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A probabilistic representation of the solution to a 1D evolution equation in a medium with negative index

Éric Bonnetier, Pierre Etoré, Miguel Martinez

TL;DR

This work addresses a one-dimensional evolution PDE with a sign-changing diffusion coefficient to model negative index materials. It combines a spectral semigroup approach with probabilistic and pseudo-probabilistic representations via Skew Brownian motion and its pseudo variant, deriving fundamental solutions and extending to finite intervals through killed Brownian motion kernels. Three complementary numerical schemes are developed: a spectral truncation method, a pseudo-random-walk-based scheme, and an implicit finite-difference scheme built from killed Brownian motion representations, each with convergence analysis or heuristic justification in the pseudo-probability setting. The results illuminate the interaction of transmission across interfaces with sign-changing diffusion and provide practical tools for simulating metamaterial diffusion problems, including finite-domain setups and robust numerical schemes. These contributions bridge PDE theory, pseudo-stochastic processes, and numerical analysis for metamaterials.

Abstract

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is changing sign, as in models for metamaterials.We focus on the construction of a fundamental solution for the evolution equation,which does not proceed as in the case of standard parabolic PDE's, since the associatedsecond order operator is not elliptic. We show that a spectral representation of the semigroup associated to the equation can be derived, which leads to a first expression of the fundamental solution. We also derive a probabilistic representation in terms of a pseudo Skew Brownian Motion (SBM).This construction generalizes that derived from the killed SBM when the diffusion coefficientis piecewise constant but remains positive.We show that the pseudo SBM can be approached by a rescaled pseudo asymmetric random walk,which allows us to derive several numerical schemes for the resolution of the PDEand we report the associated numerical test results.

A probabilistic representation of the solution to a 1D evolution equation in a medium with negative index

TL;DR

This work addresses a one-dimensional evolution PDE with a sign-changing diffusion coefficient to model negative index materials. It combines a spectral semigroup approach with probabilistic and pseudo-probabilistic representations via Skew Brownian motion and its pseudo variant, deriving fundamental solutions and extending to finite intervals through killed Brownian motion kernels. Three complementary numerical schemes are developed: a spectral truncation method, a pseudo-random-walk-based scheme, and an implicit finite-difference scheme built from killed Brownian motion representations, each with convergence analysis or heuristic justification in the pseudo-probability setting. The results illuminate the interaction of transmission across interfaces with sign-changing diffusion and provide practical tools for simulating metamaterial diffusion problems, including finite-domain setups and robust numerical schemes. These contributions bridge PDE theory, pseudo-stochastic processes, and numerical analysis for metamaterials.

Abstract

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is changing sign, as in models for metamaterials.We focus on the construction of a fundamental solution for the evolution equation,which does not proceed as in the case of standard parabolic PDE's, since the associatedsecond order operator is not elliptic. We show that a spectral representation of the semigroup associated to the equation can be derived, which leads to a first expression of the fundamental solution. We also derive a probabilistic representation in terms of a pseudo Skew Brownian Motion (SBM).This construction generalizes that derived from the killed SBM when the diffusion coefficientis piecewise constant but remains positive.We show that the pseudo SBM can be approached by a rescaled pseudo asymmetric random walk,which allows us to derive several numerical schemes for the resolution of the PDEand we report the associated numerical test results.
Paper Structure (14 sections, 10 theorems, 142 equations, 4 figures, 1 table)

This paper contains 14 sections, 10 theorems, 142 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Spectral representation of the semigroup
  3. A probabilistic construction of the fundamental solution for the evolution equation on $\mathbb{R}$
  4. The fundamental solution when $k>0$
  5. The fundamental solution in the case $k<0$ and the pseudo SBM
  6. Convergence of the scaled pseudo asymmetric random walk to the pseudo SBM
  7. The evolution equation on a finite interval
  8. The transition function of the SBM killed at $-a$ or $a$ (case $k>0$)
  9. The fundamental solution to \ref{['eq_evol']} in the general case $k\in\mathbb{R}^* \setminus \{-1\}$
  10. Various numerical schemes for the PDE \ref{['eq_evol']}
  11. Scheme inspired by the spectral representation of the semi-group
  12. Scheme inspired by the scaled pseudo asymmetric random walk
  13. Scheme inspired by the expression of the fundamental solution involving the transition function of the killed Brownian motion
  14. Numerical experiments

Key Result

Proposition 2.1

When $k < 0$, the functions $(f_{k,n}, g_{k,n})_{n \geq 1}$ form a Hilbert basis of $L^2(I)$ and of $H^1_0(I)$.

Figures (4)

  • Figure 1: Plot of an approximation of the function $u(T=0.4,\cdot)$, by $\bar{u}_{spec}^{200}(T,\cdot)$, $\bar{u}_{RW}^{2.5\times 10^5}(T,\cdot)$ and $\bar{u}_{fund}^{2\times 10^{-3}\,,\,500}(T,\cdot)$, for the initial condition $u_0(x)=(10 x^3 - 3 x^2 - 9x + 4)/2$.
  • Figure 2: Plot of the initial condition $u_0$ and of an approximation of the function $u(t,\cdot)$, by $\bar{u}_{fund}^{2\times 10^{-3}\,,\,500}(t,\cdot)$, at times $t=8\times 10^{-4}$, $t=0.12$ and $t=0.4$, for the initial condition $u_0(x)=(10 x^3 - 3 x^2 - 9x + 4)/2$.
  • Figure 3: Plot of an approximation of the function $u(T=0.4,\cdot)$, by $\bar{u}_{spec}^{200}(T,\cdot)$, $\bar{u}_{RW}^{2.5\times 10^5}(T,\cdot)$ and $\bar{u}_{fund}^{2\times 10^{-3}\,,\,500}(T,\cdot)$, for the initial condition $u_0(x)=\mathbf{1}_{x<0}-0.5$.
  • Figure 4: Plot of the initial condition $u_0$ and of an approximation of the function $u(t,\cdot)$, by $\bar{u}_{fund}^{2\times 10^{-3}\,,\,500}(t,\cdot)$, at times $t=0.012$, $t=0.12$ and $t=0.4$, for the initial condition $u_0(x)=\mathbf{1}_{x<0}-0.5$.

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 16 more