On regularity properties of Bessel flow

Abstract

We study the differentiability of Bessel flow $\rho :x\to {\rho }_t^x$, where $({\rho }_t^x{)}_{t\ge 0}$ is BES ${}^x(\delta$) process of dimension $\delta >1$ starting from $x$. For $\delta \ge 2$ we prove the existence of bicontinuous derivatives in P-a.s. sense at $x\ge 0$ and we study the asymptotic behaviour of the derivatives at $x=0$. For $1<\delta <2$ we prove the existence of a modification of Bessel flow having derivatives in probability sense at $x\ge 0$. We study the asymptotic behaviour of the derivatives at $t={\tau }_0(x)$ where ${\tau }_0(x)$ is the first zero of $({\rho }_t^x{)}_{t\ge 0}$.

21 pages, no figures