Transition probabilities for symmetric jump processes (Q2782670)

For \(x,y\in \mathbb{Z}^d\) with \(x\neq y\), let \(C_{xy}\) be positive finite numbers such that \(C_{xy}= C_{yx}\) for all \(x\), \(y\) and \(\sum_z C_{xz}\leq \infty\) for all \(x\). Set \(C_{xz}\neq 0\) for all \(x\). Define a symmetric Markov chain by \(P(X_1= y\mid X_0= x)= C_{zy}/\sum_z C_{xz}, x,y\in \mathbb{Z}^d\). Set \(p(n,x,y)= P^x(X_n= y)\). The authors assume that the index \(\alpha\) of a symmetric stable process belongs to \((0,2)\) and there exists \(\kappa\geq 1\) such that for all \(x= y\), \(\kappa^{-1}/|x-y|^{d+\alpha}\leq C_{xy}\leq \kappa/|x-y|^{d+\alpha}\). Their main result is the following theorem: There exist positive finite constants \(c_1\) and \(c_2\) such that \(p(n,x,y)\leq c_1(n^{-d/\alpha} \Lambda(n/|x-y|^{d+\alpha}))\), and for \(n\geq 2\), \(p(n,x,y)\geq c_2(n^{-d/\alpha} \Lambda(n/|x-y|^{d+\alpha}))\). The latter inequality holds for \(n=1\), as well, provided that \(x\neq y\).