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In the mathematical theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process.

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

${\displaystyle {\mathcal{ℱ}}_t=\sigma \{X_s^{-1}(B):s\in [0,t],B\in \mathcal{ℰ}\},}$

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

and then, let

${\displaystyle {\mathcal{ℱ}}_{\mathrm{\infty }}=\sigma \{X_s^{-1}(B):s\in [0,\mathrm{\infty }),B\in \mathcal{ℰ}\},}$

${\displaystyle {\mathcal{ℱ}}_{\mathrm{\infty }}^{*}=\sigma \{X_s^{-1}(B):s\in [0,\mathrm{\infty }),B\in {\mathcal{ℰ}}^{*}\}.}$

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

${\displaystyle U^{\alpha }f(x)={\mathbf{E}}^x[{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{e}^{-\alpha t}f(X_t)dt].}$

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

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

${\displaystyle U^{\alpha }f(x)={\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{e}^{-\alpha t}{P}_{t}f(x)dt.}$

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

${\displaystyle U^{\alpha }f(x)={\mathbf{E}}^x[{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}{e}^{-\alpha t}f(X_t)dt].}$

The followings are the defining properties of Borel right processes:

• Hypothesis Droite 1:

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

• Hypothesis Droite 2:

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