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-time 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. The latter 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. In a recent series of papers, we constructed such nonlinear Markov properties as the probabilistic counterparts to fundamental solutions to a list of important nonlinear PDEs, such as the porous media and the p-Laplace equation. It turns out that nonlinear Markov processes offer a new connection between PDEs and stochastic analysis, which I want to explore in the future.

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. The Leibenson process, with Viorel Barbu, Sebastian Grube and Michael Röckner, arxiv:2508.12979, 2025.
  2. Non-uniqueness of Leray–Hopf solutions for the 3D fractional Navier–Stokes equations perturbed by transport noise, with Theresa Lange and Andre Schenke, arxiv:2412.16532, 2024
  3. p-Brownian motion and the p-Laplacian, with Viorel Barbu and Michael Röckner, accepted for publication in Annals of Probability, arxiv:2409.18744, 2024
  4. 2D vorticity Euler equations: Superposition solutions and nonlinear Markov processes, with Marco Romito, accepted for publication in Stoch. Partial Differ. Equ. Anal. Comput., arxiv:2407.16609, 2024
  5. Remarks on regularization by noise, convex integration and spontaneous stochasticity, with Franco Flandoli, Milan J. Math. 92, 349–370, 2024
  6. Average dissipation for stochastic transport equations with Lévy noise, with Franco Flandoli and Andrea Papini, Stochastic Transport in Upper Ocean Dynamics III. STUOD 2023. Mathematics of Planet Earth, vol 13. Springer, Cham, 2024
  7. Weighted L1-semigroup approach for nonlinear Fokker–Planck equations and generalized Ornstein–Uhlenbeck processes, submitted, arXiv:2308.09420, 2023
  8. Nonlinear Fokker–Planck–Kolmogorov equations as gradient flows on the space of probability measures, with Michael Röckner, submitted, arXiv:2306.09530, 2023
  9. On nonlinear Markov processes in the sense of McKean, with Michael Röckner, submitted, arXiv:2212.12424, 2022, accepted for publication in Journal of Theoretical Probability
  10. 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
  11. Nonuniqueness in law for stochastic hypodissipative Navier–Stokes equations, with Andre Schenke, Nonlinear Anal., 227, 113179, 2023
  12. Flow selections for (nonlinear) Fokker–Planck–Kolmogorov equations, J. Differential Equations, 328, 105-132, 2022
  13. 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
  14. Existence of flows for linear Fokker–Planck–Kolmogorov equations and its connection to well-posedness, J. Evol. Equ., 21(1), 17-31, 2021
  15. On Cherny’s results in infinite dimensions: A theorem dual to Yamada-Watanabe, Stoch. Partial Differ. Equ. Anal. Comput., 9, 33–70, 2021

List of co-authors (alphabetical order): V. Barbu, F. Flandoli, S. Grube, M. Kang, T. Lange, A. Papini, M. Romito, M. Röckner, A. Schenke

Theses

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