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

1. Large deviations for conservative stochastic PDE and non-equilibrium fluctuations
   with Benjamin Fehrman
   available on arXiv: .
2. Synchronisation by noise for the stochastic quantisation equation in dimensions 2 and 3
   with Pavlos Tsatsoulis
   available on arXiv: .
3. Ergodicity for Stochastic Porous Media Equations
   with Konstantinos Dareiotis, Pavlos Tsatsoulis
   available on arXiv: .
4. The stochastic thin-film equation: existence of nonnegative martingale solutions
   with Manuel V. Gnann
   available on arXiv: .
5. Convergence rates for the stochastic gradient descent method for non-convex objective functions
   with Benjamin Fehrman, Arnulf Jentzen
   available on arXiv: .
6. Random attractors for locally monotone stochastic partial differential equations
   with Wei Liu, Andre Schenke
   available on arXiv: .
7. Nonlinear diffusion equations with nonlinear gradient noise
   with Konstantinos Dareiotis
   available on arXiv: .
8. Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
   with Benjamin Fehrman
   available on 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
   available on arXiv:   .

Publications in peer-reviewed journals

10. 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),
    available on arXiv:   .
11. Optimal regularity in time and space for the porous medium equation
    with Jonas Sauer, Eitan Tadmor
    to appear in Analysis and PDE (APDE),
    available on arXiv: .
12. Optimal regularity for the porous medium equation
    to appear in Journal of the European Mathematical Society (JEMS), arXiv:   .
13. Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
    with Benjamin Fehrman
    to appear in Archive for Rational Mechanics and Analysis (ARMA), arXiv:  .
14. Path-by-path regularization by noise for scalar conservation laws
    with Khalil Chouk
    to appear in J. Funct. Anal. (JFA), arXiv:   .
15. 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:  .
16. 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: .
17. Stochastic nonlinear Fokker-Planck equations
    with Michele Coghi
    to appear in Nonlinear Analysis. Theory, Methods & Applications, arXiv: .
18. Entropy solutions for stochastic porous media equations
    with Konstantinos Dareiotis, Maté Gerencsér
    to appear in J. Differential Equations (JDE), arXiv: .
19. 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: .
20. Stochastic continuity equations with conservative noise
    with Scott Smith
    to appear in J. Math. Pures Appl. (JMPA), arXiv: .
21. 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: .
22. Well-posedness by noise for scalar conservation laws
    with Mario Maurelli
    to appear in Comm. Partial Differential Equations (CPDE):   .
23. Regularization by noise for stochastic Hamilton-Jacobi equations
    with Paul Gassiat
    to appear in Probability Theory and Related Fields:   .
24. 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: .
25. Well-posedness and regularity for quasilinear degenerate
    parabolic-hyperbolic SPDE
    with Martina Hofmanova
    to appear in: The Annals of Probability .
26. Stochastic non-isotropic degenerate parabolic-hyperbolic equations
    with Panagiotis E. Souganidis
    to appear in: Stochastic Process. Appl. (SPA):   .
27. Ergodicity and local limits for stochastic local and nonlocal p-Laplace
    equations
    with Jonas M. Tölle
    to appear in: SIAM J. Math. Anal. (SIMA): .
28. Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
    with Michael Röckner
    to appear in: Transactions of the AMS: .
29. 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) .
30. 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: .
31. Synchronization by noise
    with Franco Flandoli, Michael Scheutzow
    to appear in: Probability Theory and Related Fields: .
32. Weak synchronization for isotropic flows
    with Michael Cranston, Michael Scheutzow
    to appear in: Discrete and Continuous Dynamical Systems – Series B (DCDS-B):   .
33. Synchronization by noise for order-preserving random dynamical systems
    with Franco Flandoli, Michael Scheutzow
    to appear in: The Annals of Probability:  .
34. Stability of solutions to stochastic partial differential equations
    with Jonas M. Tölle
    J. Differential Equations, 260 (2016), no. 6, 4973–5025: .
35. Singular-degenerate multivalued stochastic fast diffusion equations
    with Michael Röckner
    to appear in: SIAM J. Math. Anal. (SIMA): 
36. 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):  
37. Scalar conservation laws with multiple rough fluxes
    with Panagiotis E. Souganidis
    Commun. Math. Sci. 13 (2015), no. 6, 1569–1597.:  
38. Finite time extinction for stochastic sign fast diffusion and self-organized criticality
    Comm. Math. Phys., 335 (2015), no. 1, 309–344:  .
39. Jain-Monrad criterion for rough paths
    with Peter K. Friz, Archil Gulisashvili, Sebastian Riedel
    to appear in: The Annals of Probability, (2014): .
40. Multi-valued, singular stochastic evolution inclusions.
    with Jonas M. Tölle
    J. Math. Pures Appl. (JMPA) (9) 101 (2014), no. 6, 789–827: , arXiv:  .
41. Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise.
    The Annals of Probability, 42 (2014), no. 2, 818–864.:  .
42. Finite speed of propagation for stochastic porous media equation.
    SIAM J. Math. Anal. (SIMA), 45 (2013), no. 5, 2734–2766.:  .
43. Random attractors for singular stochastic partial differential equations.
    J. Differential Equations (JDE), 255 (2013), no. 3, 524–559:  .
44. Random Attractors for Degenerate Stochastic Partial Differential Equations.
    J. Dynam. Differential Equations (JDDE), 25(1) (2013), 121-157.  
45. Strong Solutions for Stochastic Partial Differential Equations of Gradient Type.
    J. Funct. Anal. (JFA), 263(8)(2012),2355-2383. 
46. 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.
47. 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. 
48. 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

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

Conference and workshop proceedings

51. 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
52. 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

53. Spatial rough path lifts of stochastic convolutions
    with Peter Friz, Archil Gulisashvili, Sebastian Riedel
    available on arXiv:  .
54. 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)