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

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.