On a Generalization of One-Dimensional Kinetics

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

On uniqueness of invariant measures for random walks on

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$ . In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $ . C. R. Acad. Sci. Paris250 (1960), 799–801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$ . Our work can be viewed as following on from Babillot et al [The random difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case. Ann. Probab.25(1) (1997), 478–493] and Deroin et al [Symmetric random walk on $\mathrm {HOMEO}^{+}(\mathbb {R})$ . Ann. Probab.41(3B) (2013), 2066–2089].

Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $$\varepsilon _n$$ ε n such that $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ . This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

Preferential attachment when stable

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha$th power $(\alpha >1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower-tail large deviation principle (LDP) for $L = \sum_{i=1}^{n} ({S_i^2}/{i^2})$, where $\{S_n \colon n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

Symmetric Random Walks on Regular Tetrahedra, Octahedra, and Hexahedra

We study a symmetric random walk on the vertices of three regular polyhedra. Starting from the origin, at each step the random walk moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the distributions, or at least the means and the standard deviations, of the number of steps needed (a) to return to origin, (b) to visit all vertices, and (c) to return to origin after visiting all vertices. We also find the distributions of (i) the number of vertices visited before return to origin, (ii) the last vertex visited, and (iii) the number of vertices visited during return to origin after visiting all vertices.