Related papers: Heat kernel of fractional Laplacian in cones
We study the exit time $\tau=\tau_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^\alpha$ as the Laplace transform of a positive random variable with explicit density. Consequently, $\tau$…
Let $(\mathbb M, d,\mu)$ be a metric measure space with upper and lower densities: $$ \begin{cases} |||\mu|||_{\beta}:=\sup_{(x,r)\in \mathbb M\times(0,\infty)} \mu(B(x,r))r^{-\beta}<\infty;\\ |||\mu|||_{\beta^{\star}}:=\inf_{(x,r)\in…
A refracted L\'evy process is a L\'evy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted…
An asymptotic expansion of the trace of the heat kernel on a cone where the heat coefficients have a delta function behavior at the apex is obtained. It is used to derive the renormalized effective action and total energy of a…
The fundamental solution of the heat equation on $R^n$ is known as the heat kernel which is also the transition density of a Brownian motion. Similar statements hold when $\R^n$ is replaced by a Lie group. We briefly demonstrate how the…
The nucleation and dynamics of vortices in the quasi-two-dimensional rotating dipolar Bose-Einstein condensate are explored by taking into account the Lee-Huang-Yang (LHY) correction to the mean-field (MF) theory. Assuming approximate…
We consider a complete noncompact smooth Riemannian manifold $M$ with a weighted measure and the associated drifting Laplacian. We demonstrate that whenever the $q$-Bakry-\'Emery Ricci tensor on $M$ is bounded below, then we can obtain an…
We describe the small-time heat kernel asymptotics of real powers $\Delta^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $\Delta$ acting on the sections of a hermitian vector bundle $\mathcal E$ over a closed…
The contributions to the heat kernel coefficients generated by the corners of the boundary are studied. For this purpose the internal and external sectors of a wedge and a cone are considered. These sectors are obtained by introducing,…
We study natural invariance properties of functionals defined on L\'evy processes and show that they can be described by a simplified structure of the deterministic chaos kernels in It\^o's chaos expansion. These structural properties of…
We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Levy process
Spectral theory for the transition semigroup of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction…
We obtain an off-diagonal upper bound for Green and heat kernel of Laplace type operator on symmetric spaces.
We derive explicitly the coupling property for the transition semigroup of a L\'{e}vy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the…
We investigate the heat equation corresponding to the Bessel operators on a symmetric cone $\Omega=G/K$. These operators form a one-parameter family of elliptic self-adjoint second order differential operators and occur in the Lie algebra…
We prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. This solves a long-standing open problem.
We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness…
In this work, we investigate the fine regularity of L\'evy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities…
Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\,$. The…
In this paper, we study Feynman-Kac semigroups of symmetric $\alpha$-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b|x|^{-\beta}$, where $b>0$ and…