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

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

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

and then, let

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

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

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

${\displaystyle U^{\alpha }f(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}f(X_{t})\,dt\right].}$

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

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

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

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

${\displaystyle U^{\alpha }f(x)=\mathbf {E} ^{x}\left[\int _{0}^{\infty }e^{-\alpha t}f(X_{t})dt\right].}$

The following are the defining properties of Borel right processes:[1]

• Hypothesis Droite 1:

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

• 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 ${\displaystyle t\rightarrow f(X_{t})}$ is ${\displaystyle P^{\mu }}$ almost surely right continuous on ${\displaystyle [0,\infty )}$.