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Long-time behaviour of the correlated random walk system

Joaquín Menacho, Marta Pellicer, J. Solà-Morales

TL;DR

This work analyzes the long-time behavior of the correlated random walk system, linking it to a weakly damped wave equation with dynamic boundary conditions. By casting the problem as an abstract C$0$-semigroup generated by $A$ on $X$ and proving the semigroup is eventually compact, the authors establish exponential decay of solutions at a rate given by the dominant eigenvalue $\lambda_0(S)$ of $A$, and provide a detailed spectral description and an asymptotic profile given by the corresponding eigenfunction. The spectrum is described via transcendental equations in $\nu$, with eigenfunctions of symmetric or antisymmetric type; $\lambda_0(S)$ is real, simple, and governs the asymptotic approach to zero. The results reveal Sturm-Liouville-type space oscillations and quantify how the dissipative parameter $S$ accelerates decay, with implications for applications in movement ecology and chromatography.

Abstract

In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can also be seen as a weakly damped wave equation with certain boundary conditions. We are interested in the long-time behaviour of its solutions. To be precise, we will prove that the decay of the solutions to this problem is of exponential form, where the optimal decay rate exponent is given by the dominant eigenvalue of the corresponding operator. This eigenvalue can be obtained as a particular solution of a system of transcendental equations. A complete description of the spectrum of the operator is provided, together with a comprehensive analysis of the corresponding eigenfunctions and their geometry.

Long-time behaviour of the correlated random walk system

TL;DR

This work analyzes the long-time behavior of the correlated random walk system, linking it to a weakly damped wave equation with dynamic boundary conditions. By casting the problem as an abstract C-semigroup generated by on and proving the semigroup is eventually compact, the authors establish exponential decay of solutions at a rate given by the dominant eigenvalue of , and provide a detailed spectral description and an asymptotic profile given by the corresponding eigenfunction. The spectrum is described via transcendental equations in , with eigenfunctions of symmetric or antisymmetric type; is real, simple, and governs the asymptotic approach to zero. The results reveal Sturm-Liouville-type space oscillations and quantify how the dissipative parameter accelerates decay, with implications for applications in movement ecology and chromatography.

Abstract

In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can also be seen as a weakly damped wave equation with certain boundary conditions. We are interested in the long-time behaviour of its solutions. To be precise, we will prove that the decay of the solutions to this problem is of exponential form, where the optimal decay rate exponent is given by the dominant eigenvalue of the corresponding operator. This eigenvalue can be obtained as a particular solution of a system of transcendental equations. A complete description of the spectrum of the operator is provided, together with a comprehensive analysis of the corresponding eigenfunctions and their geometry.
Paper Structure (2 sections, 5 theorems, 20 equations, 1 figure)

This paper contains 2 sections, 5 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Eigenfunctions and eigenvalues

Key Result

Theorem 1.2

[Theorem 3.4 of Hillen2010] The operator $A$ is the generator of a $C_0$-semigroup on $X$, $\{ T(t)=e^{At}, \ t\geq 0\}$. In particular, for any $U_0\in {\cal{D}}(A)$, there exists a unique solution $U=T(t)U_0\in C^1([0,\infty);X))\cap C^0([0,\infty);{\cal{D}}(A))$ satisfying opA. If $U_0\in X$, the

Figures (1)

  • Figure 1: Some eigenfunctions for $S=0.8$, real and imaginary parts respect to $x\in[-1/2,1/2]$. From left to right, and up to down, plots from $u_0,v_0$, and real and imaginary part of $u_{n,1},v_{n,1}$ for $n=2,4$ (symmetric case, top), and $n=1,3$ (antisymmetric case, bottom). In all cases, the blue-cross line corresponds to $u_{n,1}$, and the red-circle one to $v_{n,1}$. The plots for the corresponding $(u_{n,2},v_{n,2})$ are similar, so we do not include them. In these graphs, we can also see the oscillatory behaviour of the $n$-the eigenfunction, described in Proposition \ref{['prop:osc']} below.

Theorems & Definitions (10)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4