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The problem of stability of the optimal filter is revisited. The optimalfilter (or filtering process) is the conditional probability of the currentstate of some stochastic process (the signal process), given both present andpast values of another process (the observation process). Typically thefiltering process satisfies a dynamical equation, and the question investigatedhere concerns the stability of this dynamics. In contrast to previous work,signal processes given by the iterations of a deterministic mapping $f$ areconsidered, with only the initial condition being random. While the stabilityof the filter may emerge from strong randomness of the signal processes,different and more dynamical effects will be exploited in the present work.More specifically, we consider uniformly hyperbolic $f$ with stronginstabilities providing the necessary mixing. This however requires that thefiltering process is initialised with densities exhibiting already a certainlevel of smoothness. Furthermore, $f$ may also have stable directions alongwhich the filtering process will eventually not have a density, a majortechnical difficulty. Further results show that the filtering process isasymptotically concentrated on the attractor and furthermore will havedensities with respect to the invariant (SRB)~measure along instable manifoldsof $f$.

Exponential stability and asymptotic properties of the optimal filter for signals with deterministic hyperbolic dynamics