Stochastic representation of solutions for the parabolic Cauchy problem with variable exponent coefficients

TL;DR

This paper establishes the existence and uniqueness of solutions for degenerate parabolic equations with variable exponents using stochastic representations, extending classical SDEs to model complex, state-dependent dynamics.

Contribution

It introduces a novel stochastic representation approach for variable exponent parabolic equations and verifies solutions through numerical experiments.

Findings

Proved existence and uniqueness of bounded viscosity solutions.

Validated the stochastic approach with numerical experiments.

Extended SDE models to include nonlinear, state-dependent coefficients.

Abstract

In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE.