Research
Research interests
Stochastic Partial Differential Equations (SPDE), Random Dynamical Systems, Regularity for conservation laws, Regularity for SPDE, Long-time behavior of SPDE, Porous Media Equations, Scalar conservation laws, synchronization by noise, regularization by noise, averaging lemmata, singular SPDE, Rough paths.
Publications
Preprints
- Lyapunov exponents and synchronisation by noise for systems of SPDEs
with Pavlos Tsatsoulis
available on arXiv:.
- Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
with Rishabh S. Gvalani, Vitalii Konarovskyi
available on arXiv:.
- Ergodicity and random dynamical systems for conservative SPDEs
with Benjamin Fehrman, Rishabh S. Gvalani
available on arXiv:.
- Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise
with Rishabh S. Gvalani, Florian Kunick, Felix Otto
available on arXiv:.
- Well-posedness of the Dean-Kawasaki and the nonlinear Dawson-Watanabe
equation with correlated noise
with Benjamin Fehrman
available on arXiv:.
- Stochastic partial differential equations arising in self-organized criticality
with Ľubomír Baňas, Marius Neuß
available on arXiv:.
- Stabilization by transport noise and enhanced dissipation in the Kraichnan model
with Ivan Yaroslavtsev
available on arXiv:.
- Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
with Nicolas Dirr, Benjamin Fehrman
available on arXiv:.
- Numerical approximation of singular-degenerate parabolic stochastic PDEs
with Lubomír Baňas, Christian Vieth
available on arXiv:.
- Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift
with Benjamin Fehrman
available on arXiv:.
Publications in peer-reviewed journals
- Optimal regularity in time and space for stochastic porous medium equations
with Stefano Bruno, Hendrik Weber
to appear in The Annals of Probability, arXiv:.
- Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
with Sebastian Becker, Arnulf Jentzen, Peter E. Kloeden
to appear in Stochastics and Partial Differential Equations: Analysis and Computations (SPDE-AC), available on arXiv:.
- A New Correspondence Between TASEP and Burgers’ Equation
with Franco Flandoli, Francesco Grotto
to appear in Stochastics and Dynamics, arXiv:.
- Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
with Konstantinos Dareiotis, Manuel V. Gnann, Günther Grün
to appear in Archive for Rational Mechanics and Analysis (ARMA), arXiv:.
- Porous media equations with multiplicative space-time white noise
with Konstantinos Dareiotis, Máté Gerencsér
to appear in Annales de l’Institut Henri Poincaré, Probabilités et Statistiques:.
- Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
with Benjamin Fehrman
to appear in Journal de Mathématiques Pures et Appliquées:.
- Convergence rates for the stochastic gradient descent method for non-convex objective functions
with Benjamin Fehrman, Arnulf Jentzen
to appear in Journal of Machine Learning Research (JMLR), arXiv:.
- The stochastic thin-film equation: existence of nonnegative martingale solutions
with Manuel V. Gnann
to appear in Stochastic Processes and their Applications (SPA), arXiv:.
- Synchronisation by noise for the stochastic quantisation equation in dimensions 2 and 3
with Pavlos Tsatsoulis
to appear in Stochastics and Dynamics, arXiv:.
- Ergodicity for Stochastic Porous Media Equations
with Konstantinos Dareiotis, Pavlos Tsatsoulis
to appear in: SIAM J. Math. Anal. (SIMA), arXiv:.
- Random attractors for locally monotone stochastic partial differential equations
with Wei Liu, Andre Schenke
to appear in J. Differential Equations (JDE), arXiv:.
- Nonlinear diffusion equations with nonlinear gradient noise
with Konstantinos Dareiotis
to appear in Electronic Journal of Probability (EJP), arXiv:.
- Density bounds for solutions to differential equations driven by Gaussian rough paths
with Cheng Ouyang, Samy Tindel
to appear in Journal of Theoretical Probability (JOTP), arXiv:.
- Optimal regularity in time and space for the porous medium equation
with Jonas Sauer, Eitan Tadmor
to appear in Analysis and PDE (APDE), arXiv:.
- Optimal regularity for the porous medium equation
to appear in Journal of the European Mathematical Society (JEMS), arXiv:.
- Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
with Benjamin Fehrman
to appear in Archive for Rational Mechanics and Analysis (ARMA), arXiv:.
