Related papers: Fourier-Bessel heat kernel estimates

We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $\nu$ is half-integer. Moreover, still for half-integer $\nu$, we also obtain sharp estimates of all kernels…

In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators $m-(m^{2/\alpha}-\Delta)^{\alpha/2}$] in $C^{1,1}$ open sets. Here $m>0$ and…

The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a…

In this paper we study the Bessel process R_t^{(\mu)} with index \mu\neq 0 starting from x>0 and killed when it reaches a positive level a, where x>a>0. We provide sharp estimates of the transition probability density p_a^{(\mu)}(t,x,y) for…

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a…

We prove sharp estimates of the heat kernel associated with Fourier-Dini expansions on $(0,1)$ equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result…

We are concerned with the first hitting times of the Bessel processes. We give explicit expressions for the densities by means of the zeros of the Bessel functions and show their asymptotic behavior.

In the paper we consider the Bessel differential operator L^(\mu)=\dfrac{d^2}{dx^2}+\dfrac{2\mu+1}{x}\dfrac{d}{dx} in half-line (a,\infty), a>0, and its Dirichlet heat kernel p_a^(\mu)(t,x,y). For \mu=0, by combining analytical and…

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time…

In this paper, we study transition density functions for pure jump unimodal L\'evy processes killed upon leaving an open set $D$. Under some mild assumptions on the L\'evy density, we establish two-sided Dirichlet heat kernel estimates when…

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2.

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in (0, 2)$.…

In this paper, sharp two-sided estimates for the transition densities of relativistic $\alpha$-stable processes with mass $m\in (0, 1]$ in $C^{1,1}$ exterior open sets are established for all time $t>0$. These transition densities are also…

Suppose that $d\ge 1$ and $\alpha\in (0, 2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]\}$ on…

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may…

We construct in the small-time setting the upper and lower estimates for the transition probability density of a L\'evy process in $\rn$. Our approach relies on the complex analysis technique and the asymptotic analysis of the inverse…

Let $d \geq 2$, $\alpha \in (0,2)$, and $X$ be the rectilinear $\alpha$-stable process on $\mathbb{R}^d$. We first present a geometric characterization of an open subset $D\subset \mathbb{R}^d$ so that the part process $X^D$ of $X$ in $D$…

We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover…

We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a…

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively…