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

TL;DR

The work addresses a class of parabolic Cauchy problems with variable exponent coefficients, introducing nonlinear state-dependent diffusion and drift via $p(x)$ and $q(x)$. It combines viscosity-solution theory with a stochastic representation: a state-dependent SDE $dX(t)=\mu X(t)^{p(X(t))} dt + \sigma X(t)^{q(X(t))} dW(t)$ admits a unique positive strong solution, and the PDE solution $u(x,t)$ is represented by a Feynman–Kac formula $u(x,t)=\mathbb{E}_x\left[ e^{-\int_0^t V(X(s)) ds} f(X(t)) \right]$, ensuring well-posedness. Through a log-transform to achieve uniform parabolicity, the paper proves uniqueness via a comparison principle and establishes local Sobolev regularity. Numerical validation compares Crank–Nicolson on a log-grid with Euler–Maruyama Monte Carlo, finding errors on the order of $10^{-4}$ across several exponent choices, thereby corroborating the theoretical results and demonstrating the practical viability of the stochastic representation.

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.

Figures (3)

• Figure 1: Comparison of PDE and SDE solutions for variable exponents $p_1(x)$ and $q_1(x)$.
• Figure 2: Comparison of PDE and SDE solutions for variable exponents $p_2(x)$ and $q_2(x)$.
• Figure 3: Comparison of PDE and SDE solutions for the baseline case $p_3(x)=q_3(x)=1$.

Theorems & Definitions (11)

• Remark 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2
• Theorem 3.1
• proof