Research
Research interests
Stochastic partial differential equations (SPDEs), nonlinear partial differential equations, random dynamical systems, interacting particle systems, machine learning, porous media equations, scalar conservation laws, synchronization by noise, regularization by noise, rough paths.
Publications
Preprints
- Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations
with Jonas Sauer
available on arXiv.
- Landau-Lifschitz-Navier-Stokes Equations: Large Deviations and Relationship to the Energy Equality
with Daniel Heydecker, Zhengyan Wu
available on arXiv.
- Solutions to the stochastic thin-film equation for initial values with non-full support
with Konstantinos Dareiotis, Manuel V. Gnann, Max Sauerbrey
available on arXiv:.
- Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent
with Sebastian Kassing, Vitalii Konarovskyi
available on arXiv:.
- Convergence rates for momentum stochastic gradient descent with noise of machine learning type
with Sebastian Kassing
available on arXiv:.
- 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:.
- 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:.
Publications in peer-reviewed journals
- Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift
with Benjamin Fehrman
Invent. Math. (2023), arXiv:.
- Long-time behaviour of stochastic Hamilton-Jacobi equations
with Paul Gassiat, Pierre-Louis Lions, Panagiotis E. Souganidis
to appear in J. Funct. Anal. (JFA), arXiv:.
- Numerical approximation of singular-degenerate parabolic stochastic PDEs
with Lubomír Baňas, Christian Vieth
to appear in: IMA Journal of Numerical Analysis, arXiv:.
- Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise
with Rishabh S. Gvalani, Florian Kunick, Felix Otto
to appear in Mathematics of Computation, arXiv:.
- 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
Arch. Ration. Mech. Anal. (ARMA) 242 (2021), no. 1, 179–234:.
- Porous media equations with multiplicative space-time white noise
with Konstantinos Dareiotis, Máté Gerencsér
Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 4, 2354–2371:.
- Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
with Benjamin Fehrman
J. Math. Pures Appl. (9) 148 (2021), 221–266.:.
- Convergence rates for the stochastic gradient descent method for non-convex objective functions
with Benjamin Fehrman, Arnulf Jentzen
J. Mach. Learn. Res. (JMLR) 21 (2020), Paper No. 136, 48 pp.:.
- The stochastic thin-film equation: existence of nonnegative martingale solutions
with Manuel V. Gnann
Stochastic Process. Appl. (SPA) 130 (2020), no. 12, 7260–7302:.
- Synchronisation by noise for the stochastic quantisation equation in dimensions 2 and 3
with Pavlos Tsatsoulis
Stoch. Dyn. (Stochastics and Dynamics) 20 (2020), no. 6, 2040006, 17 pp.:.
- Ergodicity for Stochastic Porous Media Equations
with Konstantinos Dareiotis, Pavlos Tsatsoulis
SIAM J. Math. Anal. (SIMA) 52 (2020), no. 5, 4524–4564. :.
- Random attractors for locally monotone stochastic partial differential equations
with Wei Liu, Andre Schenke
J. Differential Equations (JDE) 269 (2020), no. 4, 3414–3455:.
- Nonlinear diffusion equations with nonlinear gradient noise
with Konstantinos Dareiotis
Electron. J. Probab. (EJP) 25 (2020), Paper No. 35, 43 pp.:.
- Density bounds for solutions to differential equations driven by Gaussian rough paths
with Cheng Ouyang, Samy Tindel
J. Theoret. Probab. (Journal of Theoretical Probability) 33 (2020), no. 2, 611–648:.
- Optimal regularity in time and space for the porous medium equation
with Jonas Sauer, Eitan Tadmor
Anal. PDE (Analysis and PDE) 13 (2020), no. 8, 2441–2480:.
- Optimal regularity for the porous medium equation
J. Eur. Math. Soc. (JEMS) 23 (2021), no. 2, 425–465:.
- Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
with Benjamin Fehrman
Arch. Ration. Mech. Anal. 233 (2019), no. 1, 249–322:.
- Path-by-path regularization by noise for scalar conservation laws
with Khalil Chouk
J. Funct. Anal. (JFA) 277 (2019), no. 5, 1469–1498:.
- Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
with Sebastian Becker, Arnulf Jentzen, Peter E. Kloeden
BIT (BIT Numerical Mathematics.) 60 (2020), no. 4, 1057–1073.:.
- Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians
with Paul Gassiat, Pierre-Louis Lions, Panagiotis E. Souganidis
Probab. Theory Related Fields 176 (2020), no. 1-2, 421–448.:.
- Stochastic nonlinear Fokker-Planck equations
with Michele Coghi
Nonlinear Anal. (Theory, Methods & Applications) 187 (2019), 259–278:.
- Entropy solutions for stochastic porous media equations
with Konstantinos Dareiotis, Maté Gerencsér
J. Differential Equations 266 (2019), no. 6, 3732–3763:.
- Supremum estimates for degenerate, quasilinear stochastic partial differential equations
with Konstantinos Dareiotis
Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 3, 1765–1796.:.
- Stochastic continuity equations with conservative noise
with Scott Smith
J. Math. Pures Appl. (JMPA) (9) 128 (2019), 225–263:.
- 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
Comm. Partial Differential Equations (CPDE) 43 (2018), no. 12, 1702–1736:.
- Regularization by noise for stochastic Hamilton-Jacobi equations
with Paul Gassiat
Probab. Theory Related Fields 173 (2019), no. 3-4, 1063–1098.:.
- 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
Ann. Probab. 46 (2018), no. 5, 2495–2544:.
- Stochastic non-isotropic degenerate parabolic-hyperbolic equations
with Panagiotis E. Souganidis
Stochastic Processes and their Applications (SPA) 127 (2017), no. 9, 2961–3004:.
- Ergodicity and local limits for stochastic local and nonlocal p-Laplace
equations
with Jonas M. Tölle
SIAM Journal on Mathematical Analysis (SIMA) 48 (2016), no. 6, 4094–4125:.
- Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
with Michael Röckner
Transactions of the AMS 369 (2017), no. 5, 3017–3045.:.
- Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes
with Benoît Perthame, Panagiotis E. Souganidis
SIAM Journal on Numerical Analysis (SINUM) 54 (2016), no. 4, 2187–2209:.
- 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
Probab. Theory Related Fields 168 (2017), no. 3-4, 511–556.:.
- Weak synchronization for isotropic flows
with Michael Cranston, Michael Scheutzow
Discrete Contin. Dyn. Syst. Ser. B (DCDS-B) 21 (2016), no. 9, 3003–3014:.
- Synchronization by noise for order-preserving random dynamical systems
with Franco Flandoli, Michael Scheutzow
The Annals of Probability 45 (2017), no. 2, 1325–1350.:.
- 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
SIAM Journal on Mathematical Analysis (SIMA) 47 (2015), no. 5, 4058–4090:.
- Stochastic scalar conservation laws driven by rough paths
with Peter K. Friz
Ann. Inst. H. Poincaré Anal. Non Linéaire (AIHP) 33 (2016), no. 4, 933–963:.
- 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)