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A probabilistic study of the set of stationary solutions to spatial kinetic-type equations

Sebastian Mentemeier, Glib Verovkin

TL;DR

The paper studies multivariate spatial kinetic-type equations, including the spatially homogeneous Boltzmann equation for Maxwellian molecules with elastic and inelastic collisions. It introduces a probabilistic representation via a continuous-time branching random walk to obtain time-dependent solutions and a detailed fixed-point analysis to classify stationary solutions. The main result characterizes stationary states as mixtures involving the Biggins martingale limit $W_\infty$ and Lévy–Khintchine components $V$, $Y$, and an invariant function $K$, with which the solution takes the form $\phi(roe_3)=\mathbb{E}\big[\exp\{-W_\infty K(r,o)-\tfrac{1}{2}r^2 V(o)+i r Y(o)\}\big]$. The framework extends previous work by removing density/centering assumptions and addressing the multivariate, rotationally invariant setting, thereby linking kinetic theory with branching-process techniques and smoothing-type fixed-point equations.

Abstract

In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Using a representation of the collision operator derived in Bassetti, Ladelli, Matthes (2015) and Dolera, Regazzini (2014), we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under assumptions as weak as possible. Our main objective is a characterisation of the set of stationary solutions, e.g. equilibrium solutions for inelastic kinetic-type equations, which we describe as mixtures of multidimensional stable laws.

A probabilistic study of the set of stationary solutions to spatial kinetic-type equations

TL;DR

The paper studies multivariate spatial kinetic-type equations, including the spatially homogeneous Boltzmann equation for Maxwellian molecules with elastic and inelastic collisions. It introduces a probabilistic representation via a continuous-time branching random walk to obtain time-dependent solutions and a detailed fixed-point analysis to classify stationary solutions. The main result characterizes stationary states as mixtures involving the Biggins martingale limit and Lévy–Khintchine components , , and an invariant function , with which the solution takes the form . The framework extends previous work by removing density/centering assumptions and addressing the multivariate, rotationally invariant setting, thereby linking kinetic theory with branching-process techniques and smoothing-type fixed-point equations.

Abstract

In this paper we study multivariate kinetic-type equations in a general setup, which includes in particular the spatially homogeneous Boltzmann equation with Maxwellian molecules, both with elastic and inelastic collisions. Using a representation of the collision operator derived in Bassetti, Ladelli, Matthes (2015) and Dolera, Regazzini (2014), we prove the existence and uniqueness of time-dependent solutions with the help of continuous-time branching random walks, under assumptions as weak as possible. Our main objective is a characterisation of the set of stationary solutions, e.g. equilibrium solutions for inelastic kinetic-type equations, which we describe as mixtures of multidimensional stable laws.
Paper Structure (21 sections, 22 theorems, 203 equations)

This paper contains 21 sections, 22 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Kinetic-type equations
  3. Setup and assumptions
  4. Adhoc statement of the main result
  5. Related works
  6. Branching processes and martingales
  7. A weighted branching process
  8. Branching random walk
  9. Associated random walks
  10. Additive martingales
  11. Main results
  12. Time-dependent solutions
  13. Stationary solutions
  14. Proofs for the time-dependent case
  15. Proofs for the stationary case
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume K:assumption_m-K:assumption_regular_variation with $\alpha \in (0,1)$. There exists a nonnegative random variable $W_\infty$ with mean one such that a characteristic function $\phi$ is a solution to K:stationary_equation if and only if for a function $K: {\mathbb{R}}_{\geqslant}\times{\mathbb{O}(3)} \to {\mathbb{C}}$ satisfying $K(r,o)=s^{-\alpha} K(rs,ou)$ for all $(s,u)\in \mathbb{S}$ an

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof : Source
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • ...and 39 more