An application of Markov operators in differential and integral equations
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.
scientific article; zbMATH DE number 94574
scientific article