Improved H\"{o}lder continuity near the boundary of one-dimensional super-Brownian motion

TL;DR

This paper investigates the boundary regularity of the local time of one-dimensional super-Brownian motion, establishing precise conditions for its Hölder continuity near the boundary.

Contribution

It provides a detailed characterization of the local time’s Hölder continuity near the boundary, identifying the exact threshold at  for continuity.

Findings

Local time is -Hölder continuous near the boundary for 0<<3

Local time fails to be -Hölder continuous for >3

Boundary regularity depends critically on the Hölder exponent 

Abstract

We show that the local time of one-dimensional super-Brownian motion is locally $\gamma$-H\"older continuous near the boundary if $0<\gamma<3$ and fails to be locally $\gamma$-H\"older continuous if $\gamma>3$.