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 Lebesgue 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).