Corollary 17.13.42.F147312B98B3 Let $\ell \in \mathcal{R}^1(X,\theta ;\phi )$. Then $\dot{\ell }_0\in L^1(X,\theta _{\phi }^ n)$ and
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17.13.43
\begin{equation} \mathbf{E}^{\phi }(\ell )=\int _ X \dot{\ell }_0\, \theta _{\phi }^ n. \end{equation}
In particular, for every $t\geq 0$,
\[ E_{\theta }^{\phi }(\ell _ t)=t\int _ X \dot{\ell }_0\, \theta _{\phi }^ n. \]
Proof.
Set $\Gamma =\ell ^*$. By theorem 17.13.21, we have $\Gamma \in \mathrm{TC}^1(X,\theta ;\phi )$ and
\[ \mathbf{E}^{\phi }(\ell )=\mathbf{E}^{\phi }(\Gamma ). \]
By proposition 17.13.38, for every $\tau <\ell ^*_{\max }$,
\[ \int _ X \theta _{\Gamma _{\tau }}^ n=\int _{\{ \dot{\ell }_0\geq \tau \} } \theta _{\phi }^ n. \]
Hence (17.12.12) gives
\[ \mathbf{E}^{\phi }(\ell ) =\ell ^*_{\max }\int _ X \theta _{\phi }^ n-\int _{-\infty }^{\ell ^*_{\max }} \int _{\{ \dot{\ell }_0<\tau \} }\theta _{\phi }^ n\, \mathrm{d}\tau . \]
For $\tau >\ell ^*_{\max }$, we have $\ell ^*_{\tau }\equiv -\infty $ by 17.13.15. Then (17.13.40) shows that
\[ \int _{\{ \dot{\ell }_0\geq \tau \} }\theta _{\phi }^ n=0. \]
The layer-cake formula therefore gives (17.13.43). The final assertion follows from proposition 17.10.65, since $E_{\theta }^{\phi }(\ell _0)=E_{\theta }^{\phi }(\phi )=0$.
$\square$
Pome
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