# A01 – Statistics for stochastic partial differential equations (SPDEs)

## Objectives

This project will contribute to the emerging field of statistics for stochastic partial differential equations (SPDEs). General principles of statistical inference for SPDEs and nonparametric methods will be developed systematically. In parallel, specific estimation problems arising in physics and the neurosciences will be treated. Mathematically founded model validation tests will enhance the impact of modelling with SPDEs in applications.

The research programme within this project follows two major directions:

**Research direction A** will be devoted to the development of general principles of statistical inference for SPDEs w.r.t. different types of available observations: full observations, discrete-in-space observations, noisy observations and in the long run also partial observations. Questions to be addressed will be (perfect) identifiability, drift and diffusion estimation.

**Research direction B** will be devoted to the development of statistical methods for SPDEs in physics and neuroscience. Particular SPDEs of interest from statistical physics, like e.g. Langevin equations on path space admitting diffusion bridges as stationary measure, and SPDEs arising in neuroscience like stochastic nerve axon equations will be analyzed. Theoretical concepts developed in research direction A will be implemented in the above mentioned model classes of SPDEs with a view towards practical applications. In addition, SPDE models derived from first principles will be validated.

### References

- Parameter estimation for stochastically perturbed Navier-Stokes Equations
Cialenco, Glatt-Holtz, Stochastic Process. Appl. 121:701-724, 2011.

- On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's
Huebner, Rozovskii, Probab. Theory Relat. Fields, 103:143-163, 1995.

- Spectral asymptotics of some functionals arising in statistical inference for SPDEs
Lototsky, Rozovskii, Stochastic Process. Appl., 79:69-94, 1999.

- Statistical inference for stochastic partial differential equations
Prakasa Rao, in Selected Proceedings of the Symposium in Inference for Stochastic Processes, pages 47-70, IMS, Beachwood, 2001.

- Stochastic partial differential equation based modelling of large space-time data sets
Sigrist, Künsch, Stahel. , J. Roy. Stat. Soc. B, 77:3-33, 2015.

## Preprints

Pasemann, G., and Stannat, W. (2019):

*Drift Estimation for Stochastic Reaction-Diffusion Systems*. ArXiv 1904.04774Altmeyer, R., and Reiß, M. (2019):

*Nonparametric estimation for linear SPDEs from local measurements.*ArXiv 1903.06984P.J. van Leeuwen, H.-R. Kuensch, L. Nerger, R. Potthast and S. Reich,

*Particle filters for applications in geosciences.*technical report, arXiv:1807.10434

## Publications

Pasemann, G., and Stannat, W. (2019):

*Drift Estimation for Stochastic Reaction-Diffusion Systems*. ArXiv 1904.04774Altmeyer, R., and Reiß, M. (2019):

*Nonparametric estimation for linear SPDEs from local measurements.*ArXiv 1903.06984P.J. van Leeuwen, H.-R. Kuensch, L. Nerger, R. Potthast and S. Reich,

*Particle filters for applications in geosciences.*technical report, arXiv:1807.10434