- Path-by-path regularization by noise for scalar conservation laws
with Khalil Chouk
to appear in J. Funct. Anal. (JFA), arXiv:.
- Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
with Sebastian Becker, Arnulf Jentzen, Peter E. Kloeden
to appear in BIT Numerical Mathematics, arXiv:.
- Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians
with Paul Gassiat, Pierre-Louis Lions, Panagiotis E. Souganidis
to appear in Probability Theory and Related Fields:.
- Stochastic nonlinear Fokker-Planck equations
with Michele Coghi
to appear in Nonlinear Analysis. Theory, Methods & Applications, arXiv:.
- Entropy solutions for stochastic porous media equations
with Konstantinos Dareiotis, Maté Gerencsér
to appear in J. Differential Equations (JDE), arXiv:.
- Supremum estimates for degenerate, quasilinear stochastic partial differential equations
with Konstantinos Dareiotis
to appear in Annales de l’Institut Henri Poincare (B) Probability and Statistics, arXiv:.
- Stochastic continuity equations with conservative noise
with Scott Smith
to appear in J. Math. Pures Appl. (JMPA), arXiv:.
- Regularity of solutions to scalar conservation laws with a force
with Xavier Lamy
Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, 505–521:.
- Well-posedness by noise for scalar conservation laws
with Mario Maurelli
to appear in Comm. Partial Differential Equations (CPDE):.
- Regularization by noise for stochastic Hamilton-Jacobi equations
with Paul Gassiat
to appear in Probability Theory and Related Fields:.
- Regularization and well-posedness by noise for ordinary and partial differential equations
to appear in: Stochastic Partial Differential Equations and Related Fields, Springer Proceedings in Mathematics & Statistics, preprint:.
- Well-posedness and regularity for quasilinear degenerate
parabolic-hyperbolic SPDE
with Martina Hofmanova
to appear in: The Annals of Probability.
- Stochastic non-isotropic degenerate parabolic-hyperbolic equations
with Panagiotis E. Souganidis
to appear in: Stochastic Process. Appl. (SPA):.
- Ergodicity and local limits for stochastic local and nonlocal p-Laplace
equations
with Jonas M. Tölle
to appear in: SIAM J. Math. Anal. (SIMA):.
- Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
with Michael Röckner
to appear in: Transactions of the AMS:.
- Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes
with Benoît Perthame, Panagiotis E. Souganidis
to appear in: SIAM Journal on Numerical Analysis (SINUM).
- Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws
with Panagiotis E. Souganidis
Comm. Pure Appl. Math. 70 (2017), no. 8, 1562–1597:.
- Synchronization by noise
with Franco Flandoli, Michael Scheutzow
to appear in: Probability Theory and Related Fields:.
- Weak synchronization for isotropic flows
with Michael Cranston, Michael Scheutzow
to appear in: Discrete and Continuous Dynamical Systems – Series B (DCDS-B):.
- Synchronization by noise for order-preserving random dynamical systems
with Franco Flandoli, Michael Scheutzow
to appear in: The Annals of Probability:.
- Stability of solutions to stochastic partial differential equations
with Jonas M. Tölle
J. Differential Equations, 260 (2016), no. 6, 4973–5025:.
- Singular-degenerate multivalued stochastic fast diffusion equations
with Michael Röckner
to appear in: SIAM J. Math. Anal. (SIMA): - Stochastic scalar conservation laws driven by rough paths
with Peter K. Friz
to appear in: Annales de l’Institut Henri Poincaré, Analyse Non Linéaire (AIHP): - Scalar conservation laws with multiple rough fluxes
with Panagiotis E. Souganidis
Commun. Math. Sci. 13 (2015), no. 6, 1569–1597.: - Finite time extinction for stochastic sign fast diffusion and self-organized criticality
Comm. Math. Phys., 335 (2015), no. 1, 309–344:.
- Jain-Monrad criterion for rough paths
with Peter K. Friz, Archil Gulisashvili, Sebastian Riedel
to appear in: The Annals of Probability, (2014):.
- Multi-valued, singular stochastic evolution inclusions.
with Jonas M. Tölle
J. Math. Pures Appl. (JMPA) (9) 101 (2014), no. 6, 789–827:, arXiv:
.
- Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise.
The Annals of Probability, 42 (2014), no. 2, 818–864.:.
- Finite speed of propagation for stochastic porous media equation.
