A01 – Statistics for stochastic partial differential equations (SPDEs)

The overarching goal of this project is to develop a general framework for statistics for stochastic
partial differential equations (SPDEs) and in parallel to work out statistical methods for concrete
models from applications, primarily from physics and biosciences.
For the third funding period the project’s focus will be on deepening and extending the scope
of statistical methods for SPDEs. The current progress in the field calls for unifying certain
methods and model assumptions. To do so, we use Le Cam theory to understand inference
questions for SPDEs in a principled way. This will on one hand reveal the information structure
provided by SPDE solution processes on the coefficients and parameters involved and, on the
other hand, allow to easily adapt standard tests and confidence sets to become applicable to
SPDE observations. Here, we shall focus on the realistic case of additional measurement errors,
which makes the inference harder, but allows to unite different observation schemes because
asymptotically sufficient statistics will rely on local averaging. Based on this understanding,
we shall develop goodness-of-fit tests for model validation, which are crucial in a practical
model building scheme, and classification methods to discriminate between different underlying
dynamics.
In the field, the main focus is so far on parabolic SPDEs, while the stochastic wave equation,
for instance, is equally important as a model for spatio-temporal stochastic dynamics. The
probabilistic and analytic properties of hyperbolic SPDEs, like the stochastic wave equation, are
quite distinct from the parabolic case, but several statistical principles from parabolic SPDEs have
been successfully conveyed to simple hyperbolic cases. Statistical estimation and inference for
hyperbolic SPDEs will now be studied with an emphasis on higher principles which explain the
sometimes surprising similarities and differences for statistics in the two model classes.
In view of applications from biosciences, our third major work concerns statistical methods
for filtering and parameter estimation of stochastic reaction-diffusion systems observed with
spatio-temporal point process observations. Our main goal here will be to develop, to analyse
and to implement efficient sequential Bayesian filtering algorithms for confocal laser scanning
microscopy (CLSM) data of intracellular activator-inhibitor dynamics.
Based on fundamental results on filtering SPDEs with point process observations obtained
in the second funding period we will develop a new generation of sequential Bayesian filtering
algorithms for CLSM recordings, that have the potential to greatly enhance the quality of statistical
inference of CLSM recordings of dynamical processes. The primary field of application will be the
Bayesian state-parameter estimation of intracellular reaction-diffusion kinetics. We will validate
the potential of our filtering algorithms in particular in experiments with the social amoeba
Dictyostelium discoideum (D. discoideum), focusing on the impact of substrate stiffness on actin wave
dynamics. But we believe that our filtering algorithms will also be of general interest for spatial-
temporal low-count Poisson denoising of CLSM recordings, where Gaussian approximation to
Poisson statistics causes comparably large approximation errors.