Borel right process

In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let [math]\displaystyle{ E }[/math] be a locally compact, separable, metric space. We denote by [math]\displaystyle{ \mathcal E }[/math] the Borel subsets of [math]\displaystyle{ E }[/math]. Let [math]\displaystyle{ \Omega }[/math] be the space of right continuous maps from [math]\displaystyle{ [0,\infty) }[/math] to [math]\displaystyle{ E }[/math] that have left limits in [math]\displaystyle{ E }[/math], and for each [math]\displaystyle{ t \in [0,\infty) }[/math], denote by [math]\displaystyle{ X_t }[/math] the coordinate map at [math]\displaystyle{ t }[/math]; for each [math]\displaystyle{ \omega \in \Omega }[/math], [math]\displaystyle{ X_t(\omega) \in E }[/math] is the value of [math]\displaystyle{ \omega }[/math] at [math]\displaystyle{ t }[/math]. We denote the universal completion of [math]\displaystyle{ \mathcal E }[/math] by [math]\displaystyle{ \mathcal E^* }[/math]. For each [math]\displaystyle{ t\in[0,\infty) }[/math], let

[math]\displaystyle{ \mathcal F_t = \sigma\left\{ X_s^{-1}(B) : s\in[0,t], B \in \mathcal E\right\}, }[/math]

[math]\displaystyle{ \mathcal F_t^* = \sigma\left\{ X_s^{-1}(B) : s\in[0,t], B \in \mathcal E^*\right\}, }[/math]

and then, let

[math]\displaystyle{ \mathcal F_\infty = \sigma\left\{ X_s^{-1}(B) : s\in[0,\infty), B \in \mathcal E\right\}, }[/math]

[math]\displaystyle{ \mathcal F_\infty^* = \sigma\left\{ X_s^{-1}(B) : s\in[0,\infty), B \in \mathcal E^*\right\}. }[/math]

For each Borel measurable function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ E }[/math], define, for each [math]\displaystyle{ x \in E }[/math],

[math]\displaystyle{ U^\alpha f(x) = \mathbf E^x\left[ \int_0^\infty e^{-\alpha t} f(X_t)\, dt \right]. }[/math]

Since [math]\displaystyle{ P_tf(x) = \mathbf E^x\left[f(X_t)\right] }[/math] and the mapping given by [math]\displaystyle{ t \rightarrow X_t }[/math] is right continuous, we see that for any uniformly continuous function [math]\displaystyle{ f }[/math], we have the mapping given by [math]\displaystyle{ t \rightarrow P_tf(x) }[/math] is right continuous.

Therefore, together with the monotone class theorem, for any universally measurable function [math]\displaystyle{ f }[/math], the mapping given by [math]\displaystyle{ (t,x) \rightarrow P_tf(x) }[/math], is jointly measurable, that is, [math]\displaystyle{ \mathcal B([0,\infty))\otimes \mathcal E^* }[/math] measurable, and subsequently, the mapping is also [math]\displaystyle{ \left(\mathcal B([0,\infty))\otimes \mathcal E^*\right)^{\lambda\otimes \mu} }[/math]-measurable for all finite measures [math]\displaystyle{ \lambda }[/math] on [math]\displaystyle{ \mathcal B([0,\infty)) }[/math] and [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ \mathcal E^* }[/math]. Here, [math]\displaystyle{ \left(\mathcal B([0,\infty))\otimes \mathcal E^*\right)^{\lambda\otimes \mu} }[/math] is the completion of [math]\displaystyle{ \mathcal B([0,\infty))\otimes \mathcal E^* }[/math] with respect to the product measure [math]\displaystyle{ \lambda \otimes \mu }[/math]. Thus, for any bounded universally measurable function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ E }[/math], the mapping [math]\displaystyle{ t\rightarrow P_tf(x) }[/math] is Lebeague measurable, and hence, for each [math]\displaystyle{ \alpha \in [0,\infty) }[/math], one can define

[math]\displaystyle{ U^\alpha f(x) = \int_0^\infty e^{-\alpha t}P_tf(x) dt. }[/math]

There is enough joint measurability to check that [math]\displaystyle{ \{U^\alpha : \alpha \in (0,\infty) \} }[/math] is a Markov resolvent on [math]\displaystyle{ (E,\mathcal E^*) }[/math], which uniquely associated with the Markovian semigroup [math]\displaystyle{ \{ P_t : t \in [0,\infty) \} }[/math]. Consequently, one may apply Fubini's theorem to see that

[math]\displaystyle{ U^\alpha f(x) = \mathbf E^x\left[ \int_0^\infty e^{-\alpha t} f(X_t) dt \right]. }[/math]

The following are the defining properties of Borel right processes:

• Hypothesis Droite 1:

For each probability measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (E, \mathcal E) }[/math], there exists a probability measure [math]\displaystyle{ \mathbf P^\mu }[/math] on [math]\displaystyle{ (\Omega, \mathcal F^*) }[/math] such that [math]\displaystyle{ (X_t, \mathcal F_t^*, P^\mu) }[/math] is a Markov process with initial measure [math]\displaystyle{ \mu }[/math] and transition semigroup [math]\displaystyle{ \{ P_t : t \in [0,\infty) \} }[/math].

• Hypothesis Droite 2:

Let [math]\displaystyle{ f }[/math] be [math]\displaystyle{ \alpha }[/math]-excessive for the resolvent on [math]\displaystyle{ (E, \mathcal E^*) }[/math]. Then, for each probability measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (E,\mathcal E) }[/math], a mapping given by [math]\displaystyle{ t \rightarrow f(X_t) }[/math] is [math]\displaystyle{ P^\mu }[/math] almost surely right continuous on [math]\displaystyle{ [0,\infty) }[/math].

References

• Sharpe, Michael (1988), General Theory of Markov Processes, ISBN 0126390606