Almost sure estimates for the concentration neighborhood of Sinai's walk

We consider Sinai's random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is almost surely bounded. As an application we get that i.o. the maximal distance between two favorite sites is almost surely bounded

The local time of a random walk on growing hypercubes

We study a random walk in a random environment (RWRE) on $\Z^d$, $1 \leq d < +\infty$. The main assumptions are that conditionned on the environment the random walk is reversible. Moreover we construct our environment in such a way that the walk can't be trapped on a single point like in some particular RWRE but in some specific d-1 surfaces. These surfaces are basic surfaces with deterministic geometry. We prove that the local time in the neighborhood of these surfaces is driven by a function of the (random) reversible measure. As an application we get the limit law of the local time as a process on these surfaces.Comment: 24 page

Time Travel and the Immutability of the Past within B-Theoretical Models

The goal of this paper is to defend the general tenet that time travelers cannot change the past within B-theoretical models of time, independently of how many temporal dimensions there are. Baron Pacific Philosophical Quarterly, 98, 129–147 offered a strong argument intended to reach this general conclusion. However, his argument does not cover a peculiar case, i.e. a B-theoretical one-dimensional model of time that allows for the presence of internal times. Loss Pacific Philosophical Quarterly, 96, 1–11 used the latter model to argue that time travelers can change the past within such model. We show a way to debunk Loss’s argument, so that the general tenet about the impossibility of changing the past within B-theoretical models is maintained

Limit law of the local time for Brox's diffusion

We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \in R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi

Renewal structure and local time for diffusions in random environment

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with 0\textless{}\kappa\textless{}1, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time t\textgreater{}0.In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable L{\'e}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.Comment: 61 page

A limit result for a system of particles in random environment

We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time $t$ and the starting point of the particles.Comment: 11 page

Back to the (Branching) Future

The future is different from the past. What is past is fixed and set in stone. The future, on the other hand, is open insofar as it holds numerous possibilities. Branching-tree models of time account for this asymmetry by positing an ontological difference between the past and the future. Given a time t, a unique unified past lies behind t, whereas multiple alternative existing futures lie ahead of t. My goal in this paper is to show that there is an incompatibility between the way branching-tree models account for the open future and the possibility of time travel. That is, I argue that once time travel enters the picture, branching time fails to model the openness of the future by means of alternative future branches. I show how this holds independently of whether branching-time models are cashed out in A-theoretic or B-theoretic terms

On the concentration of Sinai's walk

We consider Sinai's random walk in random environment. We prove that for an interval of time [1,n] Sinai's walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by $n$ converges in probability to a well defined random variable of the environment