On regularity properties of Bessel flow

TL;DR

This paper investigates the differentiability and regularity properties of Bessel flows for different dimensions, establishing existence and behavior of derivatives in various senses and analyzing their asymptotic behavior near zero and at the first zero time.

Contribution

It provides new results on the differentiability of Bessel flows for dimensions greater than one, including existence of derivatives in almost sure and probability senses, and studies their asymptotic behavior.

Findings

Existence of bicontinuous derivatives for   at xa0a0

Derivatives in probability sense for 1<<2 at xa0a0

Asymptotic behavior of derivatives at zero and first zero time

Abstract

We study the differentiability of Bessel flow $\rho : x \to \rho ^x_t$, where $(\rho ^x_t)_{t\geq 0}$ is BES $^x(\delta $) process of dimension $\delta >1$ starting from $x$. For $\delta \geq 2$ we prove the existence of bicontinuous derivatives in P-a.s. sense at $x\geq 0$ 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 $x\geq 0$. We study the asymptotic behaviour of the derivatives at $t=\tau_0(x)$ where $\tau_0(x)$ is the first zero of $(\rho ^x_t)_{t\geq 0}$.