An application of Markov operators in differential and integral equations
type: Item
instance of: scholarly article
title: An application of Markov operators in differential and integral equations
zbMATH Open document ID: 0795.60071
author: Jan Malczak
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1993-01-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108254
review text: Let \(P\) be a Markov operator on an \(L^ 1\) space. The ergodic theorem asserts the convergence of the averages \(N^{-1} \sum^{N-1}_{n=0} P^ nf\) to \(Qf\), where the range of the projection \(Q\) is the fixed point set of \(P-I\). For \(f\) a nonnegative unit vector, \(Qf\) often reduces to a singleton \(f_ *\), and often \(P^ nf\) itself converges to \(f_ *\). Call this condition \((*)\). The same theory works when the discrete semigroup \(\{P^ n:n=0,1,2,\dots\}\) is replaced by the (continuous) semigroup \(\{P(t):t \geq 0\}\). The author begins with \(\{P(t):t \geq 0\}\) and assumes that condition \((*)\) holds for \(P(t_ 0)\) for some \(t_ 0>0\). He then shows that \(P(t)f \to f_ *\) as \(t \to \infty\) for \(f \geq 0\), \(\| f \|_ 1=1\). Three examples are constructed by looking at the equation \(\partial u/ \partial t=Au\) whose solution is given by \(u(t)=P(t)f\) (for \(f=u (0))\). They are: a second order (one-dimensional) uniformly parabolic equation, a linear Boltzmann equation of Tjon-Wu type, and a third example in the context of a model arising in mathematical biology, which was studied by J. J. Tyson and K. B. Hannsgen.
reviewed by: Jerome A. Goldstein
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 94574
zbMATH Keywords: ergodic theorem
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: mathematical biology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 94574