Related papers: Dirichlet forms and ultrametric Cantor sets associ…

Asymptotic expansions as well as necessary and sufficient conditions are provided for the pointwise convergence of the spherical partial integrals of the associated Fourier transforms on the real hyperbolic space. The proposed method…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

We study the heat content for Laplacians on compact, finite metric graphs with Dirichlet conditions imposed at the "boundary" (i.e., a given set of vertices). We prove a closed formula of combinatorial flavour, as it is expressed as a sum…

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of…

Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an…

Let $\Omega$ be a bounded connected open subset in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$. Suppose that we have a system of real smooth vector fields $X=(X_{1},X_{2},$ $\cdots,X_{m})$ defined on a neighborhood of…

In this thesis, we study connections between metric and combinatorial graphs from a Dirichlet space point of view.

A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to enjoy some novel properties. The ultrametric $d_{u}$ is defined using the concept of {\em…

In this paper we study the potential theory of Dirichlet forms on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$ and the killing potential $\kappa x_d^{-\alpha}$, where $\alpha\in (0,…

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric…

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows to interpolate between the rotational invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a…

The aim of the present paper is three folds. For a reproducing kernel Hilbert space $\mathcal{A}$ (R.K.H.S) and a $\sigma-$finite measure space $(M_{1},d\mu_{1})$ for which the corresponding $L^{2}-$space is a separable Hilbert space, we…

Let $G$ be a compact Lie group and $P_{e,a}(G)=C([0,1]\to G~|~\gamma(0)=e, \gamma(1)=a)$ be the pinned path space with a pinned Brownian motion measure $\nu_{\lambda,a}$ defined by the heat kernel $p(\lambda^{-1}t,x,y)$, where $\lambda$ is…

Given a compact doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the…

Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact,…

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and…

In this thesis we describe a type of metric space called an Euclidean polyhedral complex. We define a Dirichlet form on it; this is used to give a corresponding heat kernel. We provide a uniform small time Poincare inequality for complexes…

We construct spectral triples for path spaces of stationary Bratteli diagrams and study their associated mathematical objects, in particular their zeta function, their heat kernel expansion and their Dirichlet forms. One of the main…

Let $\mu$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(\mu)$, associated to $\mu$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against…

We consider metric graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincar\'e inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from…