Heat kernel estimates for the Bessel differential operator in half‐line

Abstract

Abstract. In the paper we consider the Bessel differential operator L (µ) = d 2 dx 2 + 2µ + 1 x d dx in half-line (a, ∞), a > 0, and its Dirichlet heat kernel p (µ) a (t, x, y). For µ = 0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x, y > a and every t > 0, which complements the recent results given in [1], where the case µ = 0 was considered.

Fourier–Bessel heat kernel estimates

“…Notice that even in the classical setting of Laplacian in R n , the known estimates of related Dirichlet heat kernel for smooth domains (see [14]) are also only quantitatively sharp (see also [13] and the references therein for corresponding results on manifolds). However, in the recent papers [1] and [2] the sharp two-sided estimates for the Dirichlet heat kernel of the half-line (a, ∞) associated with the Bessel differential operator has been obtained.…”

Exit times densities of the Bessel process

“…In the context of Bessel process, the killed process was recently of special interest [2,3,5,6,10,11,17]. In the paper, we examine the exit time density of the Bessel process starting from x > 0 and killed when it reaches a fixed level a > x.…”

Asymptotic behaviour of the Bessel heat kernels

“…Theorem 2 Let μ ∈ R, a > 0, n ∈ N and t 0 (μ) is defined as in (3). We have p (μ) a (t, x, y) = a 2μ+1 (x y)…”

“…T. Takemura in [16] derived the integral representations involving highly oscillating functions. Then, in papers [2,3], sharp estimates of p (μ) a (t, x, y) in a full range of the variables x, y > a and t > 0 were obtained. In this context, providing asymptotic expansion of the considered Bessel heat kernels is a natural improvement of these results.…”