Measures on Loop Groups

Integration by parts formulas are established both for Wiener measure on the path space of a loop group and for the heat kernel measures on the loop group. The Wiener measure is defined to be the law of a certain loop group valued “Brownian motion” and the heat kernel measures are timet,t>0, distributions of this Brownian motion. A corollary of either of these integrations by parts formulas is the closability of the pre-Dirichlet form considered by B. K. Driver and T. Lohrenz [1996,J. Functional Anal.140, 381–448]. We also show that the heat kernel measures are quasi-invariant under right under right and left translations by finite energy loops.

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAdG-invariant inner product on the Lie algebra g ofG. Given this data there is a well known left invariant “H1-Riemannian structure” on L=L(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”νT(g0, ·) associated to the Laplace–Beltrami operator on L(G). HereT>0,g0∈L(G), andνT(g0,·) is a certain probability measure on L(G). For fixedg0∈L(G) andT>0, we use the measureνT(g0,·) and the Riemannian structure on L(G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.

A characterization of the closable parts of pre-Dirichlet forms by hitting distributions

A characterization of the closable parts of pre-Dirichlet forms by hitting distributions

On quasi-supports of smooth measures and closability of pre-Dirichlet forms

On quasi-supports of smooth measures and closability of pre-Dirichlet forms

Construction of a general class of Dirichlet forms in terms of white noise analysis

In the framework of white noise analysis a Gel'fand triple ( L)⊂(L 2)⊂( L) ∗ has been defined (e.g. Kubo and Yokoi, 1989), the space of smooth test functionals ( L ) and the space of Hida distributions ( L ) ∗ play some important roles. It has been shown (e.g. Yokoi, 1990) that a positive Hida distribution Φ is given by a positive measure ν Φ on the space of real tempered distributions L ∗. Thus the space ( L 2) Φ L 2( L ∗; B , ν Φ) can be defined, where B is the Borel σ-algebra on L ∗ generated by the weak topology. The present article is concerned with a special choice of pre-Dirichlet forms with domain ( L ) on ( L 2) Φ which is a generalization of the energy form (Hida, Potthoff and Streit, 1988) and of the type ε(F)= Φ; ∑ j,k H j,k·∂ jF·∂ k F ̄ , for each Fϵ( L ) and where ( H j,k ; j, kϵ N 0) is a double sequence of test functionals satisfying some natural conditions. Some closability results are given in the last section under mild conditions.

On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures

Let X be a locally compact separable metric space, <Jά be the space of positive Radon measures on X and let ι3ί'={p^JM: supp[v]=X}. Fix m^JM' and a regular Dirichlet form 6 with domain £F on L\X m), which possesses a nice core C as described in Section 3. Throughout the present paper, we assume that 6 is either irreducible or transient. Let Cap( ) be the 1-caρacity associated with S. A set A is said to be 6^-polar if Cap(A)=0. Define