Fourier–Bessel heat kernel estimates

Abstract

Abstract. We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0, 1) killed at 1 and killed or reflected at 0.

Sharp estimates of the spherical heat kernel

“…The example of S d shows that this is a difficult problem even for basic and regular Riemannian manifolds. In this connection, it is perhaps worth mentioning the recent papers [3,4,8,9] where such results were obtained for Dirichlet heat kernels related to Bessel operators in half-lines, the Dirichlet heat kernel in Euclidean balls of arbitrary dimension, and the Fourier-Bessel heat kernel on the interval (0, 1). This was achieved by a clever combination of probabilistic and analytic methods.…”

Exit times densities of the Bessel process

“…It is showed in [14] that there exists t 0 > 0 such that (40) p x/4,1 (t, x, y), whenever x, y ∈ (1/2, 1) and t < t 0 . By the absolute continuity (22) of distributions of Bessel processes with different indices we get…”

“…This kind of formulae seems to be quite general, but the mentioned singularity of the process at zero requires a separate consideration. The obtained representations enable us to use some results and methods provided in [14], where the density of the Bessel process killed when hitting the level 1 was estimated. One of the our key tools is an imitation of a reflection principle, which lets us efficiently approximate the density of the killed process for some range of parameters.…”

Dirichlet Heat Kernel for the Laplacian in a Ball

“…The main advantage of the Hunt formula compared to the series representation is the simple fact that we represent the heat kernel as a difference of two non-negative expressions, which is much simpler to deal with than with the series of oscillating components. This approach has been successfully used in [10] to study the short time behaviour of the Fourier-Bessel heat kernel. Since the Hunt formula is the starting point, we use several probabilistic tools and ideas in the proof of the main result.…”