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.
Best Answer
Actually, it is quite possible to condition Brownian motion to hit a given point at a given time. The process is a called a Brownian bridge and the distribution of the winding number of the Brownian bridge is even known explicitly. It has been computed by Marc Yor in the paper
Marc Yor: Loi de l'indice du lacet brownien, et distribution de Hartman-Watson. Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 71–95.