Consider a discrete time nonhomogeneous random walk \((X_n)_{n \in\mathbb{N}}\) (\(X_n \in \mathbb{N}\cup\{0\}\)) with right elastic barrier at 0 and probabilities of jumps \(p_{i,j}=0\) if \(|i-j|>1\), \(p_{i,i-1}=q_i\), \(p_{i,i}=r_i\), \(p_{i,i+1}=p_i\), \(p_i+q_i+r_i=1\), \(i \geq 1\), \(p_{0,0}=r_0\), \(p_{0,1}=p_0=1-r_0\) where \(p_i>0\), \(r_i>0\), \(q_{i+1}>0\), \(i \geq 0\). In this article, representations of the random walks by directed circles and weights of the corresponding Markov circuit chains are investigated. The directed circles are of the form \(c_k=(k,k+1,k)\), \(c_{k-1}=(k-1,k,k-1)\), \(c'_k=(k,k)\) and the weights \(w_c\) are defined so that the equalities \[ p_{i,j}=\frac{\sum_{c} w_c \times J_c^{(1)}(i,j)}{\sum_{c} w_c \times J_c(i)} \] hold true for all \(i \geq 0\), \(j \geq 0\). Here \(J_c^{(1)}(i,j)=1\) if \(i=c(m)\) and \(j=c(m+1)\) for some \(m \in \mathbb{N}\) and \(J_c^{(1)}(i,j)=0\) otherwise; \(J_c(i)=1\) if \(i=c(m)\) for some \(m\) and \(J_c(i)=0\) otherwise. The unique representations of given and adjoint random walks by Markov circuit chains are established and criteria of recurrence and transience of these circuit chains are obtained. The results are generalized for the case of random walks in random environments.
For the entire collection see [Zbl 1432.65003].