Curriculum Vitæ

Nils BERGLUNDBorn 14 December 1969 in Lausanne, SwitzerlandSwedish citizenFrench and german (mother tongues), english

Professional address

MAPMO - CNRS, UMR 7349Federation Denis PoissonBatiment de Mathematiques, Route de ChartresUniversite d’Orleans, BP 6759F-45067 Orleans Cedex 2nils.berglund@univ-orleans.fr

http://www.univ-orleans.fr/mapmo/membres/berglund

Education and career

Education

• 2004: Habilitation in Mathematics, University of Toulon, “Equations differentiellesstochastiques singulierement perturbees”.Referees: Gerard Ben Arous, Anton Bovier, Lawrence E. Thomas.Jury: Jean-Francois Le Gall, Etienne Pardoux, Pierre Picco, Claude-Alain Pillet.

• 1998: PhD at Institut de Physique Theorique, ETH Lausanne, “Adiabatic DynamicalSystems and Hysteresis”.Adviser: Herve Kunz.Referees: Jean-Philippe Ansermet, Serge Aubry, Oscar E. Lanford III.

• 1993: Diploma in Physics at ETH Lausanne.Diploma thesis: “Billiards magnetiques”. Direction: Herve Kunz.

• 1988–1993: Physics studies at ETH Lausanne.

Positions held

• Since September 2007: Full professor (Professeur des universites), University of Orleans.Member of MAPMO (CNRS, UMR 7349, Federation Denis Poisson).

• September 2001 – August 2007: Assistant professor (Maıtre de Conference), Mathe-matics department, University of Toulon. Member of CPT (CNRS, UMR 6207).

• October 2000 – September 2001: Postdoc with teaching, Departement Mathematik,ETH Zurich.

• April – September 2000: TMR fellow at Weierstraß-Institut fur Angewandte Analysisund Stochastik (WIAS), Berlin.

• April 1999 – March 2000: Postdoc, Swiss National Fund, Georgia Institute of Tech-nology, Atlanta.

• September 1998 – March 1999: TMR fellow at Weierstraß-Institut fur AngewandteAnalysis und Stochastik (WIAS), Berlin.

• 1994 – 1998: Assistant at Institut de Physique Theorique, ETH Lausanne.

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Award

“Prix des doctorats 1999” of ETH Lausanne.

Funded projects

• Local correspondent for ANR project MANDy, Mathematical Analysis of NeuronalDynamics, Budget for Orleans: 25’000.– Euro (September 2009 – August 2012)

• Principal investigator of research project “ACI Jeunes Chercheurs, Modelisation sto-chastique de systemes hors equilibre”, French Ministry of Research, Budget: 70’000.–Euro (August 2004 – April 2008).

Invited stays (> 1 week), Sabbaticals

• December 2015: 1 week stay at University of Warwick.• February – August 2015: Sabbatical (Chercheur delegue au CNRS), at MAPMO,

Orleans (UMR 7349).• March 2015: 2 weeks stay at TU Vienna.• One-month stays at CRC 701, Universitat Bielefeld (’07, ’08, ’09, ’10, ’11, ’12, ’13,

’14).• One-month stays at the Weierstraß-Institut fur Angewandte Analysis und Stochastik

(WIAS), Berlin (2001, ’02, ’03, ’04, ’05, ’06).• March – September 2006: Sabbatical (Chercheur delegue au CNRS), at Centre de

Physique Theorique Marseille-Luminy (CPT, UMR 6207).• March – September 2005: Sabbatical (Chercheur delegue au CNRS), WIAS Berlin

and CPT Marseille.• April 1999: Invited stay with Jacques Laskar, Bureau des Longitudes, Paris.

Peer review activities

Referee for Alea, Annales Faculte Toulouse, Annales Institut Henri Poincare, Annals ofProbability, Chaos, Comm. Nonlinear Science Numerical Simul., Discrete Contin. Dyn.Syst. A, Electron. J. Differential Equations, Inverse Problems, J. Computational Dynam-ics, J. Dynam. Diff. Eq., J. Mathematical Biology, J. Mathematical Neuroscience, J. Non-linear Science, J. Phys. A, J. Theoretical Probability, Mathematics Computers Simulation,New J. Phys., Nonlinear Analysis, Nonlinearity, Physica A, Physica D, Physical Biology,Probability Theory Related Fields, SIAM J. Appl. Dynamical Systems (SIADS), SIAMJ. Appl. Math. (SIAP), SIAM J. Scient. Comp. (SISC), Stochastic Process. Appl., Syst.Contr. Letters.

I have also reviewed grant proposals for ANR (France), NSF (United States) and NWO(Netherlands).

Curriculum Vitæ – Nils Berglund 3

List of publications

Monograph

[30] N. Berglund, B. Gentz, Noise-Induced Phenomena in Slow–Fast Dynamical Systems.A Sample-Paths Approach (Springer, Probability and its Applications, 276+xivpages, 2005)http://www.univ-orleans.fr/mapmo/membres/berglund/book.html

Refereed papers

[29] N. Berglund, C. Kuehn, Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, Preprint (2015), arxiv/1504.02953,51 pages, to appear in Electronic J. Probability.

