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Sub-exponential tails in biased run and tumble equations with unbounded velocities

Émeric Bouin, Josephine Evans, Luca Ziviani

TL;DR

This work analyzes the long-time behavior of the run-and-tumble equation with unbounded velocities, proving existence and uniqueness of a stationary state $G$ and sub-exponential convergence to equilibrium. It develops a dual approach combining Harris-type probabilistic methods with $L^2$-hypocoercivity à la Dolbeault–Mouhot–Schmeiser to handle non-explicit steady states and sub-geometric rates, aided by carefully constructed weak Lyapunov functionals and a space-splitting strategy. The analysis yields sub-exponential tails for the equilibrium density $ ho_G$ and precise moment and Poincaré-type inequalities, enabling both $L^1$ and $L^2$ convergence results and moment control. Overall, the paper extends the spectral/ergodic theory for kinetic chemotaxis models to the unbounded-velocity setting, providing rigorous tools for sub-geometric convergence and tail behavior that were unavailable in bounded-velocity frameworks.

Abstract

Run and tumble equations are widely used models for bacterial chemotaxis. In this paper, we are interested in the long time behaviour of run and tumble equations with unbounded velocities. We show existence, uniqueness and quantitative convergence towards a steady state. In contrast to the bounded velocity case, the equilibrium has sub-exponential tails and we have sub-exponential rate of convergence to equilibrium. This produces additional technical challenges. We are able to successfully adapt both Harris' type and $\sfL^2-$ hypocoercivity \textit{a la} Dolbeault-Mouhot-Schmeiser techniques.

Sub-exponential tails in biased run and tumble equations with unbounded velocities

TL;DR

This work analyzes the long-time behavior of the run-and-tumble equation with unbounded velocities, proving existence and uniqueness of a stationary state and sub-exponential convergence to equilibrium. It develops a dual approach combining Harris-type probabilistic methods with -hypocoercivity à la Dolbeault–Mouhot–Schmeiser to handle non-explicit steady states and sub-geometric rates, aided by carefully constructed weak Lyapunov functionals and a space-splitting strategy. The analysis yields sub-exponential tails for the equilibrium density and precise moment and Poincaré-type inequalities, enabling both and convergence results and moment control. Overall, the paper extends the spectral/ergodic theory for kinetic chemotaxis models to the unbounded-velocity setting, providing rigorous tools for sub-geometric convergence and tail behavior that were unavailable in bounded-velocity frameworks.

Abstract

Run and tumble equations are widely used models for bacterial chemotaxis. In this paper, we are interested in the long time behaviour of run and tumble equations with unbounded velocities. We show existence, uniqueness and quantitative convergence towards a steady state. In contrast to the bounded velocity case, the equilibrium has sub-exponential tails and we have sub-exponential rate of convergence to equilibrium. This produces additional technical challenges. We are able to successfully adapt both Harris' type and hypocoercivity \textit{a la} Dolbeault-Mouhot-Schmeiser techniques.
Paper Structure (25 sections, 35 theorems, 424 equations, 3 figures)

This paper contains 25 sections, 35 theorems, 424 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of previous contributions
  3. Assumptions and main results
  4. Organisation of the paper
  5. Weak-Lyapunov functions in $\mathsf{L}^1$
  6. A sub-exponential weak Lyapunov function
  7. A polynomial weak Lyapunov function
  8. Existence via the Harris theorem
  9. Existence via a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and the bound above on $G$ in $\mathsf{L}^\infty$
  10. The semi-group $S_{\mathcal{B} }$
  11. The iterated semi-groups $\left(\mathcal{A} S_B\right)^{\star j}$ and $S_\mathcal{B} \star \left(\mathcal{A} S_B\right)^{\star j}$
  12. Representation formula and pre-bounds above and below for the density
  13. Lower bound for the density $\rho_G$
  14. Upper bound for the density $\rho_G$
  15. Refined bounds and a Poincaré inequality under \ref{['Hyp:asympRho']}
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $\mathcal{M}$ be defined by eq:M for some $\gamma>0$ and assume that Hyp:psi holds. Then we have the following.

Figures (3)

  • Figure 1.1: Black line: Plot of the density $\rho_G$ for $\gamma=1$ and $\chi=0.8$. Blue line: expected asymptotic behaviour $x\mapsto \lvert x\rvert^{\frac{1}{4}}\exp(-2\sqrt{1.8}\,\lvert x\rvert^{\frac{1}{2}})$.
  • Figure 1.2: Plot of the stationary solution $G$ for $\gamma=1$ and $\chi=0.8$
  • Figure 1.3: Black line: Plot of $\frac{\rho_G P_G^{(4)}}{P_G^2}$ for $\gamma=1$ and $\chi=0.8$. Blue line: extrapolation of the growth rate $x\mapsto 2.35 \, \lvert x\rvert^{0.46}$.

Theorems & Definitions (67)

  • Theorem 1.2
  • Theorem 1.4
  • Lemma 2.1
  • proof : Proof of \ref{['lem:Psi']}
  • Proposition 2.2
  • proof : Proof of \ref{['Prop:Lyapunov']}
  • Proposition 2.3
  • proof : Proof of \ref{['Prop:PolyLyap']}
  • Proposition 3.1
  • Lemma 3.2
  • ...and 57 more