Green's formula, planar Brownian bridge, and Lévy's area (Q1893861)

Let \(Z = (Z_ t, t \in [0,1])\) be a planar Brownian bridge, and for every point \(z\) out of the range of \(Z\), denote by \(n_ z\) the index of \(Z\) around \(z\). For every \(\varepsilon > 0\), the Stratonovich integral \[ n^ \varepsilon_ z = {1 \over 2 \pi i} \int^ 1_ 0 \mathbf{1}_{\bigl \{| Z_ s - z | > \varepsilon \bigr\}} {\delta Z_ s \over Z_ s - z} \] coincides with the index \(n_ z\) provided that \(Z\) does not visit the ball centered at \(z\) with radius \(\varepsilon\). Though the integral \(\int_{\mathbb{R}^ 2} | n_ z | dz\) diverges a.s., the author is able to prove an ersatz of Green's formula. Namely, if \(F\) is a \({\mathcal C}^ 2\) function and \(f = \nabla F\), then the following convergence holds in probability: \[ \lim_{\varepsilon \to 0+} \int_{\mathbb{R}^ 2} n^ \varepsilon_ z f(z)dz = \int^ 1_ 0 F(Z_ s) \wedge \delta Z_ s. \] Specifying this formula to \(F(z) = z/2\) gives an approximation to Lévy's stochastic area \({\mathcal A}\) of \(Z\). A more intuitive approximation is also given. If \(S_ \varepsilon\) denotes the Wiener sausage with radius \(\varepsilon > 0\), then \[ \lim_{\varepsilon \to 0 +} \int_{\mathbb{R}^ 2-S_ \varepsilon} n_ z dz = {\mathcal A} \quad \text{(in probability}). \]