Hitting times of Bessel processes

Let $T_1^{(\mu)}$ be the first hitting time of the point 1 by the Bessel process with index $\mu\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(\mu)}(t)$ of $T_1^{(\mu)}$, obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of $T_1^{(\mu)}$ which exibit the dependence both on time and space variables. Our result provides optimal estimates for the density of the hitting time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces

Elementary theory

Potential theory of subordinated Brownian motionsPotential theory and Brownian motion Beginning of potential theory Newton (1687): Law of universal gravitation, study of F (x)-force acting on a unit mass at x ∈ R d, d � 3. Lagrange (1773): The above vector field (of forces) is a gradient of a certain function U: = U2(x) = Ad,2|x | 2−d. Green (1828) named U potential function. Gauss (1840) named U potential. Gauss: potential method is suitable to resolve many complicated problems of mathematical physics, not only problems of gravitation or electrostatics. More generally: for a field generated by a charge located according to a measure µ we define a potential of µ: U2µ(x) = Ad,2 |x − y | 2−d Γ(d/2 − 1

Poisson kernel of half spaces in real hyperbolic spaces

International audienceWe provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space \H^n. This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space \H^n. For n=3, 4 or 6 we compute explicit formulas for the Poisson kernel itself