Characterizations of random walks on random lattices and their ramifications. Issue 2 (3rd March 2020)

• Record Type: Journal Article
• Authors: White, Ryan T.; Dshalalow, Jewgeni H.
• Is Part Of: Stochastic analysis and applications. Volume 38:Issue 2(2020)
• Page Start: 307
• Page End: 342
• Publication Date: 2020-03-03
• Subjects: Random walk -- doubly-stochastic random walk -- random lattices -- independent and stationary increments processes -- fluctuations of stochastic processes -- marked point processes -- Poisson process -- exit time -- first passage time -- antagonistic games -- queueing -- queues with vacationing server -- N-Policy; 60G50 -- 60G51 -- 60G52 -- 60G55 -- 60G57 -- 60K05 -- 60K35 -- 60K40 -- 60G25 -- 90B18 -- 90B10 -- 90B15 -- 90B25; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2019.1694417
• Languages: English
• ISSNs: 0736-2994

Abstract

Abstract: We analyze the behavior of a particle moving along a d -dimensional lattice and representing a "doubly stochastic" random walk. Unlike the classical random walk, the lattice is randomly generated upon particle's landing at a node. At any time, the particle is enclosed in a rectangular cylinder R = R a × R b (i.e., with a bounded a -dimensional rectangle Ra at its base) it attempts to escape. Thus, the particle moves from one node to another at random epochs of time and, with some probability, leaves R . The particle may jump arbitrarily far beyond the boundary ∂ R at its passage time. Furthermore, the particle's location is not assumed to be known in real time, but only upon certain random epochs { τ n } . Of a key interest is the location A ( t ) of the particle at any time, where A ( t ) is the linear interpolation of particle's positions at times τn 's. If τ ρ denotes the first observed escape time (virtual first passage time), where ρ = min { n : A ( τ n ) ∈ R C }, we target the joint characteristic function of A ( τ ρ − 1 ), A ( τ ρ ), τ ρ − 1, τ ρ, and A ( t ) itself, where t ∈ [ 0, τ ρ − 1 ] or ( τ ρ − 1, τ ρ ] in tractable forms, thereby attempting to enhance as much as possible the probabilistic data lost due to the crudeness of the observations and to couple A ( τ ρ − 1 ), A ( τ ρ ), τ ρ − 1, and τ ρ with deterministic time intervals [ 0, t ] . Among various applications, we discuss and treat antagonistic games of two active and several passive players as well as situations that occur in complex queueing systems.