TY  - JOUR
A1  - Gerlach, Moritz Reinhardt
A1  - Glück, Jochen
T1  - Convergence of positive operator semigroups
JF  - Transactions of the American Mathematical Society
N2  - We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw, and Glicksberg with a purely algebraic result about positive group representations. Thus, we obtain convergence theorems not only for one-parameter semigroups but also for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive C-0-semigroup containing or dominating a kernel operator converges strongly as t ->infinity. We gain new insights into the structure theoretical background of those theorems and generalize them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.
KW  - Positive semigroups
KW  - semigroup representations
KW  - asymptotic behavior
KW  - kernel operator
Y1  - 2019
U6  - https://doi.org/10.1090/tran/7836
SN  - 0002-9947
SN  - 1088-6850
VL  - 372
IS  - 9
SP  - 6603
EP  - 6627
PB  - American Mathematical Soc.
CY  - Providence
ER  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise
JF  - SIAM Journal on Applied Dynamical Systems
N2  - 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.
KW  - data assimilation
KW  - Kalman Bucy filter
KW  - ensemble Kalman filter
KW  - stability
KW  - accuracy
KW  - asymptotic behavior
Y1  - 2018
U6  - https://doi.org/10.1137/17M1119056
SN  - 1536-0040
VL  - 17
IS  - 2
SP  - 1152
EP  - 1181
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -