Local time of Martin-Lof Brownian motion

TL;DR

This paper explores the concept of local times in Brownian motion through the lens of algorithmic randomness, introducing effective local times and linking them to Martin-Löf randomness, providing new computational insights.

Contribution

It introduces the notion of effective local time for Brownian motion and connects it to Martin-Löf randomness, offering a computational perspective on classical local times.

Findings

Martin-Löf random paths have continuous effective local times at all computable points

Provides a new simple, computationally expressed representation of classical Brownian local times

Establishes a link between algorithmic randomness and classical stochastic process properties

Abstract

In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-L\"of random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed.