@GJC20 Yes. Moreover, the distribution of $X_t$ given $X$ never reached $0$ is above (for the stochastic ordering) the distribution of $X_t$. I think that all these distributions can be computed explicitly.

If you set $x=\cos\theta$, this identity is a generalization of the formula $\cos(m\theta+n\theta) = \cos(m\theta)\cos(n\theta) - \sin(m\theta)\sin(n\theta)$.

@GJC20 The density and the distribution function of $X_t$ under $P$ and under $P[\cdot|\tau>t] $converge pointwise to 0 as $t \to +\infty$.

@Matt F. You forget the drift! Brownian motion with constant drift is transient.

@Matt F. You have $P[\tau=\infty]>0$, so there is no problem.

@ghc1997 I had omitted this condition. I have modified the example, and I hope that it is right. I have a question: where does your problem and this transformation which replaces $2 \times 2$ blocks with entries a,b,c,d by a single entry $ad-bc-(ac+bd)$ come from?

@Paul Larson I give my computations. I also could easily have made a mistake, but I hope that it is correct.

The law of $X$ is discrete, whereas the approximating normal law is absolutely continuous, so the total variation distance is $1$... So you should edit your question and precise what is the discretisation used (the speed of convergence may depend on this choice).

Representation / decomposition of elements of $Z/pZ$? Another example: if $p$ is prime, every elements of $Z/pZ$ can be written as the sum of two squares.