[28] N. Berglund, S. Dutercq, Interface dynamics of a metastable mass-conserving spa-tially extended diffusion, J. Statistical Physics 162:334–370 (2016).

[27] N. Berglund, Noise-induced phase slips, log-periodic oscillations, and the Gumbeldistribution, Preprint (2014), arxiv/1403.7393, 32 pages, to appear in MarkovProcesses Related Fields.

[26] N. Berglund, B. Gentz, C. Kuehn, From random Poincare maps to stochastic mixed-mode-oscillation patterns , J. Dynamics Differential Equations 27:83–136 (2015).

[25] N. Berglund, S. Dutercq, The Eyring-Kramers law for Markovian jump processeswith symmetries, J. Theoretical Probability Online first:1–40 (2015).

[24] N. Berglund, B. Gentz, On the noise-induced passage through an unstable periodicorbit II: General case, SIAM J. Math. Anal. 46:310–352 (2014).

[23] N. Berglund, B. Gentz, Sharp estimates for metastable lifetimes in parabolic SPDEs:Kramers’ law and beyond, Electronic J. Probability 18:no. 24,1–58 (2013).

[22] N. Berglund, Kramers’ law: Validity, derivations and generalisations, Markov Pro-cesses Relat. Fields 19:459–490 (2013).

[21] N. Berglund, D. Landon, Mixed-mode oscillations and interspike interval statisticsin the stochastic FitzHugh–Nagumo model, Nonlinearity 252:2303–2335 (2012).

[20] N. Berglund, B. Gentz, C. Kuehn, Hunting French Ducks in a Noisy Environment,J. Differential Equations 252:4786–4841 (2012).

[19] N. Berglund, B. Gentz, The Eyring-Kramers law for potentials with nonquadraticsaddles, Markov Processes Relat. Fields 16:549–598 (2010).

[18] N. Berglund, B. Gentz, Anomalous behavior of the Kramers rate at bifurcations inclassical field theories, J. Phys. A: Math. Theor. 42:052001 (9 pages) (2009).

[17] J.-P. Aguilar, N. Berglund, The effect of classical noise on a quantum two-levelsystem, J. Math. Phys. 49:102102 (23 pages) (2008).

[16] N. Berglund, B. Fernandez, B. Gentz, Metastability in interacting nonlinear stochas-tic differential equations I: From weak coupling to synchronisation, Nonlinearity20:2551–2581 (2007).

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[15] N. Berglund, B. Fernandez, B. Gentz, Metastability in interacting nonlinear stochas-tic differential equations II: Large-N regime, Nonlinearity 20:2583–2614 (2007).

[14] N. Berglund, B. Gentz, Universality of first-passage- and residence-time distributionsin nonadiabatic stochastic resonance, Europhys. Lett. 70:1–7 (2005).

[13] N. Berglund, B. Gentz, On the noise-induced passage through an unstable periodicorbit I: Two-level model, J. Statist. Phys. 114:1577–1618 (2004).

[12] N. Berglund, B. Gentz, Geometric singular perturbation theory for stochastic differ-ential equations, J. Differential equations 191:1–54 (2003).

[11] N. Berglund, B. Gentz, Metastability in simple climate models: Pathwise analysis ofslowly driven Langevin equations, Stoch. Dyn. 2:327–356 (2002).

[10] N. Berglund, B. Gentz, Beyond the Fokker–Planck equation: Pathwise control ofnoisy bistable systems, J. Phys. A 35:2057–2091 (2002).

[9] N. Berglund, B. Gentz, The effect of additive noise on dynamical hysteresis, Nonlin-earity 15:605–632 (2002).

[8] N. Berglund, B. Gentz, A sample-paths approach to noise-induced synchronization:Stochastic resonance in a double-well potential, Ann. Appl. Probab. 12:1419–1470(2002).

[7] N. Berglund, B. Gentz, Pathwise description of dynamic pitchfork bifurcations withadditive noise, Probab. Theory Related Fields 122:341–388 (2002).

[6] N. Berglund, T. Uzer, The averaged dynamics of the hydrogen atom in crossed electricand magnetic fields as a perturbed Kepler problem, Found. Phys. 31:283–326 (2001).

[5] N. Berglund, Control of dynamic Hopf bifurcations, Nonlinearity 13:225–248 (2000).

[4] N. Berglund, H. Kunz, Memory Effects and Scaling Laws in Slowly Driven Systems,J. Phys. A 32:15–39 (1999).

[3] N. Berglund, H. Kunz, Chaotic Hysteresis in an Adiabatically Oscillating DoubleWell, Phys. Rev. Letters 78:1691–1694 (1997).

[2] N. Berglund, A. Hansen, E.H. Hauge, J. Piasecki, Can a Local Repulsive PotentialTrap an Electron?, Phys. Rev. Letters 77:2149–2153 (1996).

