A finitary approach for the representation of the infinitesimal generator of a markovian semigroup

Abstract

This work is based on Nelson’s paper [1], where the central question was: under suitable regularity conditions, what is the form of the infinitesimal generator of a Markov semigroup?

In the elementary approach using IST [2]. the idea is to replace the continuous state space, such as ℝ with a finite state space X possibly containing an unlimited number of points. The topology on X arises naturally from the probability theory. For x ε X, let $\mathcal{I}_x$ be the set of all h ∈ $\mathcal{M}$ vanishing at x where $\mathcal{M}$ is the multiplier algebra of the domain $\mathcal{D}$ of the infinitesimal generator. To describe the structure of the semigroup generator A, we want to split Ah(x)=∑ _y∈X\{x} a(x,y) h(y) so that the contribution of the external set F _x of the points far from x appears separately. A definition of the quantity α _ah(x)=∑ _y∈F a(x,y) h(y) is given using the least upper bound of the sums on all internal sets W included in the external set F. This leads to the characterization of the global part of the infinitesimal generator.