  The discovery of particle filtering methods has enabled the use of nonlinear
filtering in a wide array of applications. Unfortunately, the approximation
error of particle filters typically grows exponentially in the dimension of the
underlying model. This phenomenon has rendered particle filters of limited use
in complex data assimilation problems. In this paper, we argue that it is often
possible, at least in principle, to develop local particle filtering algorithms
whose approximation error is dimension-free. The key to such developments is
the decay of correlations property, which is a spatial counterpart of the much
better understood stability property of nonlinear filters. For the simplest
possible algorithm of this type, our results provide under suitable assumptions
an approximation error bound that is uniform both in time and in the model
dimension. More broadly, our results provide a framework for the investigation
of filtering problems and algorithms in high dimension.
