A consensus Kalman filter on L2 spaces

We study the estimation problem of infinite dimensional discrete-time stochastic linear systems with finite dimensional measurements on sensor networks modeled by connected undirected graphs. The framework encompasses discretized PDEs with sampled measurements. A new scheme of distributed consensus on measurements is extended to systems evolving in L2 spaces in order to limit the information exchange to finite-dimensional vectors. We show that, in analogy to the finite-dimensional case, at each node the variance of the estimation error tends to the one of the centralized Kalman filter for systems is L2 when the number of consensus steps increases.