Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Figure 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j)\in Z_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\to\infty$. The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $Z$, four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of its neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.Comment: 11 pages, 1 figur

Tails of passage-times and an application to stochastic processes with boundary reflection in wedges

In this paper we obtain lower bounds for the tails of the distributions of the first passage-times for some stochastic processes. We consider first discrete parameter processes with asymptotically small drifts taking values in + and prove for them a general result giving lower bounds for these tails. As an application of the obtained results, we obtain lower bounds for the tails of the distributions of the first passage-times for reflected random walks in a quadrant with zero-drift in the interior. The latter bounds are then used to get explicit conditions for the finiteness or not of the moments of the first passage-time to the origin for a Brownian motion with oblique reflection in a wedge.Passage-times Recurrence classification Markov chain with boundary reflection Reflected Brownian motion

On Range and Local Time of Many-dimensional Submartingales

We consider a discrete-time process adapted to some filtration which lives on a (typically countable) subset of $\mathbb{R}^d$, $d\geq 2$. For this process, we assume that it has uniformly bounded jumps, is uniformly elliptic (can advance by at least some fixed amount with respect to any direction, with uniformly positive probability). Also, we assume that the projection of this process on some fixed vector is a submartingale, and that a stronger additional condition on the direction of the drift holds (this condition does not exclude that the drift could be equal to 0 or be arbitrarily small). The main result is that with very high probability the number of visits to any fixed site by time $n$ is less than $n^{1/2-\delta}$ for some $\delta>0$. This in its turn implies that the number of different sites visited by the process by time $n$ should be at least $n^{1/2+\delta}$.Comment: 23 pages, 8 figures; to appear in Journal of Theoretical Probabilit

Heavy-tailed random walks on complexes of half-lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution μk. If χk is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and αk is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all αkχk∈(0,1) is determined by the sign of ∑kμkcot(χkπαk). In the case of two half-lines, the model fits naturally on R and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in α1 and α2; our general setting exhibits the essential nonlinearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on R with symmetric increments of tail exponent α∈(1,2)