[1] N. Berglund, H. Kunz, Integrability and Ergodicity of Classical Billiards in a Mag-netic Field, J. Stat. Phys. 83:81–126 (1996).

Book chapters

[31] N. Berglund, B. Gentz, Stochastic dynamic bifurcations and excitability, in C. Laing,G. Lord (Eds.), Stochastic Methods in Neuroscience (Oxford University Press, Ox-ford, 2009)

[32] N. Berglund, Bifurcations, scaling laws and hysteresis in singularly perturbed sys-tems, in B. Fiedler, K. Groger, J. Sprekels (Eds.), Equadiff ’99 (World Scientific,Singapore, 2000)

Curriculum Vitæ – Nils Berglund 5

[33] N. Berglund, K.R. Schneider, Control of dynamic bifurcations, in D. Aeyels, F.Lamnabhi-Lagarrigue, A. van der Schaft (Eds.), Stability and Stabilization of Non-linear Systems (Lecture Notes in Contr. and Inform. Sciences 246, Springer, Berlin,1999)

Proceedings

[34] N. Berglund, Stochastic dynamical systems in neuroscience, Oberwolfach Reports8:2290–2293 (2011). Proceedings of the miniworkshop Dynamics of Stochastic Sys-tems and their Approximation, Oberwolfach, Germany, August 2011.

[35] N. Berglund, Dynamic bifurcations: hysteresis, scaling laws and feedback control,Prog. Theor. Phys. Suppl. 139:325–336 (2000). Proceedings of the conference “Let’sface chaos through nonlinear dynamics”, Maribor, Slovenia, June/July 1999.

[36] N. Berglund, Classical Billiards in a Magnetic Field and a Potential, NonlinearPhenomena in Complex Systems 3:61–70 (2000). Proceedings of the conference“Let’s face chaos through nonlinear dynamics”, Maribor, Slovenia, June/July 1996.

[37] N. Berglund, On the Reduction of Adiabatic Dynamical Systems near EquilibriumCurves (1998). Proceedings of the conference “Celestial Mechanics, Separatrix Split-ting, Diffusion”, Aussois, France, June 1998.

Diploma thesis, Phd thesis, habilitation thesis

[38] N. Berglund, Equations differentielles stochastiques singulierement perturbees,Habilitation , University of Toulon (2004).http://www.univ-orleans.fr/mapmo/membres/berglund/hdr.html

[39] N. Berglund, Adiabatic Dynamical Systems and Hysteresis,Phd thesis 1800, ETH Lausanne (1998).http://www.univ-orleans.fr/mapmo/membres/berglund/these.html

[40] N. Berglund, Billards magnetiques, Diploma thesis, ETH Lausanne (1993).

Popularisation

[41] Probabiliser, Images des Mathematiques, CNRS, 2015

[42] Des canards dans mes neurones, Images des Mathematiques, CNRS, 2015

[43] Mathematiques du quotidien, Fete de la Science, Octobre 2014

[44] Qu’est-ce qu’une Equation aux Derives Partielles Stochastique ?, Images des Mathe-matiques, CNRS, 2014

[45] Mathematiques et biologie, Centre Galois, 2014

[46] Le climat et ses aleas, Microscoop CNRS, Hors-serie 22, 2013

[47] Notre univers est-il irreversible ?, Images des Mathematiques, CNRS, 2013

[48] Hasard et glaciations, Mathematiques de la Planete Terre, 2013

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[49] La probabilite d’extinction d’une espece menacee, Images des Mathematiques, CNRS,2013

[50] Rangez-moi ces bouquins !, Images des Mathematiques, CNRS, 2012

[51] Probabilites et physique statistique, Centre Galois, 2011

[52] Recreation mathematique : La suite de Fibonacci, USTV, 2005

Lecture notes

See http://www.univ-orleans.fr/mapmo/membres/berglund/teaching.html

Teaching

• 2007-2016: University of Orleans (lectures):

– Master of mathematics, 2nd year: Martingales and stochastic calculus– Master of mathematics, 2nd year: Stochastic processes– Master of mathematics, 2nd year: Preparation to French “Agregation” exams– Master of mathematics, 2nd year: Financial mathematics– Master of mathematics, 1st year: Probability theory– Master of mathematics, 1st year: Financial mathematics– Master of economics, 1st year: Data analysis– Master of economics, 1st year: Qualitative data analysis– Bachelor of economics, 3rd year: Linear optimisation– Bachelor of mathematics, 3rd year: Probability and statistics

• 2001-2007: University of Toulon:

– Lectures:

∗ Master of mathematics, 2nd year: Mathematical physics∗ Master of mathematics, 1st year: Probability theory∗ Master of mathematics, 1st year: Mathematical physics∗ Bachelor of economics, 3rd year: Probability∗ Engineering school, 3rd year: Stochastic processes∗ Engineering school, 2nd year: Financial mathematics∗ Bachelor in mathematics/computer science, 1st year: Algebra∗ Bachelor in physics, 1st year: Matrices

– Exercise classes:

