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 $$E$$ be a locally compact, separable, metric space. We denote by $$\mathcal E$$ the Borel subsets of $$E$$. Let $$\Omega$$ be the space of right continuous maps from $$[0,\infty)$$ to $$E$$ that have left limits in $$E$$, and for each $$t \in [0,\infty)$$, denote by $$X_t$$ the coordinate map at $$t$$; for each $$\omega \in \Omega $$, $$X_t(\omega) \in E$$ is the value of $$\omega$$ at $$t$$. We denote the universal completion of $$\mathcal E$$ by $$\mathcal E^*$$. For each $$t\in[0,\infty)$$, let



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



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

and then, let



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



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

For each Borel measurable function $$ f $$ on $$ E$$, define, for each $$x \in E$$,



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

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

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



U^\alpha f(x) = \int_0^\infty e^{-\alpha t}P_tf(x) dt. $$

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



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

The following are the defining properties of Borel right processes:


 *  Hypothesis Droite 1:


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


 *  Hypothesis Droite 2:


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