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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

  1. Stochastic Modified Flows for Riemannian Stochastic Gradient Descent
    with Sebastian Kassing, Nimit Rana
    available on arXiv .
  2. SVI solutions to stochastic nonlinear diffusion equations on general measure spaces
    with Michael Röckner, Weina Wu
    available on arXiv .
  3. Optimal Regularity in Time and Space for Nonlocal Porous Medium Type Equations
    with Jonas Sauer
    available on arXiv .
  4. Landau-Lifschitz-Navier-Stokes Equations: Large Deviations and Relationship to the Energy Equality
    with Daniel Heydecker, Zhengyan Wu
    available on arXiv .
  5. 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: .
  6. Convergence rates for momentum stochastic gradient descent with noise of machine learning type
    with Sebastian Kassing
    available on arXiv: .
  7. Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
    with Rishabh S. Gvalani, Vitalii Konarovskyi
    available on arXiv: .
  8. Ergodicity and random dynamical systems for conservative SPDEs
    with Benjamin Fehrman, Rishabh S. Gvalani
    available on arXiv: .
  9. Stochastic partial differential equations arising in self-organized criticality
    with Ľubomír Baňas, Marius Neuß
    available on arXiv: .
  10. Stabilization by transport noise and enhanced dissipation in the Kraichnan model
    with Ivan Yaroslavtsev
    available on arXiv: .
  11. Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
    with Nicolas Dirr, Benjamin Fehrman
    available on arXiv: .

