Related papers: Rate functions for symmetric markov processes via …
We prove upper and lower bounds of the heat kernel for the operator $\Delta-\nabla (\frac{1}{|x|^{\alpha}})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $\alpha >0$. We obtain these bounds from an isoperimetric inequality for a…
We suggest a method of reduction of mixed absolute and relative boundary conditions to pure ones. The case of rank two tensor is studied in detail. For four-dimensional disk the corresponding heat kernel is expressed in terms of scalar heat…
In this paper we establish the existence and uniqueness of heat kernels to a large class of time-inhomogenous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates,…
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o}…
In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a…
We provide a condition for f-ergodicity of strong Markov processes at a subgeometric rate. This condition is couched in terms of a supermartingale property for a functional of the Markov process. Equivalent formulations in terms of a drift…
In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal…
On complete, non-compact manifolds and infinite graphs, Faber-Krahn inequalities have been used to estimate the rate of decay of the heat kernel. We develop this technique in the setting of finite Markov chains, proving upper and lower…
For $d\geq 1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}; a \in [0, 1]\}$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+…
We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary…
Let $X$ be a symmetric jump process on $\R^d$ such that the corresponding jumping kernel $J(x,y)$ satisfies $$J(x,y)\le \frac{c}{|x-y|^{d+2}\log^{1+\varepsilon}(e+|x-y|)}$$ for all $x,y\in\R^d$ with $|x-y|\ge1$ and some constants…
We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$,…
The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In…
Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are…
When studying non-symmetric nonlocal operators $$ {\cal L} f(x) = \int_{{\bf R}^d} \left( f(x+z)-f(x)-\nabla f(x)\cdot z 1_{\{|z|\leq 1\}} \right) \frac{\kappa (x, z)}{|z|^{d+\alpha}} d z , $$ where $0<\alpha<2$ and $\kappa (x, z)$ is a…
In this note we apply heat kernels to derive some localization formula in sympletcic geometry, to study moduli spaces of flat connections on a Riemann surface, to obtain the push-forward measures for certain maps between Lie groups and to…
We prove sharp estimates on heat kernels and Green functions for subordinate Markov processes with both discrete an continuous time, under relatively weak assumptions about original processes as well as Laplace exponents of subordinators.…
We give large-time asymptotic estimates, both in uniform and $L^1$ norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of $\mathbb{R}^d$ satisfying certain technical assumptions. We always assume that…
In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ \Delta^{\alpha/2}_v + v \cdot \nabla_x, \quad \alpha \in (0, 2),\ (x,v)\in {\mathbb…
Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma, where the output entropy of a binary symmetric channel is bounded in terms of the average entropy of the input projected on a random subset of coordinates. Here,…