My main area of research is Stochastic Analysis. I focus on the analytical study of nonlinear stochastic partial differential equations (SPDEs), primarily SPDEs driven by transport noise which are used in ocean and climate modelling. I am strongly motivated by the challenge posed by randomness in multi-scale fluid dynamics and by the probabilistic descriptions that can be derived. In addition, I am interested in nonlinear filtering and high-dimensional data assimilation problems which involve SPDEs driven by different types of noise.
I have obtained existence and uniqueness of solutions for several nonlinear stochastic transport models: the two-dimensional Euler equation, the great lake equation, two new stochastic rotating shallow water (SRSW) models with similar types of stochasticity. I have also proven the applicability of one of these SRSW models in a data assimilation setting. I am currently working on Eulerian and pathwise calibration approaches for transport SPDEs, blow-up issues, and Hurst parameter estimation problems.
Fields of interest: Stochastic Partial Differential Equations, Probability Theory, Nonlinear Filtering & Data Assimilation, Statistical Aspects of Stochastic Fluid Dynamics.
Publications and preprints:
Published papers:
1. Well-posedness for a stochastic 2D Euler equation with transport noise, Stoch PDE: Anal Comp, https://doi.org/10.1007/s40072-021-00233-7 (joint with D. Crisan).
2. Well-posednessfor the great lake equation with transport noise, Rev. Roumaine Math. Pures Appl. 66 (2021), 1, 131–155 (joint with D. Crisan).
3. Bayesian Inference for fluid dynamics: a case study for the stochastic rotating shallow water model, Frontiers in Applied Mathematics and Statistics, 8:949354, doi: 10.3389/fams.2022.949354 (joint with P. J. van Leeuwen, D. Crisan, R. Potthast).
4. A pathwise parameterisation for stochastic transport, STUOD Proceedings by Springer Nature (joint with Wei Pan).
5. Well-posedness Properties for a Stochastic Rotating Shallow Water Model, J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-022-10243-1 (joint with D. Crisan).
6. Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty, J. Math. Fluid Mech. 25, 29 (2023). https://doi.org/10.1007/s00021-023-00769-9 (joint with D. Crisan, E. Mémin).
7. Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model, arXiv:2207.07457, to appear in Stochastics and Dynamics (joint with R. Mensah, D. Crisan, W. Pan, D. Holm).
PhD Thesis:
O. Lang, Nonlinear stochastic transport partial differential equations: well-posedness and data assimilation, PhD Thesis (2020), https://doi.org/10.25560/89816.
Preprints:
1. Existence and uniqueness of maximal solutions to SPDEs with applications to viscous fluid equations, arXiv:2209.09137v1, (joint with D. Goodair, D. Crisan).
2. Comparison of Stochastic Parametrization Schemes using Data Assimilation on Triad Models, preprint (joint with A. Lobbe, D. Crisan, D. Holm, E. Mémin, B. Chapron).
3. Noise calibration for the stochastic rotating shallow water model, preprint (joint with A. Lobbe, D. Crisan, P. J. van Leeuwen, R. Potthast).