Lyapunov Exponents for Branching Processes in a Random Environment: The Effect of Information
We consider multitype branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define Markov chains associated to the branching process, and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound, and we show that adding less information improves the lower bound. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bounds.
KeywordsProduct of random matrices Lyapunov exponent Multitype branching process Markovian random environment Extinction criterion
Mathematics Subject Classification60J80 60J22 60J27