Title: Error Analysis of the Stochastic Linear Feedback Particle Filter,
Abstract: This paper is concerned with the convergence and long-term stability analysis
of the feedback particle filter (FPF) algorithm. The FPF is an interacting
system of $N$ particles where the interaction is designed such that the
empirical distribution of the particles approximates the posterior
distribution. It is known that in the mean-field limit ($N=\infty$), the
distribution of the particles is equal to the posterior distribution. However
little is known about the convergence to the mean-field limit. In this paper,
we consider the FPF algorithm for the linear Gaussian setting. In this setting,
the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although
these algorithms have been numerically evaluated and widely used in
applications, their convergence and long-term stability analysis remains an
active area of research. In this paper, we show that, (i) the mean-field limit
is well-defined with a unique strong solution; (ii) the mean-field process is
stable with respect to the initial condition; (iii) we provide conditions such
that the finite-$N$ system is long term stable and we obtain some mean-squared
error estimates that are uniform in time.