  We consider matrix-valued stochastic processes known as isotropic Brownian
motions, and show that these can be solved exactly over complex fields. While
these processes appear in a variety of questions in mathematical physics, our
main motivation is their relation to a May-Wigner-like stability analysis, for
which we obtain a stability phase diagram. The exact results establish the full
joint probability distribution of the finite-time Lyapunov exponents, and may
be used as a starting point for a more detailed analysis of the
stability-instability phase transition. Our derivations rest on an explicit
formulation of a Fokker-Planck equation for the Lyapunov exponents. This
formulation happens to coincide with an exactly solvable class of models of the
Calgero-Sutherland type, originally encountered for a model of phase-coherent
transport. The exact solution over complex fields describes a determinantal
point process of biorthogonal type similar to recent results for products of
random matrices, and is also closely related to Hermitian matrix models with an
external source.
