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

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

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

and then, let

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

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

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

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

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

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

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

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

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

The following are the defining properties of Borel right processes:

• Hypothesis Droite 1:

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

• Hypothesis Droite 2:

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

References

• Sharpe, Michael (1988), General Theory of Markov Processes, ISBN 0126390606