Profile
Profile

Prof. Dr. Massimiliano Gubinelli

Hausdorff Chair

E-mail: gubinelli(at)iam.uni-bonn.de
Phone: +49 228 73 62359
Homepage: http://www.iam.uni-bonn.de/abteilung-gubinelli/home/
Room: 3.027
Location: Mathematics Center
Institute: Institute for Applied Mathematics
Research Areas: Research Area C
Research Area G

Academic Career

1998

Diploma in Physics, University of Pisa, Italy

2001 - 2006

Ricercatore (Assistant Professor), Institute of Applied Mathematics, University of Pisa, Italy

2003

PhD in Theoretical Physics, University of Pisa, Italy

2006 - 2008

Maître de conférences, Paris-Sud University, Orsay, France

2008 - 2011

Professeur des Universités (2ème classe), Paris Dauphine University, Paris, France

2011 - 2015

Professeur des Universités (1ère classe), Paris Dauphine University, Paris, France

2012 - 2015

Part-time Professor, École Polytechnique, Palaiseau, France

Since 2015

Hausdorff Chair (W3), University of Bonn

Research Profile

I’m interested in problems of mathematical physics in connection with stochastic analysis. More generally in the description and analysis of random influences in evolutionary systems inspired by physics. In recent years I’ve been working in developing Rough Path Theory, which is a set of ideas and tools which allows a detailed analysis of irregular signals on non-linear systems. I’ve generalised the original theory, introduced by T. Lyons, to a wider class of signals, Branched Rough Paths and proposed various other theories in order to handle more complex dynamics like those underlying parabolic and hyperbolic PDEs. Rough paths and their generalisations have inspired the theory of Regularity Structures, invented by Hairer to describe the local structure of solutions to singular PDEs of the kind appearing in mathematical physics: the Stochastic Quantisation Equation, the Kardar—Parisi—Zhang equation, the parabolic Anderson model. In a parallel development, in collaboration with Imkeller and Perkowski, I introduced tools of harmonic analysis also applicable to such singular SPDEs. In collaboration with Flandoli and Priola and subsequently with some PhD students we analysed the effect of random perturbation in non-linear infinite dimensional dynamics modelled by PDEs and we showed some situations where the presence of the noise improves the behaviour of solutions for hyperbolic and dispersive PDEs.

Research Projects and Activities

Project Blanc ANR ECRU “Explorations on rough paths”
Coordinator, 2009 - 2012

Projet Jeunes Chercheurs ANR MAGIX Mathématiques, Algèbre, Géométrie Exactes
Member, 2009 – 2012

Project B09 “Large scale modeling of non-linear microscopic dynamics via singular SPDEs”
within DFG Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”
Principal investigator

Contribution to Research Areas

Research Area G
My main research interest is the study of partial differential equations in presence of noise.
In collaboration with P. Imkeller and N. Perkowski we developed [1] a new method to define and analyze stochastic partial differential equation with quite singular non-linearities like the Kardar-Parisi-Zhang equation, the cubic reaction--diffusion model in three dimension with additive white noise and the parabolic Anderson model in two and three dimensions, among others. In the context of the KPZ equation together with N. Perkowski we proved [2] uniqueness of energy solutions [3] for the KPZ equation. Energy solutions is a flexible tool to prove universality of fluctuations in weakly asymmetric particle systems (see e.g. [4]).
Another line of research concerns the regularizing effects of the noise in the dynamics of partial differential equations. Together with F. Flandoli and E. Priola we gave one of the first examples of regularization by noise in the context of stochastic partial differential equations [5]. In collaboration with my former Ph.D. student Khalil Chouk [6] we proved regularization by stochastic modulation in nonlinear Schrödinger equations.
Research Area C
I studied path integral techniques in the analysis of certain quantum systems.
In particular, in collaboration with F. Hiroshima and J. Lörinczi [7] we explicitly constructed a measure on path space for the ground state of the renormalized Nelson Hamiltonian. This Hamiltonian describes the interaction of a Boson field with a quantum non-relativistic particle. With J. Lörinczi [8] we constructed path integral representations for the interaction of particles with bosonic vector quantum fields, making a link with the theory of stochastic currents and rough paths.
With H. Koch and T. Oh we started the investigation of singular stochastic hyperbolic PDEs [9] which can be seen as simple models mimicking the behavior of the renormalization for quantum fields in Minkowski space or as large scale descriptions of fluctuations for nonlinear waves.

