We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let @math and @math be two centered, weakly dependent random walks. We establish that @math for a non-random @math . In the classical setting, @math , it is well-known that @math . We prove that for any non-trivial @math one has @math and the exponent @math depends only on @math . Our result holds also in the continuous setting, when @math and @math are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.
This perspective was the initial motivation for analyzing the problems in this paper (more precisely the result given in Theorem ). In fact, the question arises from studies of extremal particles of a branching random walk in a time-inhomogeneous random environment. In the companion paper @cite_1 we show that the randomness of the environment has a slowing effect on the position of the maximal particle. Namely, the logarithmic correction to the speed is bigger than in the standard (time-homogenous) case, which is a consequence of ).
@cite_1
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In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations. Abstract The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form V n − φ log n + o P ( log n ) , where V n is a function of the environment that behaves as a random walk and φ > 0 is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk. Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length n of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time n. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order n1 3, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement. A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.