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A new look to branching Brownian motion from a particle based reaction diffusion dynamics point of view

Alberto Lanconelli, Berk Tan Perçin

Abstract

Aim of this note is to analyse branching Brownian motion within the class of models introduced in the recent paper [4] and called chemical diffusion master equations. These models provide a description for the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. We derive an infinite system of Fokker-Planck equations that rules the probabilistic evolution of the single particles generated by the branching mechanism and analyse its properties using Malliavin Calculus techniques, following the ideas proposed in [13]. Another key ingredient of our approach is the McKean representation for the solution of the Fisher-Kolmogorov-Petrovskii-Piskunov equation and a stochastic counterpart of that equation. We also derive the reaction-diffusion partial differential equation solved by the average concentration field of the branching Brownian system of particles.

A new look to branching Brownian motion from a particle based reaction diffusion dynamics point of view

Abstract

Aim of this note is to analyse branching Brownian motion within the class of models introduced in the recent paper [4] and called chemical diffusion master equations. These models provide a description for the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. We derive an infinite system of Fokker-Planck equations that rules the probabilistic evolution of the single particles generated by the branching mechanism and analyse its properties using Malliavin Calculus techniques, following the ideas proposed in [13]. Another key ingredient of our approach is the McKean representation for the solution of the Fisher-Kolmogorov-Petrovskii-Piskunov equation and a stochastic counterpart of that equation. We also derive the reaction-diffusion partial differential equation solved by the average concentration field of the branching Brownian system of particles.
Paper Structure (3 sections, 5 theorems, 51 equations)

This paper contains 3 sections, 5 theorems, 51 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Proof of Theorem \ref{['main 1']}
  3. Proof of Theorem \ref{['main 2']}

Key Result

Theorem 1.1

The sequence $\{\rho_n\}_{n\geq 1}$ solves the system of equations here, $\delta_0$ stands for the Dirac delta function concentrated at the origin while $\hat{\otimes}$ denotes symmetric tensor product with respect to the variables $(y_1,...,y_n)$. Moreover, we have the representation where $\{g_n\}_{n\geq 1}$ is a sequence of functions with and defined recursively as The symbol $\mathtt{p}_{t

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 2 more