Nonlinear PDEs and measure-valued branching type processes
type: Item
instance of: scholarly article
title: Nonlinear PDEs and measure-valued branching type processes
zbMATH Open document ID: 1252.35286
author: Lucian Beznea
author: Andrei-George Oprina
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-09-12T00:00:00Z
review text: Let \(E\) be a topological space homeomorphic to a universally measurable set in a metrizable compact, \(E^{(m)}\) be \(E^{m}\) factorized by the action of the permutation group \(S_{m}\). Let \(S=\oplus_{m\geq 0}E^{(m)}\). \(S\) appears as the set of all finite sums of Dirac measures \(\delta_{x}\) on \(E\), this defines an addition for elements in \(S\) and a convolution for measures on \(S\). A multiplicative function on \(S\) is a \(\varphi^{\wedge}(\mu )=e^{\int \log\varphi d\mu}\), where \(\varphi :E\rightarrow (0,1]\). A kernel \(N\) on \(S\) is called branching if \(N_{\mu +\nu}=N_{\mu}\ast N_{\nu}\), where \(N_{\mu}\) is \(\mu N\). The branching kernels are in a bijective correspondence with the submarkovian ones \(B\) taking functions on \(S\) to ones on \(E\) by \(B^{\wedge}(\varphi^{\wedge})=((B\varphi )^{\wedge})^{\wedge}\). Every kernel \(P\) on \(E\) defines a \(P\otimes^{m}\) on \(E^{m}\) and, by restriction, one \(P^{(m)}\) on \(E^{(m)}\). Then, if \(P\) is submarkovian, a kernel \(P^{\sim}\) on \(S\) follows. The family of transition kernels of a right Markov process \(X\) with state space \(E\) defines in this way such a process on \(S\), with the corresponding resolvent. Suppose that \(E\) is Lusin and let \((U_{\alpha})\) be the resolvent of \(X\). Let \(A\) be a continuous additive functional of \(X\) with bounded \((U_{A}^{q}f)(x)=E^{x}\int e^{- qt}f(X_{t})dA_{t}\), \(A_{t}^{m}(\omega_{1},\dots ,\omega_{n})= \sum_{i=1}^{n}A_{t}(\omega_{i})\), \(A_{t}^{'m}=A_{t}^{m}/m\). Consider the resolvents \((K_{\alpha}^{q,m\sim})_{\alpha >0}\) and \((K_{\alpha}^{'q,m\sim})_{\alpha >0}\) with initial kernels the restrictions, to \(E^{(m)}\), of \(U_{A^{m}}^{q}\), \(U_{A'{}^{m}}{}^{q}\) respectively, \[ K_{1}^{q\sim}=\oplus_{m\geq 1}K_{1}^{q,m\sim}, \quad K_{1}^{'q\sim}=\oplus_{m\geq 1}K_{1}^{'q,m\sim}. \] Let \((W_{\alpha}^{\sim})\), \((W_{\alpha}^{'\sim})\) be the (subordinate) resolvents with initial kernels \[ U_{q}^{\sim}-K_{1}^{q\sim}U_{q}^{\sim}, \qu…
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 5947106
zbMATH Keywords: branching kernel
zbMATH Keywords: continuous additive functional
zbMATH Keywords: killing kernel
zbMATH Keywords: resolvent
zbMATH Keywords: right process
zbMATH Keywords: infinitesimal operator
zbMATH Keywords: nonlinear semigroup of operators
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.10.034
OpenAlex ID: W2084983529
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DOI: 10.1016/J.JMAA.2010.10.034
description: scientific article; zbMATH DE number 5947106