Random holonomy of Hopf fibrations (Q5939931)

Given a Riemannian fiber bundle \(U\) over a Riemannian manifold \(M\), consider the initial value problem for the horizontal heat equation \((\frac{\partial}{\partial t}+\frac{1}{2}\Delta^H_{x}) u(x,t)=0\), \(u(x,0)=f(x)\), where \(f\) is a fixed section of \(U\), \(u(x,t)\) is a section of \(U\) for each \(t\), and \(\Delta^H\) is the horizontal connection Laplacian on sections. The solution can be represented as \(u(x,t)=E_x[ P_t^{-1}f ( x_t) ]\), where \(x_{t}\) is the Brownian motion on the base manifold \(M\) and \(P_{t}\) is stochastic parallel transport along this motion. Using conditional expectation, we may rewrite the solution as NEWLINE\[NEWLINE u(x,t)=\int_M E_x[ P_t^{-1}f(y) |x_{t}=y ] p_{t}(x,y)d \text{vol}(y), NEWLINE\]NEWLINE where \(p_{t}\) is the scalar heat kernel on \(M\). In general one would expect the norm of \(F(x,y)=E_x[ P_t^{-1}f(y) |x_{t}=y ]\) to be less than the norm of \(f(y)\) in the presence of nontrivial holonomy, due to cancellation while averaging. The operator norm of \(f(x)\to \int_{M}F(x,y)d \text{vol}(y)\) on sections measures the amount of cancellation, and the kernel \(f(x) \to F(x,y)\) is studied in this paper. One expects that the magnitude of cancellation depends on curvature, and the authors make precise calculations for some particular constant curvature examples, the Hopf fibrations and tautological line bundles over complex and quaternionic projective space. NEWLINENEWLINENEWLINEThe authors explicitly calculate the Itô decomposition of Brownian motion on each projective space; in fact the Brownian motion on complex projective space is a local martingale. Likewise, the Itô-stochastic differential equation for stochastic parallel transport on each of these bundles is calculated, and the desired expectation is shown to reduce to a calculation for a one-dimensional radial stochastic differential equation -- a harmonic oscillator. The norm of the kernel described above is shown to decrease exponentially in time, and the rate of decay is the explicitly determined eigenvalue of the ground state of the harmonic oscillator equation.