Selected Publications

[1] Massimiliano Gubinelli, Peter Imkeller, Nicolas Perkowski
Paracontrolled distributions and singular PDEs
Forum Math. Pi , 3: : e6, 75
2015
DOI: 10.1017/fmp.2015.2
[2] Massimiliano Gubinelli, Nicolas Perkowski
Energy solutions of KPZ are unique
Journal of the American Mathematical Society
2017
DOI: 10.1090/jams/889
[3] M. Gubinelli, M. Jara
Regularization by noise and stochastic Burgers equations
Stoch. Partial Differ. Equ. Anal. Comput.
, 1: (2): 325--350
2013
DOI: 10.1007/s40072-013-0011-5
[4] Joscha Diehl, Massimiliano Gubinelli, Nicolas Perkowski
The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
Comm. Math. Phys. , 354: (2): 549--589
2017
DOI: 10.1007/s00220-017-2918-6
[5] F. Flandoli, M. Gubinelli, E. Priola
Well-posedness of the transport equation by stochastic perturbation
Invent. Math. , 180: (1): 1--53
2010
DOI: 10.1007/s00222-009-0224-4
[6] K. Chouk, M. Gubinelli
Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations
Comm. Partial Differential Equations , 40: (11): 2047--2081
2015
DOI: 10.1080/03605302.2015.1073300
[7] Massimiliano Gubinelli, Fumio Hiroshima, József Lörinczi
Ultraviolet renormalization of the Nelson Hamiltonian through functional integration
J. Funct. Anal. , 267: (9): 3125--3153
2014
DOI: 10.1016/j.jfa.2014.08.002
[8] Massimiliano Gubinelli, József Lörinczi
Gibbs measures on Brownian currents
Comm. Pure Appl. Math. , 62: (1): 1--56
2009
DOI: 10.1002/cpa.20260
[9] Massimiliano Gubinelli, Herbert Koch, Tadahiro Oh
Renormalization of the two-dimensional stochastic nonlinear wave equation
arXiv preprint arXiv:1703.05461, to appear in Trans. A.M.S.
2017
[10] Massimiliano Gubinelli, Nicolas Perkowski
KPZ reloaded
Comm. Math. Phys.
, 349: (1): 165--269
2017
DOI: 10.1007/s00220-016-2788-3
[11] Massimiliano Gubinelli, Samy Tindel
Rough evolution equations
Ann. Probab. , 38: (1): 1--75
2010
DOI: 10.1214/08-AOP437
[12] Massimiliano Gubinelli
Ramification of rough paths
J. Differential Equations , 248: (4): 693--721
2010
DOI: 10.1016/j.jde.2009.11.015
[13] M. Gubinelli
Controlling rough paths
J. Funct. Anal. , 216: (1): 86--140
2004
DOI: 10.1016/j.jfa.2004.01.002

Publication List

Editorships

• Electronic Journal of Probability (Associate Editor, since 2011)
• Electronic Communications in Probability (Associate Editor, since 2011)
• Discrete and Continuous Dynamical Systems A (2015 - 2017)
• Bernoulli Journal (Area Editor, since 2015)
• Annals of Applied Probability (since 2015)
• SIAM Journal of Mathematical Analysis (since 2015)

Awards

2003

Invited professor, Institut E. Cartan, Université Nancy 1, France

2013 - 2018

Junior member of the Institute Universitaire de France

Selected Invited Lectures

2009

Minicourse on “Rough Paths”, École Polytechnique, Palaiseau, France

2012

Minicourses in Marseille, Berlin and Rome

2014

Invited lecturer at Escola Brasileira de Probabilidade, Mambucaba, Brasil and at Centro Ennio de Giorgi, Pisa, Italy

2015

Winter school “Recent Breakthroughs in Singular SPDEs”, University of Milano-Bicocca, Italy

2016

CIME-EMS Summer school “Singular random dynamics”, Cetraro, Italy

2016

Minicourse at the school “Young women in probability”, Bonn

2018

Invited Section Lecture, International Congress of Mathematicians, Rio de Janeiro, Brasil

Selected PhD students

Khalil Chouk (2013): “Trois chemins controlés”,
now Postdoc, TU Berlin

Rémi Catellier (2014): “Perturbations irrégulières et systèmes différentiels rugueux”,
now Maître de Conferences, Université Nice Sophia Antipolis, France

Supervised Theses

  • PhD theses: 5, currently 3
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