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.
“…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.…”
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.
“…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.…”
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.
“…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.…”
We examine the density functions of the first exit times of the Bessel process from the intervals [0, 1) and (0, 1). First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the n-dimensional Brownian motion from a ball.2010 Mathematics Subject Classification. 60J60, 60J65.
“…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)…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
We consider Dirichlet heat kernel p (μ) a (t, x, y) for the Bessel differential operator L (μ) = d 2 dx 2 + 2μ+1 2x , μ ∈ R, in half-line (a, ∞), a > 0, and provide its asymptotic expansions for x y/t → ∞.
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