My research interest is in probability theory and stochastic analysis. More precisely, currently I am mostly interested in the following topics.

Nonlinear Fokker-Planck-Kolmogorov equations and nonlinear Markov processes

FPK equations are parabolic equations for measures whose solutions are curves of the 1D-marginal distributions of stochastic processes, usually of solutions to stochastic differential equations (SDEs). If the FPKE is nonlinear, its solutions are 1D-marginals of distribution-dependent SDEs. I am interested in nonlinear FPKEs with singular coefficients and initial data, their geometric interpretation and their relation to nonlinear Markov processes. These are processes satisfying a generalized, in a sense ’non-convex‘ notion of Markov property and are typically given by a family of solutions to distribution-dependent SDEs

Convex integration for fluid dynamical PDEs

The method of convex integration (CI) originated from John Nash’s proof of the isometric embedding theorem from differential geometry (1954). Nowadays, CI is used to construct exotic energy solutions to fluid dynamical equations, most prominently culminating in a proof of the Onsager conjecture and a non-uniqueness result for the Navier–Stokes equations. Recently, CI has also been applied to stochastic PDEs. Currently, I am interested in the question whether CI can lead to non-uniqueness results for (S)PDEs in physically relevant classes such as Leray-solutions.

Publications and preprints

  1. p-Brownian motion and the p-Laplacian, with Viorel Barbu and Michael Röckner, arxiv:2409.18744, 2024
  2. 2D vorticity Euler equations: Superposition solutions and nonlinear Markov processes, with Marco Romito, submitted, arxiv:2407.16609, 2024
  3. Remarks on regularization by noise, convex integration and spontaneous stochasticity, with Franco Flandoli, accepted for publication in Milan J. Math, arxiv:2402.16525, 2024
  4. Average dissipation for stochastic transport equations with Lévy noise, with Franco Flandoli and Andrea Papini, submitted, arXiv:2402.08461, 2024
  5. Weighted L1-semigroup approach for nonlinear Fokker–Planck equations and generalized Ornstein–Uhlenbeck processes, submitted, arXiv:2308.09420, 2023
  6. Nonlinear Fokker–Planck–Kolmogorov equations as gradient flows on the space of probability measures, with Michael Röckner, submitted, arXiv:2306.09530, 2023
  7. On nonlinear Markov processes in the sense of McKean, with Michael Röckner, submitted, arXiv:2212.12424, 2022
  8. Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia, with Myeongju Kang, Commun. Math. Sci., 21(7), 1875-1894, 2023
  9. Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations, with Andre Schenke, Nonlinear Anal., 227, 113179, 2023
  10. Flow selections for (nonlinear) Fokker–Planck–Kolmogorov equations, J. Differential Equations, 328, 105-132, 2022
  11. Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker–Planck–Kolmogorov equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24(3), 1705-1739, 2023
  12. Existence of flows for linear Fokker–Planck–Kolmogorov equations and its connection to well-posedness, J. Evol. Equ., 21(1), 17-31, 2021
  13. On Cherny’s results in infinite dimensions: A theorem dual to Yamada-Watanabe, Stoch. Partial Differ. Equ. Anal. Comput., 9, 33–70, 2021

Theses

My PhD thesis is available here. For my bachelor’s and master’s thesis, please contact me.