Components and Exit Times of Brownian Motion in two or more $p$-Adic Dimensions

TL;DR

This paper studies the behavior of p-adic Brownian motions in multiple dimensions, revealing component dependency and analyzing exit times through asymptotic probability analysis.

Contribution

It introduces a p-adic analogue of the Wiener process in multiple dimensions and analyzes the dependency of components and exit time probabilities.

Findings

Components are dependent for all time.

Exit time probabilities reflect component dependency.

Asymptotic analysis characterizes the process behavior.

Abstract

The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.