A01 – Statistics for stochastic partial differential equations (SPDEs)

Stochastic partial differential equations (SPDEs) give a natural description of dynamical phenomena in space and time. With the availability of high resolution observations, the understanding of the statistical properties of SPDE leads to accessible tools for calibrating as well as validating these models.

During the first funding period, we have developed a new approach to statistical inference for SPDEs, which interprets point observations in space as localized averages in a physically meaningful way. In that context, we have established a theory for nonparametric diffusivity estimation. We have focused on statistical inference for linear and semilinear SPDE, including the class of stochastic reaction-diffusion processes. Special emphasis has been put on the impact of lower order reaction terms and possible model misspecification. We applied our theoretical findings to experimentally observed data on traveling actin waves and cell  repolarization, providing first steps towards a quantitative understanding of SPDE models in cell biology.

In the second funding period we now turn to the problem of nonparametric estimation of transport and reaction terms. In addition, we study the impact of measurement noise on previous estimation results in detail. The investigation of change points and interfaces present in the diffusivity function will lead to new theoretical insights. We will systematically analyze the spatial and temporal  correlation present in the dynamical noise, and we shall provide tests for validating the presence (or absence) of dynamical noise in the generating equations. Our setting includes local and discrete observations in space. The different techniques and results will be combined with fluorescence microscopy observations of D. discoideum in order to obtain powerful models for intracellular pattern formation based on SPDEs.

  • Gaudlitz, S. and Reiß, M. (2022). Estimation for the reaction term in semi-linear SPDEs under small diffusivity. arXiv:2203.10527

  • Cialenco, I. and Kim, H.-J. and Pasemann, G. (2021). Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach. arXiv:2103.04211

  • Altmeyer, R. and Bretschneider, T. and Janák, J. and Reiß, M. (2020). Parameter Estimation in an SPDE Model for Cell RepolarisationarXiv 2010.06340

  • Altmeyer, R. and Cialenco, I. and Pasemann, G. (2020). Parameter estimation for semilinear SPDEs from local measurements. arXiv 2004.14728

  • Pasemann, G. and Flemming, S. and Alonso, S. and Beta, C. and Stannat, W. (2020). Diffusivity Estimation for Activator-Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton. Journal of Nonlinear Science 31, 59 (2021). doi:10.1007/s00332-021-09714-4  arXiv 2005.09421

  • Altmeyer, R. and Reiß, M. (2020). Nonparametric estimation for linear SPDEs from local measurements. Annals of Applied Probability, to appear. arXiv 1903.06984

  • Pasemann, G. and Stannat, W. (2019). Drift Estimation for Stochastic Reaction-Diffusion Systems. Electron. J. Statist. 14 (2020), no. 1, 547-579. doi:10.1214/19-EJS1665  arXiv 1904.04774