∗ Master in mathematics, 1st year: Algebra∗ Bachelor in economics, 3rd year: Probability∗ Bachelor in mathematics/computer science, 2nd year: Analysis, geometry,

series∗ Bachelor in biology, 2nd year: Probability and statistics∗ Bachelor in biology, 1st year: Calculus

• 2000-2001: ETH Zurich (lectures):

– Geometrische Theorie dynamischer Systeme– Storungstheorie dynamischer Systeme

• 1994-1998: ETH Lausanne (exercise classes):

– General mechanics

Curriculum Vitæ – Nils Berglund 7

– General relativity and cosmology– Nonlinear phenomena and chaos– Electrodynamics

Thesis supervision

• PhD students:

– Manon Baudel (since 2014)– Sebastien Dutercq (2011–2015)– Damien Landon (2008–2012)– Jean-Philippe Aguilar (2005–2008)

• Diploma students:

– Sebastien Dutercq (2011)– Kanal Hun (2009)– Damien Landon (2008)– Antoine Loyer (2005, in collaboration with Barbara Gentz, Berlin)– Jean-Philippe Aguilar (2005)– Evelyne Vas (2004, in collaboration with Emmanuel Buffet, Dublin)

Habilitation committee• Jonathan Touboul (College de France, June 2015)

Doctoral committees• Christophe Poquet (Universite Paris Diderot, October 2013): reviewer• Julian Tugaut (Universite Nancy 1, July 2010): reviewer• Benedicte Aguer (Universite Lille 1, June 2010): reviewer• Guillaume Voisin (Orleans, December 2009): president of jury• Laurent Marin (Orleans, November 2009)

Research interests

• Probability theory: Stochastic differential equations, stochastic partial differentialequations, stochastic exit problem, metastability, large deviations, potential theory,regularity structures

• Dynamical Systems: Singular perturbation theory, dynamic bifurcations, mixed-mode oscillations, spatially extended systems, Hamiltonian systems, symplectic maps

• Applications: Statistical physics, classical and quantum open systems, neuro-science, climate models

My research interests mainly lie at the intersection of different fields, such as probabilitytheory and dynamical systems. These intersections lead to interesting new approachesand results. For instance, with Barbara Gentz I have developed new methods for thequantitative analysis of stochastic differential equations with several timescales, based ona combination of stochastic analysis and singular perturbation theory.

My current research projects mainly deal with the following themes:

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• Metastability in spatially extended systems: Characterisation of metastabletransitions (transition time, optimal transition path) in systems of coupled oscillatorsand stochastic partial differential equations. These problems require the extensionof known results to infinite-dimensional situations and situations with bifurcations.

• Applications in the neurosciences: The creation of action potentials (“spiking”)in neurons is often modeled by slow–fast ordinary and stochastic differential equa-tions. A particularly interesting feature is the phenomenon of excitability, in whichsmall deterministic or stochastic perturbations can trigger spikes. The distributionof interspike time intervals is still poorly understood in irreversible cases.

International collaborations

• Herve Kunz (Lausanne), Alex Hansen, Eivind Hauge (Trondheim), Jaroslav Piasecki(Warschau): Billiards.

• Herve Kunz (Lausanne): adiabatic dynamical systems.• Klaus Schneider (Berlin): control theory.• Turgay Uzer (Atlanta): Hamiltonian systems.• Barbara Gentz (Berlin/Bielefeld): stochastic differential equations.• Bastien Fernandez (Marseille): spatially extended systems.• Christian Kuehn (Vienna): fast–slow dynamical systems, regularity structures.• Giacomo Di Gesu (Paris), Hendrik Weber (Warwick): stochastic partial differential

equations.

These collaborations have been partly funded by the Swiss National Fund, the Euro-pean Union (TMR Project), the Forschungsinstitut Mathematik (FIM) at ETH Zurich, theEuropean Science Foundation (ESF), and the Deutsche Forschungsgemeinschaft (DFG).

Administration

Committees

• Since 2014: Responsible for communication committee - Mathematics, University ofOrleans.

• 2014: External member of appointment committee, University of Poitiers.• 2013: External member of appointment committee, University of Tours.• Since 2012: President of appointment committee, mathematics, Orleans.• 2010: External member of appointment committee, University of Poitiers.• 2010: External member of appointment committee, University Paris 7.• Since 2009: Elected member of appointment committee, mathematics, Orleans.• Since 2008: Elected member of institute committee, MAPMO–CNRS.• Since 2008: Member of thesis committee Orleans–Tours.• 2004-2007: Member of appointment committee, mathematics, University of Toulon.• 2005-2007: External member of appointment committee, mathematics, University Aix-

Marseille I.• 2005-2007: Nominated member of institute committee, CPT-CNRS.• 2004-2007: Responsible for teaching attribution, department of mathematics, Univer-

sity of Toulon.

