International audienceLet $(Z_{n})$ be a supercritical branching process in a random environment $\xi $, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\log Z_n$ are also established

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International audienceWe consider a branching random walk on $\mathbb{R}$ with a random environment in time (denoted by $\xi$). Let $Z_n$ be the counting measure of particles of generation $n$ and $\tilde Z_n (t)$ be its Laplace transform.We show the convergence of the free energy $ n^{-1}{\log \tilde Z_n(t)}$, large deviation principles and central limit theorems for the sequence of measures $\{Z_n\}$, and a necessary and sufficient condition for the existence of moments of the limit of the martingale ${\tilde Z_n(t)}/{\mathbb E[\tilde Z_n(t)|\xi]}$

We consider a R d -valued discrete time branching random walk with a stationary and ergodic environment in time. Let Zn be the counting measure of particles of generation n. With the help of the uniform convergence of martingales and multifractal analysis, we show a large deviation result associated to the measures Zn as well as the corresponding moderate deviations.AMS 2010 subject classifications. 60J80, 60K37, 60F10.