General position of a projection and its image under a free unitary Brownian motion

Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ina$W^{\star}$-noncommutativeprobabilityspacesuchthat$Y$and$P$are$\star$-free in Voiculescu’s sense, the main result of this paper states that $P$ and $Y_{t}P{Y}_t^{\star }$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $S{Y}_{t}S{Y}_t^{\star }$ where $S=2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say ${\mu }_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of ${\mu }_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \infty)$distributions\mathrm{.}Thisfactallowstocomputetheweightthat$\mu_t$assignsto$z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $S{Y}_{t}S{Y}_t^{\star }$.