Characterizations of random walks on random lattices and their ramifications

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 $\mathcal{R}=R_a\times {\mathbb{R}}^b$ (i.e., with a bounded a-dimensional rectangle R_a 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 $\mathcal{R}.$ The particle may jump arbitrarily far beyond the boundary $\partial \mathcal{R}$ at its passage time. Furthermore, the particle’s location is not assumed to be known in real time, but only upon certain random epochs $\left\{{\tau }_n\right\}.$ Of a key interest is the location $\mathbf{A}(t)$ of the particle at any time, where $\mathbf{A}(t)$ is the linear interpolation of particle’s positions at times τ_n’s. If ${\tau }_{\rho }$ denotes the first observed escape time (virtual first passage time), where $\rho =\min \left\{n:\mathbf{A}({\tau }_n)\in {\mathcal{R}}^C\right\},$ we target the joint characteristic function of $\mathbf{A}({\tau }_{\rho -1}),\mathbf{A}({\tau }_{\rho }),{\tau }_{\rho -1},{\tau }_{\rho },$ and $\mathbf{A}(t)$ itself, where $t\in [0,{\tau }_{\rho -1}]$ or $({\tau }_{\rho -1},{\tau }_{\rho }]$ 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 $\mathbf{A}({\tau }_{\rho -1}),\mathbf{A}({\tau }_{\rho }),{\tau }_{\rho -1},$ and ${\tau }_{\rho }$ with deterministic time intervals $\left[0,t\right].$ 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.

Keywords:

• 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

AMS CLASSIFICATION:

• 60G50
• 60G51
• 60G52
• 60G55
• 60G57
• 60K05
• 60K35
• 60K40
• 60G25
• 90B18
• 90B10
• 90B15
• 90B25