Collection: Results on stopping times for Brownian motion (with drift)

Below, $(X_t)_{t \geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.

$\tau = \tau_a:=\inf\{t \geq 0; X_t = a\}$ for $a>0$.

Note: We have $\tau=\inf\{t \geq 0; X_t \geq a\}$ a.s. if $(X_t)_{t \geq 0}$ is a BM, see this answer.

(BM) $\tau_a = \tfrac{1}{c} \tau_{a \sqrt{c}}$ in distribution
(BM) $\mathbb{P}(\tau<\infty)=1$ (via martingale methods)
(BM with non-negative drift) $\mathbb{P}(\tau<\infty)=1$ (via law of iterated logarithm)
(BM with negative drift) Computation of $\mathbb{P}(\tau<\infty)$
(BM) $\mathbb{E}\tau = \infty$
(BM with positive drift) $\mathbb{E}(\tau^n)<\infty$ for all $n \in \mathbb{N}$ and computation of these moments
(BM with drift) density of the distribution of $\tau$
(BM) Laplace transform of $\tau$
(BM with positive drift) Laplace transform of $\tau$

$\tau= \inf\{t \geq 0; X_t \notin [a,b]\}$

(BM) $\mathbb{P}(\tau<\infty)=1$ (for $d$-dimensional BM)
(BM) Distribution of $X_{\tau}$
(BM with drift) Distribution of $X_{\tau}$ (i.e. computation of $\mathbb{P}(X_{\tau}=a)$ and $\mathbb{P}(X_{\tau}=b)$.)
(BM) Computation of $\mathbb{E}(\tau)$ (see this question for BM started at $X_0=x$)
(BM) Computation of $\mathbb{E}(\tau^2)$
(BM) $\mathbb{E}(\tau^p) < \infty$ for all $p \geq 1$
(BM; $a=b$) Laplace transform of $\tau$ (see also this question)
(BM) distribution of $\tau$

Hitting times for some curves

(BM) $\mathbb{E}\tau=\infty$ for $\tau := \inf\{t; B_t^2 \geq 1+t\}$ (see also this question)
(BM) $\mathbb{E}(\tau) =\tfrac{1}{2}$ for $\tau = \inf\{t; B_t^2 = 1-t\}$
(BM) $\mathbb{E}\tau<\infty$ for $\tau:=\inf\{t; |B_t| = \tfrac{1}{2}(1+\sqrt{1+t})\}$
(BM) $\mathbb{E}(\tau)=\infty$ for $\tau=\inf\{t; B_t \geq e^{-\lambda t}\}$

Random variables which are not stopping times

(BM) $\tau=\inf\{t; X_t = \max_{s \in [0,1]} X_s\}$ is not a stopping time (see also this question)
(BM) hitting time of an open set is not a stopping time with respect to the canonical filtration

Miscellaneous

(BM) $\mathbb{E}(\tau)$ for $\tau=\inf\{t; |X_t-x| \geq r\}$ where $X$ is a multi-dimensional Brownian motion
(BM) $\mathbb{E}(\tau)$ for $\tau=\inf\{t; \sup_{s \leq t} X_s \geq b, \inf_{s \leq t} X_s \leq a\}$ with $a<0<b$
(BM) Proof of Wald's identity (see also this question and this version for $\mathbb{E}(\sqrt{\tau})<\infty$)