Curriculum Vitæ – Nils Berglund 9

Organisation of seminars and conferences

• “Journees de Probabilites 2013”, Orleans (June 2013).• “Third meeting of the GDR Quantum Dynamics”, Orleans (February 2011).• “Fourth Workshop on Random Dynamical Systems”, Bielefeld (November 2010), with

Barbara Gentz.• Conference “Stochastic Models in Neuroscience”, CIRM, Marseille-Luminy (January

2010), with Simona Mancini and Michele Thieullen (Paris 6).• Responsible for institute seminar, MAPMO (since 2008), with Simona Mancini and

Aurelien Alvarez (since 2011), and Luc Hillairet (since 2013).• Conference “Maths and billiards” (Orleans, March 2008), with Stephane Cordier

(Orleans) and Emmanuel Lesigne (Tours).• Closing meeting of the ACI Jeunes Chercheurs “Modelisation stochastique de systemes

hors equilibre” (Orleans, February 2008).• “Journees de Probabilites 2007”(La Londe, September 2007), with K. Bahlali, Y.

Lacroix (Toulon), P. Picco (Marseille).• Minisymposium “Recent developments in the stochastic exit problem” at the interna-

tional conference ICIAM07 (Zurich, July 2007), with Samuel Herrmann (Nancy).• International conference QMath9 (Giens, September 2004), with J. Asch, J.-M. Bar-

baroux, J.-M. Combes, J.-M. Ghez (Toulon), A. Joye (Grenoble).• Colloquium “Equations Differentielles Stochastiques”, University of Toulon (2003),

with K. Bahlali, P. Briet (Toulon), P. Picco (Marseille).• “Seminaire de Dynamique Quantique et Classique”, CPT (2002).

Talks and conferences (since 1999)

Some slides are available athttp://www.univ-orleans.fr/mapmo/membres/berglund/restalks.html

Invited talks at conferences

• 2.2.2016. “Computational statistics and molecular simulation”, Paris, Sharp esti-mates on metastable lifetimes for one- and two-dimensional AllenCahn SPDEs (Invi-tation: Arnaud Guyader, Tony Lelievre, Gabriel Stoltz).

• 5.11.2015. “8th Workshop on random dynamical systems”, Bielefeld, Regularity struc-tures and renormalisation of FitzHugh–Nagumo SPDEs in three space dimensions (In-vitation: Barbara Gentz, Peter Imkeller).

• 8.7.2015. “Equadiff 2015”, Lyon, On FitzHugh–Nagumo SDEs and SPDEs (Invita-tion: Hans Crauel, Ale Jan Homburg).

• 16.6.2015. ENS Lyon, “Statistical mechanics and computation of large deviation ratefunctions”, Metastability in systems of coupled multistable SDE (Invitation: FreddyBouchet).

• 20.3.2015. Paris, Institut des Systemes Complexes, ANR project meeting, A tool-box to quantify effects of noise on slow-fast dynamical systems (Invitation: MathieuDesroches).

• 12.12.2014. “7th Workshop on random dynamical systems”, Bielefeld, Extreme-valuetheory and the stochastic exit problem (Invitation: Barbara Gentz).

• 18.11.2014. “Multi-scale models, slow-fast differential equations, averaging in ecol-ogy”, Centre Bernoulli, Lausanne, Noise-induced transitions in slowfast dynamicalsystems (Invitation: Mathieu Desroches).

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• 28.7.2014. “37th Conference on Stochastic Processes and their Applications”, BuenosAires, Some rigorous results on interspike interval distributions in stochastic neuronmodels (Invitation: Jonathan Touboul).

• 8.7.2014. “10th AIMS Conference on Dynamical Systems, Differential Equations andApplications”, Madrid, A group-theoretic approach to metastability in networks ofinteracting SDEs (Invitation: Georgi Medvedev).

• 28.1.2014. “Inhomogeneous Random Systems”, Institut Henri Poincare, Paris, Noise-induced phase slips, log-periodic oscillations and the Gumbel distribution (Invitation:Giambattista Giacomin).

• 31.10.2013. “6th Workshop on random dynamical systems”, Bielefeld, From ran-dom Poincare maps to stochastic mixed-mode-oscillation patterns (Invitation: BarbaraGentz).

• 6.6.2013. “Workshop on Slow-Fast Dynamics: Theory, Numerics, Application to Lifeand Earth Sciences”, CRM, Barcelona, Random Poincare maps and noise-inducedmixed-mode-oscillation patterns (Invitation: Mathieu Desroches).

• 8.2.2013. “5eme Rencontre du GDR Dynamique Quantique”, Lille, Irreversible diffu-sions and non-selfadjoint operators: Results and open problems (Invitation: StephanDe Bievre, Gabriel Riviere).

• 5.10.2012. “5th Workshop on Random Dynamical Systems”, Bielefeld, Some applica-tions of quasistationary distributions to random Poincare maps (Invitation: BarbaraGentz).

• 5.7.2012. “Random models in neuroscience”, Universite Paris 6, Some results oninterspike interval statistics in conductance-based models for neuron action potentials(Invitation: Michele Thieullen).

• 21.3.2012. “Critical Transitions in Complex Systems”, Imperial College London,Sample-path behaviour of noisy systems near critical transitions (Invitation: JeroenLamb, Greg Pavliotis, Martin Rasmussen).

