  This paper is concerned with the mathematical analysis of continuous time
Ensemble Kalman Filters (EnKBFs) and their mean field limit in an infinite
dimensional setting. The signal is determined by a nonlinear Stochastic Partial
Differential Equation (SPDE), which is posed in the standard variational
setting. Assuming global one-sided Lipschitz conditions and finite dimensional
observations we first prove the well posedness of both the EnKBF and its
corresponding mean field version. We then investigate the quantitative
convergence of the EnKBF towards its mean field limit, recovering the rates
suggested by the law of large numbers for bounded observation functions. The
main tool hereby are exponential moment estimates for the empirical covariance
of the EnKBF, which may be of independent interest. In the appendix of the
paper we investigate the connection of the mean field EnKBF and the Stochastic
Filtering Problem. In particular we derive the Feedback Particle Filter for
infinite dimensional signals and show that the mean field EnKBF can viewed as
its constant gain approximation.
