Decompositions

Abstract

Note first that with the energy e, we can associate a time-rescaled Markov chain $\widehat{x}_{t}$ in which holding times at any point x are exponential times of parameters $\lambda_{x}:\,\widehat{x}_{t}=x_{\tau_{t}}\,{\rm{with}}\, \tau_{t}=\inf(s,\;\int_{0}^{s}\frac{1}{\lambda_{x_{u}}}du=t).$ For the time-rescaled Markov chain, local times coincide with the time spent in a point and the duality measure is simply the counting measure. The potential operator then essentially coincides with the Green function. The Markov loops can be time-rescaled as well and we did it in fact already when we introduced pointedloops. More generally we may introduce different holding time parameters but it would be rather useless as the randomvariables we are interested in are intrinsic, i.e. depend only on e.