• 12.12.2011. EPSRC Symposium Workshop “Multiscale Systems: Theory and Ap-plications”, Warwick Mathematics Institute, Canards, mixed-mode oscillations andinterspike distributions in stochastic systems (Invitation: Martin Hairer, Greg Pavli-otis, Andrew Stuart).

• 2.11.2011. “Stochastic Dynamics in Mathematics, Physics and Engineering”, ZiF,Bielefeld, Does noise create or suppress mixed-mode oscillations? (Invitation: BarbaraGentz, Max-Olivier Hongler, Peter Reimann).

• 22.8.2011. “Dynamics of Stochastic Systems and their Approximation”, Mathematis-ches Forschungszentrum Oberwolfach, Stochastic dynamical systems in neuroscience(Invitation: Evelyn Buckwar, Barbara Gentz, Erika Hausenblas).

• 21.7.2011. ICIAM 2011, Vancouver, On the Interspike Time Statistics in the Stochas-tic FitzHugh–Nagumo Equation (Invitation: Jinqiao Duan, Barbara Gentz).

• 26.1.2011. “Inhomogeneous Random Systems”, Institut Henri Poincare, Paris, TheKramers law: Validity, derivations and generalizations (Invitation: Christian Maes).

• 2.12.2010. “Open quantum systems”, Institut Fourier, Grenoble, Metastability in aclass of parabolic SPDEs (Invitation: Alain Joye).

• 19.11.2009. 3rd Workshop “Random Dynamical Systems”, Universitat Bielefeld,Metastability for Ginzburg-Landau-type SPDEs with space-time white noise (Invita-tion: Barbara Gentz).

• 14.5.2009. “Deterministic and Stochastic Modeling in Computational Neuroscienceand Other Biological Topics”, CRM, Barcelona, Stochastic models for excitable systems(Invitation: Antony Guillamon).

Curriculum Vitæ – Nils Berglund 11

• 12.3.2009. “Journees Systemes Ouverts”, Institut Fourier, Grenoble, Metastabilityfor the Ginzburg–Landau equation with space-time noise (Invitation: Alain Joye).

• 18.11.2008. 2nd Workshop “Random Dynamical Systems”, Universitat Bielefeld,Metastability in systems with bifurcations (Invitation: Barbara Gentz).

• 19.6.2008. Workshop “Applications of spatio-temporal dynamical systems in biol-ogy”, Laboratoire Dieudonne, Nice, Metastability in a system of coupled nonlineardiffusions (Invitation: Elisabeth Pecou).

• 30.11.2007. Workshop “Random Dynamical Systems”, Universitat Bielefeld, Meta-stable lifetimes and optimal transition paths in spatially extended systems (Invitation:Barbara Gentz).

• 23.2.2007. Workshop “Multiscale Analysis and Computations in Stochastic Differ-ential Equation Modelling 2007”, University of Sussex, Noise-induced phenomena inslow–fast dynamical systems (Invitation: Greg Pavliotis).

• 12.10.2006. Conference “Chaos and Complex Systems”, Novacella, Desynchronisa-tion of coupled bistable oscillators perturbed by additive noise (Invitation: Marco Rob-nik).

• 27.6.2006. AIMS conference “Dynamical Systems and Differential Equations”, Poi-tiers, Metastability and stochastic resonance in slow–fast systems with noise (Invita-tion: Jianzhong Su).

• 9.6.2005. Stochastic Partial Differential Equations and Climatology, ZIF, UniversitatBielefeld, Geometric singular perturbation theory for stochastic differential equations(Invitation: Friedrich Gotze, Yuri Kondratiev, Michael Rockner).

• 31.5.2005. Period of Concentration, Stochastic climate models, Max-Planck Institut,Leipzig, Geometric singular perturbation theory applied to stochastic climate models(Invitation: Peter Imkeller).

• 19.5.2005. Period of Concentration, Metastability, ageing, and anomalous diffusion,Max-Planck Institute, Leipzig, Metastability in irreversible diffusion processes andstochastic resonance (Invitation: Anton Bovier).

• 22.6.2004. Coupled Map Lattices 2004 , Institut Henri Poincare, Paris, Special sessionon effect of noise on synchronization (Invitation: Bastien Fernandez).

• 28.4.2004. European Geophysical Union meeting (EGU04), Nice, Universal propertiesof noise-induced transitions (Invitation: Daniel Scherzer).

• 31.5.2003. 4th European Advanced Studies Conference, Novacella, On the noise-induced passage through an unstable periodic orbit (Invitation: Marko Robnik).

Invited talks at seminars

• 17.12.2015. TU Darmstadt, Oberseminar Stochastik, Synchronisation, noise-inducedphase slips and the Gumbel distribution (Invitation: Volker Betz).

• 9.12.2015. Warwick, Probability Seminar, An Eyring–Kramers formula for one-dimensional parabolic SPDEs with space-time white noise (Invitation: Hendrik We-ber).

