Let \(\theta>0\) and \(X\) be the sticky Brownian motion with \(X_{0}=0\).

(1) Prove that \(L_{t}^{x}(X)=0\), for every \(x<0\); then, prove that \(X_{t} \geq 0\), a.s.

(2) Let \(A_{t}^{+}=\int_{0}^{t} d s \mathbb{1}_{\left\{X_{s}>0\right\}}, A_{t}^{0}=\int_{0}^{t} d s \mathbb{1}_{\left\{X_{s}=0\right\}}\), and define their inverses \(\alpha_{u}^{+}=\inf \left\{t: A_{t}^{+}>u\right\}\) and \(\alpha_{u}^{0}=\inf \left\{t: A_{t}^{0}>u\right\}\). Identify the law of \(\left(X_{\alpha^{+}}, u \geq 0\right)\).

(3) Let \(G\) be a Gaussian variable, with unit variance and 0 expectation. Prove that, for any \(u>0\) and \(t>0\)

\[\alpha_{u}^{+} \stackrel{\text { law }}{=} u+\frac{1}{\theta} \sqrt{u}|G| ; A_{t}^{+ \text {law }}\left(\sqrt{t+\frac{G^{2}}{4 \theta^{2}}}-\frac{|G|}{2 \theta}\right)^{2}\]

deduce that

\[A_{t}^{0} \stackrel{\text { law }}{=} \frac{|G|}{\theta} \sqrt{t+\frac{G^{2}}{4 \theta^{2}}}-\frac{G^{2}}{2 \theta^{2}}\]

and compute \(\mathbb{E}\left(A_{t}^{0}\right)\).

The process \(X_{\alpha_{u}^{+}}=W_{u}^{+}+\theta A_{\alpha_{u}^{+}}^{0}\) where \(W_{u}^{+}\)is a BM and \(A_{\alpha_{u}^{+}}^{0}\) is an increasing process, constant on \(\left\{u: X_{\alpha_{u}^{+}}>0\right\}\), solves Skorokhod equation. Therefore it is a reflected BM. The obvious equality \(t=A_{t}^{+}+A_{t}^{0}\) leads to \(\alpha_{u}^{+}=u+A_{\alpha_{u}^{+}}^{0}\), and \(a A_{\alpha_{u}^{+}}^{0} \stackrel{\text { law }}{=} L_{u}^{0}\).