TITLE: Brownian Motion Winding Number
QUESTION [7 upvotes]: Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the usual way for complex curves. 
If instead we have a 2D Brownian motion $Z=X+iY$, then this definition becomes more complicated. For example, if we have a Brownian motion starting at the origin, we can talk about the winding number $\theta_t$ at time $t$ around the origin by solving the stochastic differential equation
$d\theta_t=\frac{X_tdY_t-Y_tdX_t}{|Z_t|^2}$
with initial condition $\theta_0=0$. 
The issue is that for Brownian motion, we cannot condition on the path $Z$ to hit a particular point, because this has probability zero. Moreover, by considering annuli around $b$ and the fact that planar Brownian motion moves between concentric annuli with positive probability, it seems to me the situation becomes rather singular. 

Question: Is there a sensible generalization for the winding number of a Brownian motion conditioned to hit a single point?

For example, can we look at the limit of the winding number around an annulus about point $b$ whose radius shrinks to zero? I would imagine we would require the Brownian motion to be conditioned to hit some region of positive area just outside the shrinking annulus.

REPLY [2 votes]: I think you need to take the log of the Brownian motion and look at the imaginary coordinate.  
$$R + i\theta = \log(X+iY)$$
Since $z \mapsto \log z$ is conformal, there exists a time change taking one Brownian motion to the other.
Also, notice that $R_t$ is related to the Cauchy distribution.