On the spectrum and Martin boundary of homogeneous spaces (Q1347192)

Let \(({\mathcal X}, d)\) be a connected separable metric space with isometry group \(\Gamma\) acting transitively on \({\mathcal X}\) and \(X = \{X_ t : t \geq 0\}\) be a conservative Markov process on \({\mathcal X}\) with the corresponding semigroup \(\{P_ t : t \geq 0\}\). The process \(X\) is assumed to be spatially homogeneous in the sense that for all \(t \geq 0\), \(g \in \Gamma\) and any continuous real function \(F\) on \({\mathcal X}\) we have \(g \cdot P_ t F = P_ t (g \cdot F)\), where \(g \cdot G(x) = G(g^{- 1}x)\), \(x \in {\mathcal X}\), by definition. Let \(\Delta\) be the generator of \(X\) and \({\mathcal D} (\Delta)\) denote the domain of \(\Delta\). Put \[ \lambda_{\mathcal X} = \sup \bigl\{ \lambda : \exists f \in {\mathcal D} (\Delta) : f > 0, \;\Delta f + \lambda f \leq 0 \bigr\}. \] The number \(\lambda_{\mathcal X}\) is called the bottom of the spectrum of the operator \(\Delta\). Under some additional conditions on the process \(X\) and the group \(\Gamma\) the properties of \(\lambda_{\mathcal X}\) are investigated. The main result is: if \(\lambda_{\mathcal X} = 0\), then there exists a unique point in the Martin boundary fixed by \(\Gamma\) (i.e. \(g \xi = \xi\) for all \(g \in \Gamma\) where \(\xi\) is the Martin boundary of \({\mathcal X})\).