M2 Report in Mathematics proposed by ´ & K. Raschel M. Peigne

Random walks in cones of the Euclidean space We consider here a random walk (Sn )n≥1 on IRd , whose increments have law µ, and assume that µ is adapted and aperiodic. We fix a cone C ∈ IRd and study the asymptotic behavior of (Sn )n≥1 inside C. For instance, if d = 1, the cone C will be the half line [0, +∞[; in dimension 2, we can consider the quarter plane [0, +∞[×[0, +∞. We will assume that the origin o belongs to C an that the probability starting from o to stay inside the cone is > 0, (otherwise we will change the starting point of the walk). Under suitable moment conditions on µ, we will study the asymptotic behaviour of the random walk inside C: 1. if τ is the stopping time defined by τ := inf{n ≥ 1 : Sn ∈ / C}, we will precise the tail asymptotics of τ that is the behaviour at infinity of the sequence (P[τ > n])n≥1 2. we will also study a local limit theorem for this walk, i-e find an asymptotic equivalent of the sequence (P[Sn ∈ K ∩ C, τ > n])n where K is a compact of IRd . These question have been studied for a long time in dimension 1; one may cite for instance Kozlov [4], Iglehard citeI, .... their approach uses fine Fourier analysis and is based on the famous Wiener Hopf factorization, which allows to control the fluctuation of a 1-dimensionnal random walk between IR− and IR∗+ (see [2] and [5] for more details and references about fluctuations of random walks). A spectacular progress has been made recently by D. Denisov and V. Watchel ([1]) who consider the general situation of a cone in any dimension; their approach is quite different from the previous ones and is based on a construction of an harmonic function for the random walk staying in the cone and an approximation of random walks by the Brownian motion (with a control of the quality of this normal approximation). In this report, we will (try) read the paper by D. Denisov and V. Watchel and to detail the different steps of their proof. Contacts. [email protected], [email protected]

References [1] Denisov D & Watchel V. Random walks in cone, arxiv.org/pdf/1110.1254. [2] Feller W. An introduction to Probability Theory and Its Applications, Vol. II, J. Wiley, (1970). [3] Iglehart D.L. Random walks with negative drift conditionned to stay positive J. Ap.. Prob. 1974, vol. 11, pp 742-751 [4] Kozlov,M.V. On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment J. Th. Prob. 1976, vol. 21, n. 4, pp 791-804 [5] Spitzer F. Principles of random walks , Springer, (1964)