  This paper proposes a probabilistic approach to the problem of intrinsic
filtering of a system on a matrix Lie group with invariance properties. The
problem of an invariant continuous-time model with discrete-time measurements
is cast into a rigorous stochastic and geometric framework. Building upon the
theory of continuous-time invariant observers, we show that, as in the linear
case, the error equation is a Markov chain that does not depend on the state
estimate. Thus, when the filter's gains are held fixed, and the filter admits
almost-global convergence properties with noise turned off, the noisy error's
distribution is proved to converge to a stationary distribution, providing
insight into the mathematical theory of filtering on Lie groups. For
engineering purposes we also introduce the discrete-time Invariant Extended
Kalman Filter, for which the trusted covariance matrix is shown to
asymptotically converge, and some numerically more involved sample-based
methods as well to compute the Kalman gains. The methods are applied to
attitude estimation, allowing to derive novel theoretical results in this
field, and illustrated through simulations on synthetic data.
