Alexander Grigor'yan

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds etc. which are related to recurrence and non-explosion.

Heat kernel upper bounds on a complete non-compact manifold

Let M be a smooth connected non-compact geodesically complete Riemannian manifold, ? denote the Laplace operator associated with the Riemannian metric, n = 2 be the dimension of M. Consider the heat equation on the manifold ut - ?u = 0, where u = u(x,t), x I M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t ? +8 and r = dist(x,y) ? +8.

Sub-Gaussian estimates of heat kernels on infinite graphs

We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

Heat kernel upper bounds for jump processes and the first exit time

for all t > 0, x ∈ M , and f ∈ L2(M,μ). The function pt(x, y) can be considered as the transition density of the associated Markov processX = {Xt}t≥0, and the question of estimating of pt(x, y), which is the main subject of this paper, is closely related to the properties of X. The function pt(x, y) is also referred to as a heat kernel, and this terminology goes back to the classical Gauss-Weierstrass heat kernel associated with the heat semigroup {e}t≥0 in R n, whose Markov process is Brownian motion. A somewhat more general but still well treated case is when (M,d, μ) is a Riemannian metric measure space, that is, M is a Riemannian manifold,

The Heat Kernel on Hyperbolic Space

The purpose of this note is to provide a new proof for the explicit formulas of the heat kernel on hyperbolic space. By definition, the hyperbolic space H is a (unique) simply connected complete n-dimensional Riemannian manifold with a constant negative sectional curvature −1. Let ∆ denote the Laplacian on a Riemannian manifold X. The heat kernel on X is a function p(x, y, t) on X ×X × (0,∞) which is the minimal positive fundamental solution to the heat equation

Integral maximum principle and its applications

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

Hitting probabilities for Brownian motion on Riemannian manifolds

4 Specific estimates of hitting probabilities 17 4.1 Upper estimates I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Upper estimates II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Two-sided estimates in the non-parabolic case . . . . . . . . . . . . . . . . . . . . . 22 4.4 Two-sided estimates in the parabolic case . . . . . . . . . . . . . . . . . . . . . . . 25

Escape rate of brownian motion on riemanian manifolds

2 Analysis on weighted manifolds 5 2.1 Weighted manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Faber-Krahn inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Mean value inequality for subsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Heat Kernels and Green Functions on Metric Measure Spaces

We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball.

Fundamental solution of the heat equation on an arbitrary Riemannian manifold

We consider upper estimates for the Green function of the heat equation on an arbitrary smooth connected Riemannian manifold M. We define the Green function G(t, x; y) as the limit of the Green functions G~ for the precompact domains ~CM as ~ ~ M. If the manifold M has boundary, then we will always assume the Neumann homogeneous condition to be fulfilled on the boundary and also consider only those ~ for which a~ is transversed to 3M. Let us denote the geodesic distance between two points x, y~M by Ix yJ and the geodesic ball of radius r with center at x by Br x. If N is a submanifold, then we will denote its volume, corresponding to the dimension, by JN I . THEOREM I. Let the following isoperimetric inequality be fulfilled in a precompact geodesic ball Bpx: for each domain Q ~ Box that has smooth boundary 3Q, transversal to aM,