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Iterated random walks in random scenery (PAPAPA)

Nadine Guillotin-Plantard, Françoise Pène, Frédérique Watbled

TL;DR

The article analyzes a 3D random walk in a stratified lattice with random horizontal orientations, introducing the PAPAPA construction as a random walk in random scenery evolving in a second random scenery. It proves a joint functional limit for the triple (X, Y, Z), showing convergence to the KS process Δ, a second-level integral Γ against the local time of Δ, and Brownian motion B, with a precise hierarchy of diffusion exponents (n^{3/4}, n^{5/8}, n^{1/2}). The authors develop a general framework for joint convergence involving a process and its local time, extend it via local-limit theorems to iterated random scenery, and establish a concrete joint limit for the PAPAPA with the PA and PAPA under coprimality, while proposing conjectures for higher-order iterates and outlining related 3D models and open problems. These results deepen understanding of diffusion in randomly oriented and inhomogeneous media and provide tools for analyzing iterated random walks in complex random environments.

Abstract

We establish a limit theorem for a new model of 3-dimensional random walk in an inhomogeneous lattice with random orientations. This model can be seen as a 3dimensional version of the Matheron and de Marsily model [12]. This new model leads us naturally to the study of iterated random walk in random scenery, which is a new process that can be described as a random walk in random scenery evolving in a second random scenery. We use the french acronym PAPAPA for this new process and answer a question about its stochastic behaviour asked about twenty years ago by St{é}phane Le Borgne.

Iterated random walks in random scenery (PAPAPA)

TL;DR

The article analyzes a 3D random walk in a stratified lattice with random horizontal orientations, introducing the PAPAPA construction as a random walk in random scenery evolving in a second random scenery. It proves a joint functional limit for the triple (X, Y, Z), showing convergence to the KS process Δ, a second-level integral Γ against the local time of Δ, and Brownian motion B, with a precise hierarchy of diffusion exponents (n^{3/4}, n^{5/8}, n^{1/2}). The authors develop a general framework for joint convergence involving a process and its local time, extend it via local-limit theorems to iterated random scenery, and establish a concrete joint limit for the PAPAPA with the PA and PAPA under coprimality, while proposing conjectures for higher-order iterates and outlining related 3D models and open problems. These results deepen understanding of diffusion in randomly oriented and inhomogeneous media and provide tools for analyzing iterated random walks in complex random environments.

Abstract

We establish a limit theorem for a new model of 3-dimensional random walk in an inhomogeneous lattice with random orientations. This model can be seen as a 3dimensional version of the Matheron and de Marsily model [12]. This new model leads us naturally to the study of iterated random walk in random scenery, which is a new process that can be described as a random walk in random scenery evolving in a second random scenery. We use the french acronym PAPAPA for this new process and answer a question about its stochastic behaviour asked about twenty years ago by St{é}phane Le Borgne.

Paper Structure

This paper contains 6 sections, 7 theorems, 169 equations, 2 figures.

Table of Contents

  1. Model and main result
  2. A joint convergence result
  3. A joint convergence result using local limit theorems
  4. Joint Convergence of the PAPAPA together with the PA and the PAPA
  5. Conjecture and properties of the limit processes
  6. Other three-dimensional randomly oriented models

Key Result

Theorem 1.1

The sequence of 3-uples of processes converges in distribution in $\mathcal{D}([0,+\infty))^3$ to the 3-uple of processes $(\Delta,\Gamma,B)$, where $B$ and $\Delta$ are as above and where $\Gamma$ is given by where $W^{(2)}$ is a bilateral Brownian motion, independent of $(B,W^{(1)})$.

Figures (2)

  • Figure 1: Example of orientations of horizontal and vertical planes (in black and red) and at a point (in blue).
  • Figure 2: A trajectory of $\widetilde{M}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['PRO1']}
  • Theorem 4.1
  • proof : Proof of Theorem \ref{['THMtriple']}
  • ...and 6 more