Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process (LD-QBD)

2023/02/04 by Małgorzata M. O’Reilly, O'Reilly, Małgorzata M., Zbigniew Palmowski +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2302.02225

openalex publication_date 2023/02/04 · openalex created_date 2023/02/09 · openalex updated_date 2026/07/28

Abstract

We consider a neighbourhood random walk on a quadrant, \(X₁(t),X₂(t),φ(t)):t≥ 0\, with state space Samp;=amp;\(n,m,i):n,m=0,1,2,…;i=1,2,…,k(n,m)\. Assuming start in state (n,m,i), the process spends exponentially distributed amount of time in (n,m,i) according to some parameter λ_i^(n,m). Upon leaving state (n,m,i) the process moves to some state (n^',m^',j) with j∈\1,…,k(n^',m^')\ and n^'∈\n-1,n,n+1\, m^'∈\m-1,m,m+1\, according to some probabilities (p_n;a^m;b)_i,j with a,b∈\+,-,0\. We transform this process into a one-dimensional LD-QBD \(Z(t),χ(t)):t≥ 0\ with level variable Z(t) and phase variable χ(t). Using this transform we find its transient and stationary analysis using matrix-analytic methods, as well as the distribution at first hitting times.

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