Related papers: Subordinated Markov processes: sharp estimates for…

We prove sharp near-diagonal pointwise bounds for the Green function $G_\Omega(x,y)$ for nonlocal operators of fractional order $\alpha \in (0,2)$. The novelty of our results is two-fold: the estimates are robust as $\alpha \to 2-$ and we…

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the…

We study distributions of meeting times for finite symmetric Markov chains. For Markov kernels defined on large state spaces which satisfy certain weak inhomogeneity in return probabilities of points up to large numbers of steps, we obtain…

We consider heat kernels on Weyl chambers corresponding to Laplacians subject to mixed Dirichlet-Neumann boundary conditions imposed on the boundary. Using purely analytic tools we prove genuinely sharp two-sided global estimates in the…

We establish pointwise estimates for the ground states of some classes of posi- tivity preserving operators. The considered operators are negatively perturbed (by measures) strongly local Dirichlet operators. These estimates will be written…

In this note we show the optimal gradient estimate for heat kernels of stable-like operators by providing a counterexample.

We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $\alpha \in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$…

In this paper, we consider a symmetric pure jump Markov process $X$ on a metric measure space with volume doubling conditions. Our focus is on estimating the transition density $p(t,x,y)$ of $X$ and studying its stability when the jumping…

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…

For a class of Laplace exponents we derive the heat trace asymptotics of the generator of the corresponding subordinate Brownian motion on Euclidean space. The terms in the asymptotic expansion are found to depend both on the geometry of…

We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain…

We prove pointwise and $L^p$ gradient estimates for the heat kernel on the bounded and unbounded Vicsek set and applications to Sobolev inequalities are given. We also define a Hodge semigroup in that setting and prove estimates for its…

Under some mild assumptions on the Levy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure-jump isotropic unimodal Levy processes in $R^d$.

We obtain an off-diagonal upper bound for Green and heat kernel of Laplace type operator on symmetric spaces.

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp…

We construct a conservative and strongly local regular symmetric Dirichlet form on the pointed Gromov--Hausdorff limit space and demonstrate the stability of heat kernel estimates under this convergence. Furthermore, we establish the Mosco…

We consider heat kernel for higher-order operators with constant coefficients in $d$-dimensio\-nal Euclidean space and its asymptotic behavior. For arbitrary operators which are invariant with respect to $O(d)$-rotations we obtain exact…

We give sharp asymptotic estimates at infinity of all radial partial derivatives of the heat kernel on H-type groups. As an application, we give a new proof of the discreteness of the spectrum of some natural sub-Riemannian…

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 consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a…