Publications in peer-reviewed journals

  1. Lyapunov exponents and synchronisation by noise for systems of SPDEs
    with Pavlos Tsatsoulis
    to appear in Ann. Probab., arXiv: .
  2. Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent
    with Sebastian Kassing, Vitalii Konarovskyi
    to appear in J. Mach. Learn. Res. (JMLR), arXiv: .
  3. Well-posedness of the Dean-Kawasaki and the nonlinear Dawson-Watanabe
    equation with correlated noise
    with Benjamin Fehrman
    to appear in Arch. Ration. Mech. Anal. (ARMA), arXiv: .
  4. Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift
    with Benjamin Fehrman,
    Invent. Math. 234 (2023), no. 2, 573–636, arXiv: .
  5. Long-time behavior of stochastic Hamilton-Jacobi equations
    with Paul Gassiat, Pierre-Louis Lions, Panagiotis E. Souganidis
    J. Funct. Anal. 286 (2024), no. 4, Paper No. 110269, 49 pp, arXiv: .
  6. Numerical approximation of singular-degenerate parabolic stochastic PDEs
    with Lubomír Baňas, Christian Vieth
    to appear in: IMA Journal of Numerical Analysis, arXiv: .
  7. Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise
    with Rishabh Gvalani, Florian Kunick, Felix Otto
    Math. Comp. 92 (2023), no. 343, 1931–1976, arXiv: .
  8. Optimal regularity in time and space for stochastic porous medium equations
    with Stefano Bruno, Hendrik Weber
    Ann. Probab. 50 (2022), no. 6, 2288–2343,, arXiv: .
  9. Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
    with Sebastian Becker, Arnulf Jentzen, Peter E. Kloeden
    Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023), no. 1, 211–268., arXiv: .
  10. An example of intrinsic randomness in deterministic PDEs
    Franco Flandoli, Francesco Grotto
    Stoch. Dyn. 22 (2022), no. 7, Paper No. 2240023, 30 pp, arXiv:  .
  11. 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: .
  12. 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: .
  13. Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
    with Benjamin Fehrman
    J. Math. Pures Appl. (9) 148 (2021), 221–266.: .
  14. 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.: .
  15. The stochastic thin-film equation: existence of nonnegative martingale solutions
    with Manuel V. Gnann
    Stochastic Process. Appl. (SPA) 130 (2020), no. 12, 7260–7302: .
  16. 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.: .
  17. Ergodicity for Stochastic Porous Media Equations
    with Konstantinos Dareiotis, Pavlos Tsatsoulis
    SIAM J. Math. Anal. (SIMA) 52 (2020), no. 5, 4524–4564. : .
  18. Random attractors for locally monotone stochastic partial differential equations
    with Wei Liu, Andre Schenke
    J. Differential Equations (JDE) 269 (2020), no. 4, 3414–3455: .
  19. Nonlinear diffusion equations with nonlinear gradient noise
    with Konstantinos Dareiotis
    Electron. J. Probab. (EJP) 25 (2020), Paper No. 35, 43 pp.: .
  20. 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: .
  21. 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: .
  22. Optimal regularity for the porous medium equation
    J. Eur. Math. Soc. (JEMS) 23 (2021), no. 2, 425–465: .
  23. Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
    with Benjamin Fehrman
    Arch. Ration. Mech. Anal. 233 (2019), no. 1, 249–322:  .
  24. Path-by-path regularization by noise for scalar conservation laws
    with Khalil Chouk
    J. Funct. Anal. (JFA) 277 (2019), no. 5, 1469–1498:   .
  25. 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.:  .
  26. 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.: .
  27. Stochastic nonlinear Fokker-Planck equations
    with Michele Coghi
    Nonlinear Anal. (Theory, Methods & Applications) 187 (2019), 259–278: .
  28. Entropy solutions for stochastic porous media equations
    with Konstantinos Dareiotis, Maté Gerencsér
    J. Differential Equations 266 (2019), no. 6, 3732–3763: .
  29. Supremum estimates for degenerate, quasilinear stochastic partial differential equations
    with Konstantinos Dareiotis
    Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 3, 1765–1796.: .
  30. Stochastic continuity equations with conservative noise
    with Scott Smith
    J. Math. Pures Appl. (JMPA) (9) 128 (2019), 225–263: .
  31. 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: .
  32. Well-posedness by noise for scalar conservation laws
    with Mario Maurelli
    Comm. Partial Differential Equations (CPDE) 43 (2018), no. 12, 1702–1736: .
  33. Regularization by noise for stochastic Hamilton-Jacobi equations
    with Paul Gassiat
    Probab. Theory Related Fields 173 (2019), no. 3-4, 1063–1098.:   .
  34. 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: .
  35. Well-posedness and regularity for quasilinear degenerate
    parabolic-hyperbolic SPDE
    with Martina Hofmanova
    Ann. Probab. 46 (2018), no. 5, 2495–2544: .
  36. Stochastic non-isotropic degenerate parabolic-hyperbolic equations
    with Panagiotis E. Souganidis
    Stochastic Processes and their Applications (SPA) 127 (2017), no. 9, 2961–3004: .
  37. 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: .
  38. Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
    with Michael Röckner
    Transactions of the AMS 369 (2017), no. 5, 3017–3045.: .
  39. 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: .
  40. 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: .
  41. Synchronization by noise
    with Franco Flandoli, Michael Scheutzow
    Probab. Theory Related Fields 168 (2017), no. 3-4, 511–556.: .
  42. Weak synchronization for isotropic flows
    with Michael Cranston, Michael Scheutzow
    Discrete Contin. Dyn. Syst. Ser. B (DCDS-B) 21 (2016), no. 9, 3003–3014:   .
  43. Synchronization by noise for order-preserving random dynamical systems
    with Franco Flandoli, Michael Scheutzow
    The Annals of Probability 45 (2017), no. 2, 1325–1350.:  .
  44. Stability of solutions to stochastic partial differential equations
    with Jonas M. Tölle
    J. Differential Equations, 260 (2016), no. 6, 4973–5025: .
  45. Singular-degenerate multivalued stochastic fast diffusion equations
    with Michael Röckner
    SIAM Journal on Mathematical Analysis (SIMA) 47 (2015), no. 5, 4058–4090: .
  46. 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:  .
  47. Scalar conservation laws with multiple rough fluxes
     with Panagiotis E. Souganidis
    Commun. Math. Sci. 13 (2015), no. 6, 1569–1597.:  .
  48. Finite time extinction for stochastic sign fast diffusion and self-organized criticality
    Comm. Math. Phys., 335 (2015), no. 1, 309–344:  .
  49. Jain-Monrad criterion for rough paths
    with Peter K. Friz, Archil Gulisashvili, Sebastian Riedel
    to appear in: The Annals of Probability, (2014): .
  50. Multi-valued, singular stochastic evolution inclusions.
    with Jonas M. Tölle
    J. Math. Pures Appl. (JMPA) (9) 101 (2014), no. 6, 789–827: , arXiv:  .
  51. Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise.
    The Annals of Probability, 42 (2014), no. 2, 818–864.:  .
  52. Finite speed of propagation for stochastic porous media equation.
    SIAM J. Math. Anal. (SIMA), 45 (2013), no. 5, 2734–2766.:  .
  53. Random attractors for singular stochastic partial differential equations.
    J. Differential Equations (JDE), 255 (2013), no. 3, 524–559:  .
  54. Random Attractors for Degenerate Stochastic Partial Differential Equations.
    J. Dynam. Differential Equations (JDDE), 25(1) (2013), 121-157.  
  55. Strong Solutions for Stochastic Partial Differential Equations of Gradient Type.
     J. Funct. Anal. (JFA), 263(8)(2012),2355-2383. 
  56. 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.
  57. 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. 
  58. 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

  1. Stochastic Flows induced by Stochastic Partial Differential Equations, Dissertation, University of Bielefeld, 2011
  2. Convexity of Chebyshev sets, Master thesis, Warwick University, 2009

Conference and workshop proceedings

  1. 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
  2. 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

  1. Spatial rough path lifts of stochastic convolutions
    with Peter Friz, Archil Gulisashvili, Sebastian Riedel
    available on arXiv:  .
  2. 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)