We present an introduction (also for non{experts) to a new framework for the analysis of (up to) second order di erential operators (with merely measurable coe cients and in possibly in nitely many variables) on L 2 {spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, so{called generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C 0 {semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. FOT], MR1]) as well as time dependent Dirichlet forms (cf. O1]). We discuss many applications to di erential operators that can be treated within the new framework only, e.g. parabolic di erential operators with unbounded drifts satisfying no L p {conditions, singular and fractional di usion operators. Subsequently, we analyze the probabilistic counterpart. More precisely, we identify necessary and su cient analytic properties of a generalized Dirichlet form on an arbitrary topological state space to be associated with a nice strong Markov process, e.g. a di usion. These results extend previous results on the existence of associated processes in the elliptic as well as the parabolic case.

Abstract. The ensemble Kalman filter has become a popular data assimilation technique in the geosciences.However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf {R}^m$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf {Z}^d$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<\alpha

We introduce the Lyapunov approach to optimal control problems of average risk-sensitive Markov control processes with general risk maps. Motivated by applications in particular to behavioral economics, we consider possibly nonconvex risk maps, modeling behavior with mixed risk preference. We introduce classical objective functions to the risk-sensitive setting and we are in particular interested in optimizing the average risk in the infinite-time horizon for Markov Control Processes on general, possibly non-compact, state spaces allowing also unbounded cost. Existence and uniqueness of an optimal control is obtained with a fixed point theorem applied to the nonlinear map modeling the risksensitive expected total cost. The necessary contraction is obtained in a suitable chosen seminorm under a new set of conditions: 1) Lyapunov-type conditions on both risk maps and cost functions that control the growth of iterations, and 2) Doeblin-type conditions, known for Markov chains, generalized to nonlinear mappings. In the particular case of the entropic risk map, the above conditions can be replaced by the existence of a Lyapunov function, a local Doeblin-type condition for the underlying Markov chain, and a growth condition on the cost functions.

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