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Existence of martingale solutions to a stochastic kinetic model of chemotaxis

Benjamin Gess, Sebastian Herr, Anne Niesdroy

TL;DR

This work extends the mesoscopic chemotaxis model by incorporating a stochastic drift in velocity and proves the existence of weak martingale solutions to the stochastic kinetic chemotaxis system. The authors develop novel pathwise Strichartz estimates for stochastic kinetic transport, leveraging a stochastic flow framework and a time-splitting approach to achieve dispersion both locally and globally under suitable assumptions on the noise and turning kernel. Through regularization, a priori estimates in mixed $L^p$ spaces, and a careful compactness/renormalization scheme, they establish local and global existence results for small initial data, and they provide a rigorous martingale-structure formulation. This yields a mathematically solid foundation for stochastic chemotaxis models and offers techniques potentially applicable to other kinetic-type SPDEs with nonlocal reaction terms.

Abstract

We show the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on new Strichartz estimates for stochastic kinetic transport. The derivation of these estimates involves a local in time dispersion analysis using properties of stochastic flows, and a time-splitting argument to extend the local in time results to arbitrary time intervals.

Existence of martingale solutions to a stochastic kinetic model of chemotaxis

TL;DR

This work extends the mesoscopic chemotaxis model by incorporating a stochastic drift in velocity and proves the existence of weak martingale solutions to the stochastic kinetic chemotaxis system. The authors develop novel pathwise Strichartz estimates for stochastic kinetic transport, leveraging a stochastic flow framework and a time-splitting approach to achieve dispersion both locally and globally under suitable assumptions on the noise and turning kernel. Through regularization, a priori estimates in mixed spaces, and a careful compactness/renormalization scheme, they establish local and global existence results for small initial data, and they provide a rigorous martingale-structure formulation. This yields a mathematically solid foundation for stochastic chemotaxis models and offers techniques potentially applicable to other kinetic-type SPDEs with nonlocal reaction terms.

Abstract

We show the existence of local and global in time weak martingale solutions for a stochastic version of the Othmer-Dunbar-Alt kinetic model of chemotaxis under suitable assumptions on the turning kernel and stochastic drift coefficients, using dispersion and stochastic Strichartz estimates. The analysis is based on new Strichartz estimates for stochastic kinetic transport. The derivation of these estimates involves a local in time dispersion analysis using properties of stochastic flows, and a time-splitting argument to extend the local in time results to arbitrary time intervals.

Paper Structure

This paper contains 16 sections, 12 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Comments on the literature
  3. Structure of the paper
  4. Notation and preliminaries
  5. Stochastic kinetic transport and Strichartz estimates
  6. Stochastic kinetic transport
  7. Dispersion estimates for stochastic kinetic transport
  8. Strichartz estimates for stochastic kinetic transport
  9. Different types of external random force
  10. External random force that allows for local in time dispersion
  11. External random force that allows for global in time dispersion
  12. Stochastic versions of chemotactic movement
  13. Solution of a regularized chemotactic equation
  14. A-priori-estimates and bootstrapping
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 1 more sections

Key Result

Theorem 1.6

Let $d \ge 2$. Fix $T \in (0, \infty)$ and consider parameters $r,a,p,q$ such that Consider IVP_stoch nonnegative initial data $f_0:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ that is supported in $\mathbb{R}^d \times V$ such that $f_0 \in L^1(\mathbb{R}^{2d}) \cap L^{a}(\mathbb{R}^{2d})$ and $\Vert f_0\Vert_{L^a(\mathbb{R}^{2d})}$ is sufficiently small. Assume that f

Theorems & Definitions (39)

  • Remark 1.2
  • Theorem 1.6
  • Definition 2.1: Weak martingale solution
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1
  • proof
  • ...and 29 more