• 13.11.2015. Strasbourg, Seminaire de Calcul stochastique, Structures de regulariteet renormalisation d’EDPS de FitzHugh–Nagumo (Invitation: Nicolas Juillet).

• 26.3.2015. Le Havre, Seminaire du LMAH, Phenomenes induits par le bruit dans lessystemes dynamiques lent–rapides (Invitation: Benjamin Ambrosio).

• 12.3.2015. LMA Poitiers, Seminaire de probabilites, statistique et applications, Dis-tribution de spikes pour des modeles stochastiques de neurones et chaınes de Markova espace continu (Invitation: Hermine Bierme).

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• 4.3.2015. TU Vienna, Institute for Analysis und Scientific Computing, An Eyring–Kramers formula for parabolic SPDEs with space-time white noise (Invitation: Chris-tian Kuehn).

• 5.12.2014. LMPT Tours, Seminaire de probabilites et theorie ergodique, Distributionde spikes pour des modeles stochastiques de neurones et chaınes de Markov a espacecontinu (Invitation: Kilian Raschel).

• 16.4.2014. Ruhr-Universitat Bochum, Synchronization, noise-induced phase slips andextreme-value theory (Invitation: Christof Kulske).

• 13.3.2014. Evry, Seminaire de Probabilites et mathematiques financieres, Problme desortie stochastique travers une orbite instable (Invitation: Dasha Loukinaova).

• 9.12.2013. IRMAR, Universite de Rennes, Seminaire de Probabilites, Chaınes deMarkov a espace continu, distributions quasi-stationnaires, et diffusions irreversibles(Invitation: Jean-Christophe Breton).

• 1.2.2013. Rhein-Main Kolloquium Stochastik, Gutenberg-Universitat Mainz, Quan-tifying neuronal spiking patterns using continuous-space Markov chains (Invitation:Reinhard Hopfner).

• 2.11.2012. Heriot-Watt University, Edinburgh, Quantifying the effect of noise onneuronal spiking patterns (Invitation: Sebastien Loisel, Gabriel Lord).

• 7.2.2012. Seminaire de Probabilites, Institut de Mathematiques de Toulouse, Oscil-lations multimodales et chasse aux canards stochastiques (Invitation: Clement Pelle-grini).

• 5.5.2011. Groupe de travail modelisation, LPMA, Paris 6, Oscillations multimodalesdans les equations differentielles stochastiques (Invitation: Gianbattista Giacomin).

• 15.3.2011. Groupe de travail Neuromathematiques et modeles de perception, IHP,Paris, Geometric singular perturbation theory for stochastic differential equations withapplications to neuroscience (Invitation: Alessandro Sarti).

• 14.3.2011. Groupe de travail mathematiques et neurosciences, IHP, Paris, Chasse auxcanards en environnement bruite (Invitation: Michele Thieullen).

• 11.2.2011. Laboratoire Dieudonne, Nice, Chasse aux canards en environnement bruite(Invitation: Jean-Marc Gambaudo, Bruno Marcos).

• 18.1.2011. Institut du Fer a Moulin, Paris, Quantifying the effect of noise on oscil-latory patterns (Invitation: Jacques Droulez).

• 12.11.2010. Seminaire de probabilites, EPF Lausanne, Metastability in a class ofparabolic SPDEs (Invitation: Charles Pfister).

• 18.6.2010. IGK-Seminar, Bielefeld, Stochastic differential equations in neuroscience(Invitation: Sven Wiesinger).

• 1.4.2010. Groupe de travail systemes dynamiques, LJLL, Paris 6, Systemes lents-rapides perturbes par un bruit — Applications dans les neurosciences (Invitation:Jean-Pierre Francoise, Alain Haraux).

• 26.6.2009. Journee en l’honneur d’Herve Kunz, EPF Lausanne, Metastabilite dans desequations de Ginzburg–Landau avec bruit blanc spatiotemporel (Invitation: VincenzoSavona).

• 27.11.2008. Groupe de Travail Modelisation de l’Evolution du Vivant, Ecole Poly-technique, Paris/Palaiseau, Equations differentielles stochastiques lentes–rapides (In-vitation: Sylvie Meleard).

• 27.10.2008. Theoretical Physics Seminar, KU Leuven, Kramers rate theory at bifur-cations (Invitation: Christian Maes).

• 24.6.2008. Seminaire de physique mathematique, Lille, L’effet d’un bruit classiquesur un systeme quantique a deux niveaux (Invitation: Stephan de Bievre).

Curriculum Vitæ – Nils Berglund 13

• 27.5.2008. Seminaire de probabilites, Paris 6 & 7, Metastabilite dans un systeme dediffusions couplees (Invitation: Jean Bertoin).

• 10.3.2008. Seminaire de probabilites Nord-Ouest, Angers, Metastabilite dans un sys-teme de diffusions couplees (Invitation: Loıc Chaumont).

• 13.6.2007. Seminar on Stochastic Analysis, University of Warwick, Metastability ina chain of coupled nonlinear diffusions (Invitation: Martin Hairer).

