Large deviations for Branching Processes in Random Environment

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A branching process in random environment (Zₙ, n ∈) is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of large deviations. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as Zₙ ≤ ecn when c is smaller than the typical geometric growth rate L and Zₙ ≥ ecn when c > L. One way to obtain such an atypical rate of growth is to have a typical realization of the branching process in an atypical sequence of environments. This gives us a general lower bound for the rate of decrease of their probability. When each individual leaves at least one offspring in the next generation almost surely, we compute the exact rate function of these events and we show that conditionally on the large deviation event, the trajectory t 1n Z[nt], t∈ [0,1] converges to a deterministic function fc:[0,1] + in probability in the sense of the uniform norm. The most interesting case is when c < L and we authorize individuals to have only one offspring in the next generation. In this situation, conditionally on Zₙ ≤ ecn, the population size stays fixed at 1 until a time n tc. After time n tc an atypical sequence of environments let Zₙ grow with the appropriate rate (≠ L) to reach c. The corresponding map fc(t) is piecewise linear and is 0 on [0,tc] and fc(t) = c(t-tc)/(1-tc) on [tc,1].