SIAM J. Math. Anal. (SIMA), 45 (2013), no. 5, 2734–2766.:.
- Random attractors for singular stochastic partial differential equations.
J. Differential Equations (JDE), 255 (2013), no. 3, 524–559:.
- Random Attractors for Degenerate Stochastic Partial Differential Equations.
J. Dynam. Differential Equations (JDDE), 25(1) (2013), 121-157. - Strong Solutions for Stochastic Partial Differential Equations of Gradient Type.
J. Funct. Anal. (JFA), 263(8)(2012),2355-2383. - Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise. (Concise announcement of the results)
Comptes Rendus Mathematique, 350(5–6)(2012), 299–302. - Random attractors for a class of stochastic partial differential equations driven by general additive noise.
with Wei Liu, Michael Röckner
J. Differential Equations (JDE), 251(4-5)(2011), 1225 – 1253. - The Global Random Attractor for a Class of Stochastic Porous Media Equations
with Wolf-Jürgen Beyn, Michael Röckner, Paul Lescot
Comm. Partial Differential Equations (CPDE), 36 (3) (2011), 446 – 469.
Theses
- Stochastic Flows induced by Stochastic Partial Differential Equations, Dissertation, University of Bielefeld, 2011
- Convexity of Chebyshev sets, Master thesis, Warwick University, 2009
Conference and workshop proceedings
- Regularization by noise for stochastic Hamilton-Jacobi equations
based on joint work with Paul Gassiat
Oberwolfach Reports 24 (2016), 1349 – 1353.
Proceedings for the workshop: Rough Paths, Regularity Structures and Related Topics, May 1-7th, 2016,
Mathematisches Forschungsinstitut Oberwolfach - Spatial rough path lifts of stochastic convolutions
based on joint work with Peter Friz, Archil Gulisashvili, Sebastian Riedel
Oberwolfach Reports 41 (2012), 17 – 21.
Proceedings for the workshop: Rough Paths and PDEs, August 19-25th, 2012,
Mathematisches Forschungsinstitut Oberwolfach
Electronic publications
- Spatial rough path lifts of stochastic convolutions
with Peter Friz, Archil Gulisashvili, Sebastian Riedel
available on arXiv:.
- On the Variational Regularity of Cameron-Martin paths
with Peter K. Friz, Sebastian Riedel
available on arXiv:.
Coauthors
(in chronological order)
- Michael Röckner (Universität Bielefeld)
- Wolf-Jürgen Beyn (Universität Bielefeld)
- Paul Lescot (Université de Rouen)
- Wei Liu (Universität Bielefeld)
- Jonas M. Tölle (Technische Universität Berlin)
- Peter Friz (Technische Universität Berlin)
- Archil Gulisashvili (Ohio University)
- Sebastian Riedel (Technische Universität Berlin)
- Panagiotis E. Souganidis (University of Chicago)
- Franco Flandoli (University of Pisa)
- Michael Scheutzow (Technische Universität Berlin)
- Michael Cranston (University of California, Irvine)
- Benoît Perthame (Université Pierre et Marie Curie)
- Paul Gassiat (Université Paris Dauphine)
- Martina Hofmanova (Universität Bielefeld)
- Mario Maurelli (TU Berlin)
- Xavier Lamy (Université Toulouse)
- Khalil Chouk (TU Berlin)
- Scott Smith (MPI MIS Leipzig)
- Sebastian Becker
- Arnulf Jentzen (ETH Zürich)
- Peter E. Kloeden (Universität Frankfurt)
- Konstantinos Dareiotis (MPI MIS Leipzig)
- Benjamin Fehrman (MPI MIS Leipzig)
- Cheng Ouyang (University of Illinois at Chicago)
- Samy Tindel (Purdue University)
- Maté Gerencsér (IST Austria)
- Pierre-Louis Lions (Collège de France)
- Felix Otto (MPI MIS Leipzig)
- Rishabh Gvalani (MPI MIS Leipzig)
- Florian Kunick (MPI MIS Leipzig)
- Ľubomír Baňas (Universität Bielefeld)
- Ivan Yaroslavtsev (Universität Hamburg)
- Christian Vieth (Universität Bielefeld)
- Stefano Bruno (University Bath)
- Hendrik Weber (University Bath)
- Nicolas Dirr (University Cardiff)