Spectral distribution of the free Jacobi process (Q2841859)

The authors characterize the spectral distribution of the free Jacobi process with parameter values \(\lambda = 1/2\) and \(\theta=1\). The case where \(0<\lambda<2\) and \(\theta = 1/2\) is to be discussed in a forthcoming paper. The starting point is the description of the free Jacobi process by the following free SDE NEWLINE\[NEWLINE dJ_t = \sqrt{\lambda\theta}\sqrt{J_t} \,dW_t\sqrt{P-J_t} + \sqrt{\lambda\theta}\sqrt{J_t} \,dW_t^*\sqrt{P-J_t} + (\theta P - J_t)\,dt,\quad t\in [0,T], NEWLINE\]NEWLINE where \(T\) is the infimum of all \(t\)'s such that \(J_t\) or \(P-J_t\) fails to be injective, \(W\) is a \(P\mathcal{A}P\)-complex free Brownian motion on the \(W^*\)-probability space \((\mathcal A, \tau)\) and \(P,Q\) are projections such that \(\tau(P)=\lambda\theta\), \(\tau(Q)=\theta\), \(PQ=QP=P\) if \(P\leq Q\) and \(=Q\) if \(P\geq Q\). This allows a recurrence relation of the moments \(m_n(t)\) of the free random variable \(J_t\) which can be explicitly solved using the generating function. In fact, NEWLINE\[NEWLINE m_n(t) = 2^{-2n}\binom{2n}{n} + 2^{-2n+1} \sum_{k=1}^n \binom{2n}{n-k} \frac 1k L_{k-1}^1(2kt) e^{-kt} NEWLINE\]NEWLINE with the \(n\)th Laguerre polynomial \(L_n^1\) of index \(1\). The main result of the paper states that the spectral distribution \(\mu_t\) of \(J_t\) in the compressed space fits the distribution of the free random variable \(1/4(Y_{2t}+Y_{2t}^*+2\mathbf{1})\) in \((\mathcal{A},\tau)\) (\(\mathbf 1\) is the unit and \(Y\) is the free unitary Brownian motion in the von Neumann algebra \(\mathcal A\)). Moreover, \(\mu_t\) is absolutely continuous with respect to the Lebesgue measure on \(\mathbb R\) and its support fills \((0,1)\) at time \(t=2\).