In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.
Let E be a locally compact, separable, metric space. We denote by E the Borel subsets of E . Let Ω be the space of right continuous maps from [ 0 , ∞ ) to E that have left limits in E , and for each t ∈ [ 0 , ∞ ) , denote by X t the coordinate map at t ; for each ω ∈ Ω , X t ( ω ) ∈ E is the value of ω at t . We denote the universal completion of E by E ∗ . For each t ∈ [ 0 , ∞ ) , let
F t = σ { X s − 1 ( B ) : s ∈ [ 0 , t ] , B ∈ E } , F t ∗ = σ { X s − 1 ( B ) : s ∈ [ 0 , t ] , B ∈ E ∗ } , and then, let
F ∞ = σ { X s − 1 ( B ) : s ∈ [ 0 , ∞ ) , B ∈ E } , F ∞ ∗ = σ { X s − 1 ( B ) : s ∈ [ 0 , ∞ ) , B ∈ E ∗ } . For each Borel measurable function f on E , define, for each x ∈ E ,
U α f ( x ) = E x [ ∫ 0 ∞ e − α t f ( X t ) d t ] . Since P t f ( x ) = E x [ f ( X t ) ] and the mapping given by t → X t is right continuous, we see that for any uniformly continuous function f , we have the mapping given by t → P t f ( x ) is right continuous.
Therefore, together with the monotone class theorem, for any universally measurable function f , the mapping given by ( t , x ) → P t f ( x ) , is jointly measurable, that is, B ( [ 0 , ∞ ) ) ⊗ E ∗ measurable, and subsequently, the mapping is also ( B ( [ 0 , ∞ ) ) ⊗ E ∗ ) λ ⊗ μ -measurable for all finite measures λ on B ( [ 0 , ∞ ) ) and μ on E ∗ . Here, ( B ( [ 0 , ∞ ) ) ⊗ E ∗ ) λ ⊗ μ is the completion of B ( [ 0 , ∞ ) ) ⊗ E ∗ with respect to the product measure λ ⊗ μ . Thus, for any bounded universally measurable function f on E , the mapping t → P t f ( x ) is Lebeague measurable, and hence, for each α ∈ [ 0 , ∞ ) , one can define
U α f ( x ) = ∫ 0 ∞ e − α t P t f ( x ) d t . There is enough joint measurability to check that { U α : α ∈ ( 0 , ∞ ) } is a Markov resolvent on ( E , E ∗ ) , which uniquely associated with the Markovian semigroup { P t : t ∈ [ 0 , ∞ ) } . Consequently, one may apply Fubini's theorem to see that
U α f ( x ) = E x [ ∫ 0 ∞ e − α t f ( X t ) d t ] . The followings are the defining properties of Borel right processes:
Hypothesis Droite 1:For each probability measure
μ on
( E , E ) , there exists a probability measure
P μ on
( Ω , F ∗ ) such that
( X t , F t ∗ , P μ ) is a Markov process with initial measure
μ and transition semigroup
{ P t : t ∈ [ 0 , ∞ ) } .
Hypothesis Droite 2:Let
f be
α -excessive for the resolvent on
( E , E ∗ ) . Then, for each probability measure
μ on
( E , E ) , a mapping given by
t → f ( X t ) is
P μ almost surely right continuous on
[ 0 , ∞ ) .
