Current Trends in Random Walks on Random Lattices

Abstract

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

Dependent Competing Failure Processes in Reliability Systems

“…The convergence of the series is due to ∥γ(xyu, zvw, θ + ϑ)∥ < 1 as established in [36]. Finally, we arrive at Formula (32), proving that…”

“…Lastly, Formula (31) for Φ ρ (s, y, z, θ, u, v, ϑ; M, N) = Es ρ y A ρ z B ρ u A ρ−1 v B ρ−1 e −θτ ρ −ϑτ ρ−1 follows from summing up expressions (32)- (34) as per (44) and a straightforward algebra. Formulas (35) and (36) are also subject to the summation of the pairs of (32), ( 34) and ( 33), (34), respectively.…”

Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks

“…In a more recent work, random walk is considered on a variety of configurations, more general than rectangular, cf. Dshalalow and White [27]. Now the random walk modeling is not just an interpretation, but it suggested fluctuation analysis as a primary tool for dealing with this class of models, with the benefit of obtaining a closed form distribution of targeted parameters such as the failure time of the system and the total damage to the system at the failure time.…”

“…We earlier worked on multidimensional walks (cf. Dshalalow and White [27]), only this time the walker's movements are not rectangular. The present paper points to ways of their modifications.…”

“…In our present model, the walker moves along a randomly generated grid that is not even rectangular. (See more on different variants of random walks in Dshalalow and White [27].) Here, the walker moves in R 2 within a random rectangular set B = (0, τ ν ] × (0, M] ⊆ R 2 + and it escapes at the random point (τ n u, M) or (t ν , A n u).…”