Related papers: Markov properties for Gaussian fields associated w…
We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple…
We provide an introduction to Dirichlet forms on discrete spaces and study their global properties such as recurrence, stochastic completeness and regularity of the Neumann form. In this setting we compare the notion of a recurrent…
It was shown many times in the literature that a Markov random field is equivalent to a Gibbs random field when all realizations of the field have non-zero probabilities; the proofs are rather complicated. A simpler proof, which is based…
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…
We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.
We introduce Riemannian-like structures associated with strong local Dirichlet forms on general state spaces. Such structures justify the principle that the pointwise index of the Dirichlet form represents the effective dimension of the…
We derive an explicit link between Gaussian Markov random fields on metric graphs and graphical models, and in particular show that a Markov random field restricted to the vertices of the graph is, under mild regularity conditions, a…
We prove that stable-like non-local Dirichlet forms converge to local Dirichlet form in the sense of Mosco on metric measure spaces. We prove that subordinated Dirichlet forms converge to the original Dirichlet form in the sense of Mosco on…
In this paper, we shall explore the Mosco convergence on regular subspaces of one-dimensional irreducible and strongly local Dirichlet forms. We find that if the characteristic sets of regular subspaces are convergent, then their associated…
In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general…
We compute explicitly traces of the Dirichlet form related to the Bessel process with respect to discrete measures as well as measures of mixed type. Then some global properties of the obtained Dirichlet forms, such as conservativeness,…
We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces,…
In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express…
We prove that the domain of the local Dirichlet form is strictly contained in the domain of any stable-like non-local Dirichlet form on general metric measure spaces.
We introduce an asymptotic Markov's exponent and show that it is equal to Markov's exponent for a wide class of norms. As a consequence we obtain a lower bound for the optimal exponent in Markov's inequality considered with the norms…
We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green's formula for functions…
We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence…
We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces,…
In this short article, we shall study one-dimensional local Dirichlet spaces. One result, which has its independent interest, is to prove that irreducibility implies the uniqueness of symmetrizing measure for right Markov processes. The…
In this paper, we introduce an analog of Gauss sums over function fields in positive characteristic. We establish several fundamental properties, including reflection formula, Stickelberger's theorem, and Hasse-Davenport relations. In…