Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

TL;DR

This paper studies the evolution of network structures under stochastic mean curvature flow using Ito calculus, deriving SDEs and analyzing stability and pattern formation through numerical simulations.

Contribution

It introduces a novel application of Ito calculus to stochastic mean curvature flow on networks, deriving new SDE models and analyzing their dynamics.

Findings

Derived SDEs governing network edge evolution

Analyzed stability and long-term behavior of stochastic networks

Observed pattern formation influenced by stochastic perturbations

Abstract

In this paper, we investigate the stochastic mean curvature flow (SMCF) on networks, a niche area within stochastic processes and geometric analysis. By applying Ito calculus, we analyze the evolution of network structures influenced by random perturbations. We derive a stochastic differential equation (SDE) for the network edges and utilize numerical simulations to study the stability, long-term behavior, and pattern formation in these systems. Our results offer new insights into the dynamics of complex networks under stochastic influences and open pathways for future research in stochastic geometry.