Martin boundary of polynomial random walks on the \(d\)-dimensional lattice of nonnegative integers (Q1827464)

Random walks on \(\mathbb{N}^d\), viewed as a \(d\)-dimensional (product) hypergroup, are homogeneous Markov chains with transition kernels of the form \(T_{\mu} = \sum_{n \in \mathbb{N}^d} \mu(n) T_n\) where \(T_n\) are the hypergroup translations and \(\mu\) is a probability measure on \(\mathbb{N}^d\) (``the law of the walk''), see \textit{L. Gallardo} [Stochastics Stochastics Rep. 73, 1--23 (2002; Zbl 1007.60007)] and \textit{L. Gallardo} and \textit{O. Gebuhrer} [Expo. Math. 5, 41--73 (1987; Zbl 0618.60014)]. The hypergroup structure is based on \(d\) families \(\{P_i^k(x)\}_{k=1}^{\infty}\), \(1\leq i \leq d\}\), of polynomials in \(x \in \mathbb{R}\), each orthogonal w.r.t. a due symmetric measure supported in \([-\beta_i,\beta_i] \subset [-1,1]\). Assuming that \(\mu\) has finite support and supports of its convolution powers cover \(\mathbb{N}^d\), the paper establishes, for \(r \in \mathbb{R}^+\), that \[ r \geq \widehat{\mu}(\beta_1,\ldots,\beta_d) \equiv \sum_{n_1,\ldots,n_d} P_{n_1}^1(\beta_1)\ldots P_{n_d}^d(\beta_d)\tag{1} \] \(\Leftrightarrow\) \[ T_{\mu} h = rh \text{ with a nonzero function } h \geq 0\text{ on }\mathbb{N}^d.\tag{2} \] Moreover, the set \(\mathcal{C}_{\mu}^r\) of positive \(r\)-eigenvectors of \(T_{\mu}\) is an ordered cone and a one-to-one correspondence between \(\mathcal{C}_{\mu}^r\) and the set of positive finite Borel measures \(\nu\) on \(\mathcal{S}_{\mu}^r = \{(x_1,\ldots,x_d) \in [\beta_1,+\infty)\times \ldots \times [\beta_d,+\infty)\), \(\widehat{\mu}(x_1,\ldots,x_d) = r\}\) (with Euclidean topology) is given by the Martin representation \(h(n_1,\ldots,n_d) = \int_{\mathcal{S}_{\mu}^r} P_{n_1}^1(x_1)\ldots P_{n_d}^d(x_d) \,d\nu(x_1,\ldots,x_d)\). Some applications are considered as well as an example of the random walk to the nearest neighbour with reflecting barriers at coordinate hyperplanes, cf. \textit{I. A. Kurkova} and \textit{V. A. Malyshev} [Markov Process. Relat. Fields 4, 203--272 (1988; Zbl 0929.60055)], on \(2\)-dimensional walks without reflection conditions.