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 $\backslash mathcal\; E$ the Borel subsets of $E$.

Let $\backslash Omega$ be the space of right continuous maps from $[0,\backslash infty)$ to $E$ that have left limits in $E$,

and for each $t\; \backslash in\; [0,\backslash infty)$, denote by $X\_t$ the coordinate map at $t$; for

each $\backslash omega\; \backslash in\; \backslash Omega$, $X\_t(\backslash omega)\; \backslash in\; E$ is the value of $\backslash omega$ at $t$.

We denote the universal completion of $\backslash mathcal\; E$ by $\backslash mathcal\; E^*$.

For each $t\backslash in[0,\backslash infty)$, 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 $x\; \backslash in\; E$,

: $$

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

Since $P\_tf(x)\; =\; \backslash mathbf\; E^x\backslash left[f(X\_t)\backslash right]$ and the mapping given by $t\; \backslash rightarrow\; X\_t$ is right continuous, we see that

for any uniformly continuous function $f$, we have the mapping given by $t\; \backslash rightarrow\; P\_tf(x)$ is right continuous.

Therefore, together with the monotone class theorem, for any universally measurable function $f$, the mapping given by $(t,x)\; \backslash rightarrow\; P\_tf(x)$, is jointly measurable, that is, $\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^*$ measurable, and subsequently, the mapping is also $\backslash left(\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^*\backslash right)^\{\backslash lambda\backslash otimes\; \backslash mu\}$-measurable for all finite measures $\backslash lambda$ on $\backslash mathcal\; B([0,\backslash infty))$ and $\backslash mu$ on $\backslash mathcal\; E^*$.

Here,

$\backslash left(\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^*\backslash right)^\{\backslash lambda\backslash otimes\; \backslash mu\}$ is the completion of

$\backslash mathcal\; B([0,\backslash infty))\backslash otimes\; \backslash mathcal\; E^*$ with respect

to the product measure $\backslash lambda\; \backslash otimes\; \backslash mu$.

Thus, for any bounded universally measurable function $f$ on $E$,

the mapping $t\backslash rightarrow\; P\_tf(x)$ is Lebeague measurable, and hence,

for each $\backslash alpha\; \backslash in\; [0,\backslash infty)$, 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 $\backslash \{U^\backslash alpha\; :\; \backslash alpha\; \backslash in\; (0,\backslash infty)$

\} is a Markov resolvent on $(E,\backslash mathcal\; E^*)$,

which uniquely associated with the Markovian semigroup $\backslash \{\; P\_t\; :\; t\; \backslash in\; [0,\backslash infty)\; \backslash \}$.

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:{{harvnb|Sharpe|1988|loc=Sect. 20}}

• Hypothesis Droite 1:

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

• Hypothesis Droite 2:

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