Coordinate changed random fields on manifolds

TL;DR

This paper introduces a new class of time-dependent random fields on compact Riemannian manifolds, modeled via time-changed Brownian motions, which are solutions to fractional diffusion equations involving the Laplace-Beltrami operator.

Contribution

It presents a novel framework for representing random fields on manifolds using time-changed diffusion processes governed by fractional derivatives.

Findings

Representation of random fields via time-changed Brownian motions

Connection to fractional diffusion equations on manifolds

Modeling of random fields on varying manifolds

Abstract

We introduce a class of time dependent random fields on compact Riemannian monifolds. These are represented by time-changed Brownian motions. These processes are time-changed diffusion, or the stochastic solution to the equation involving the Laplace-Beltrami operator and a time-fractional derivative of order $\beta\in (0,1)$. The time dependent random fields we present in this work can therefore be realized through composition and can be viewed as random fields on randomly varying manifolds.