  The stability properties of matrix-valued Riccati diffusions are
investigated. The matrix-valued Riccati diffusion processes considered in this
work are of interest in their own right, as a rather prototypical model of a
matrix-valued quadratic stochastic process. Under rather natural observability
and controllability conditions, we derive time-uniform moment and fluctuation
estimates and exponential contraction inequalities. Our approach combines
spectral theory with nonlinear semigroup methods and stochastic matrix
calculus. This analysis seem to be the first of its kind for this class of
matrix-valued stochastic differential equation. This class of stochastic models
arise in signal processing and data assimilation, and more particularly in
ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion
represents the flow of the sample covariance matrices associated with
McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here
applies to filtering problems with unstable signals.
