A pure jump Markov process associated with Smoluchowski's coagulation equation
type: Item
instance of: scholarly article
title: A pure jump Markov process associated with Smoluchowski's coagulation equation
zbMATH Open document ID: 1018.60067
author: Madalina Deaconu
author: Nicolas Fournier
author: Etienne Tanré
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The Smoluchowski's coagulation equation is considered. Let \[ \begin{cases} \frac{d}{dt} n(k,t)=\tfrac 12 \sum^{k-1}_{j=1} K(j,k-j)n(j,t)n(k-j,t)-n(k,t)\sum^\infty_{j=1} K(j,k)n(j,t),\\ n(k,0)=n_0(k)\end{cases}\tag{SD} \] for \(k\in\mathbb{N}^*\) and \[ \begin{cases} \frac{\partial}{\partial t} n(x,t)=\tfrac 12 \int^x_0 K(y,x-y)n(y,t)n(x-y,t) dy-n(x,t)\int^\infty_0 K(x,y)n(y,t) dy,\\ n(x,0)=n_0(x)\end{cases}\tag{SC} \] for all \(x\in\mathbb{R}_+\), where the coagulation kernel \(K\) is nonnegative and symmetric, be the coagulation equations in the discrete and continuous case, respectively. The authors construct a pure jump stochastic process \((X_t)\), \(t\geq 0\), whose law is the solution of the coagulation equation in the following sense: in the discrete case \(P(X_t= k) = k_n(k,t)\) for all \(t\geq 0\) and all \(k\in\mathbb{N}^*\), in the continuous case \(P(X_t\in dx)= xn(x,t)dx\) for all \(t\geq 0\). This jump process satisfies a nonlinear Poisson driven stochastic differential equation. Existence, uniqueness and pathwise regularity for the solution of this equation are studied. It is also proved that the nonlinear process \(X\) can be obtained as the limit of Marcus-Lushnikov procedure.
reviewed by: Maria Stolarczyk
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1906099
zbMATH Keywords: Smoluchowski's coagulation equation
zbMATH Keywords: nonlinear stochastic differential equation
zbMATH Keywords: Poisson measure
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1039548371
OpenAlex ID: W1994094057
Wikidata QID: Q60920743
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DOI: 10.1214/AOP/1039548371
description: scientific article; zbMATH DE number 1906099