Let $(B_t, \mathcal{F}_t, P_x)$ be the Markov process associated with standard Brownian motion and let $\theta_t : \Omega \to \Omega$ (,$t \geq 0$) be the associated semigroup of translations with
a)$\theta_0 = Id_{\Omega}$, b)$\theta_{s+t} = \theta_s \circ \theta_t$, $\forall s,t \geq 0$, c) $X_s \circ \theta_t = X_{s+t}$, $\forall s,t \geq 0$
Furthermore, we defined $\tau_a := \inf\{t\geq 0: B_t=a\}$ and $\tau_a (t) := \min\{s\geq t: B_s=a\}$, if there exists an $s$ with $B_s = a$ (else: $\tau_a (t) = \infty$) . Can you tell me why the following equalities hold:
$\{ \tau_{-a} < \infty \} \cap \theta_{\tau_{-a}}^{-1} \{ \tau_a < \infty \} = \{ \omega \in \Omega_{\tau_{-a}} : \tau_a (\theta_{\tau_{-a} (\omega)} (\omega)) < \infty \} = \{ \tau_{-a} < \infty \} \cap \{ \tau_a (\tau_{-a}) < \infty \}$
Thank you very much for any hint!
I'm assuming $\tau-a$s are typos and should be replaced with $\tau_{-a}$. Also, what is $\Omega_{\tau-a}$ (or $\Omega_{\tau_{-a}}$)?
$\Omega = C([0,\infty),\mathbb{R}^d)$ and $(\theta_t(\omega))_s = \omega_{s+t}$ for $\omega \in \Omega$.
Note that the path $\theta_{\tau_{-a}(\omega)}(\omega)$ starts from $-a$ for each $\omega \in \Omega$. Then $$\{\omega \in \Omega \mid \tau_{-a}(\omega)<\infty\}\cap\{\omega \in \Omega \mid \tau_a (\theta_{\tau_{-a}(\omega)}(\omega)) < \infty\} = \text{All paths which hit $-a$ in finite time and after that hit $a$ agian in finite time }.$$
The other terms in the equality are just restatements of the middle term, e.g. $$\theta_{\tau_{-a}(\cdot)}^{-1}\{\tau_{a} < \infty\} = \{\omega \in \Omega \mid \theta_{\tau_{-a}(\omega)}(\omega) \in \{\tau_{a} < \infty\} \} = \{\omega \in \Omega \mid \tau_a (\theta_{\tau_{-a}(\omega)}(\omega)) < \infty\}$$