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Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

Roman Bahadursingh

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.

Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

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.

Paper Structure

This paper contains 14 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Formulation of the Stochastic Mean Curvature Flow on Networks
  4. Applying Ito Calculus
  5. Numerical Simulation
  6. Code Implementation
  7. Results
  8. Discussion
  9. Impact of Noise Intensity
  10. Pattern Formation and Stability
  11. Comparison with Deterministic Mean Curvature Flow
  12. Applications and Future Research Directions
  13. Conclusion
  14. Future Work

Figures (1)

  • Figure 1: Stochastic Mean Curvature Flow on Network: Each line represents the position of an edge over time, demonstrating the impact of stochastic perturbations on the deterministic mean curvature flow.