A Representation for Non-Colliding Random Walks

A Representation for Non-colliding Random Walks

Let D0(R+ ) denote the space of cadlag paths f : R+ → R with f(0) = 0. For f, g ∈ D0(R+ ), define f ⊗ g ∈ D0(R+ ) and f g ∈ D0(R+ ) by (f ⊗ g)(t) = inf 0≤s≤t [f(s) + g(t) − g(s)], and (f g)(t) = sup 0≤s≤t [f(s) + g(t) − g(s)]. Unless otherwise deleniated by parentheses, the default order of operations is from left to right; for example, when we write f ⊗ g ⊗ h, we mean (f ⊗ g) ⊗ h. Define a seq...

Random walks colliding before getting trapped

Let P be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain X has transition matrix P and speed λ if it jumps at rate λ according to the matrix P . Fix λX , λY , λZ ≥ 0, then let X,Y and Z be independent Markov chains with transition matrix P and speeds λX , λY and λZ respectively, all started from the stationary distribution. Wha...

Dependent percolation and colliding random walks

Large Deviations for Random Walks in a Mixing Random Environment and Other (Non-Markov) Random Walks

We extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. c © 2004 Wiley Periodicals, Inc.

Non - Colliding Random Walks , Tandem Queues , and Discrete Orthogonal Polynomial Ensembles

We show that the function h(x) = ∏ i<j(xj − xi) is harmonic for any random walk in Rk with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = {x : x1 < x2 < · · · < xk} onto a point where h vanishes, we define the corresponding Doob h-transform. For certain special cases, we show that the marginal distribution ...