• 18.4.2007. Berliner Kolloquium Wahrscheinlichkeitstheorie, WIAS Berlin, Metastabi-lity in a chain of coupled nonlinear diffusions (Invitation: Anton Bovier).

• 2.4.2007. Seminaire de Probabilites, Universite Rennes 1, Metastabilite dans un sys-teme de diffusions non-lineaires couplees (Invitation: Ying Hu, Florent Malrieu).

• 10.1.2007. Oberseminar Math. Statistik & Wahrscheinlichkeitstheorie, UniversitatBielefeld, Metastability in a system of interacting nonlinear diffusions (Invitation:Barbara Gentz).

• 7.12.2006. Seminaire de probabilites, Nancy, Synchronisation de diffusions couplees(Invitation: Samy Tindel).

• 15.11.2006. Seminar on Stochastic processes, Zurich, Metastability in a system ofinteracting diffusions (Invitation: Erwin Bolthausen).

• 10.3.2006. Seminaire de probabilites, LATP Marseille, Synchronisation de diffusionscouplees (Invitation: Laurent Miclo).

• 18.11.2003. Institut Fourier, Grenoble, Passage sous l’effet d’un bruit a travers uneorbite periodique instable (Invitation: Alain Joye).

• 29.10.2003. WIAS Berlin, On the noise-induced passage through an unstable periodicorbit (Invitation: Barbara Gentz).

• 8.11.2002. LATP, Marseille, Theorie geometrique des perturbations singulieres pourequations differentielles stochastiques (Invitation: Yvan Velenik).

• 26.6.2002. Humboldt-Universitat zu Berlin, Geometric singular perturbation theoryfor stochastic differential equations (Invitation: Peter Imkeller).

• 8.5.2001. EPFL, Resonance stochastique et hysterese (Invitation: Charles Pfister).• 21.3.2001. CPT Marseille, Resonance stochastique et synchronisation induite par

bruit (Invitation: Claude-Alain Pillet).• 14.11.2000. Institut Fourier, Grenoble, L’effet de bruit additif sur les bifurcations

dynamiques (Invitation: Alain Joye).• 18.10.2000. WIAS Berlin, Dynamic pitchfork bifurcations with additive noise (Invi-

tation: Barbara Gentz).• 5.9.2000. Universitat Augsburg, The effect of additive noise on dynamic pitchfork

bifurcations (Invitation: Fritz Colonius).• 23.9.1999. University of Victoria, BC, Hysteresis and scaling laws in dynamic bifur-

cations (Invitation: Florin Diacu).• 6.5.1999. Bureau des Longitudes, Paris, Bifurcations dynamiques (Invitation: Jac-

ques Laskar).• 24.2.1999. Universitat Augsburg, Control of dynamic Hopf bifurcations (Invitation:

Fritz Colonius).

Minicourses

• November 2015. Nice, Laboratoire Dieudonne, 2 lecture, Modeles neuronaux etstructures de regularite (Invitation: Francois Delarue).

• October 2011. CIRM, Marseille Luminy, 5 lectures, Bifurcations in stochastic sys-tems with multiple timescales (Invitation: Olivier Faugeras).

14 February 23, 2016

• April 2011. IGK, Bielefeld, 5 lectures, Stochastic differential equations with multipletimescales (Invitation: Barbara Gentz).

Talks at local seminars

• 6.2015. Tours, Seminaire tournant de probabilites et statistiques, Theorie des valeursextremes et probleme de sortie stochastique.

• 4.2015. Seminaire du MAPMO, Orleans, Introduction aux EDPS et aux structures deregularite.

• 2.2013. Groupe de travail probabilites, Orleans, Markov loops, paths and fields I-III .• 14.12.2010. Evaluation AERES, Orleans, Quantifying the effect of noise on oscilla-

tory patterns.• 10/12.2008. Groupe de travail de probabilites, Orleans, Metastablite.• 15.11.2007. Journee de la Federation Denis Poisson, Orleans, EDS lentes-rapides,

metastablite et resonance stochastique.• 2/3.2007. Groupe de travail Systemes Ouverts, CPT Marseille, Grandes deviations

et processus d’exclusion I-III .• 10.11.2006. Open Systems Days, Marseille, Metastabilite dans une chaıne classique

bruitee.• 17.11.2003. Colloquium Equations Differentielles Stochastiques, Universite de Tou-

lon, Modeles climatiques stochastiques.• 14.11.2003. Seminaire de Travail en Physique Mathematique, CPT Marseille, L’eva-

sion stochastique a travers une orbite periodique instable.• 2.10.2003. Colloquium Equations Differentielles Stochastiques, Universite de Toulon,

Mouvement Brownien, integration stochastique.• 13.6.2003. Seminaire de Travail en Physique Mathematique, CPT Marseille, Equa-

tions differentielles stochastiques singulierement perturbees.• 6.12.2002. Seminaire de Travail en Physique Mathematique, CPT Marseille, Equa-

tions differentielles singulierement perturbees.• 25.1.2002. Colloquium du Groupe 3, CPT Marseille, Atome d’hydrogene en champs

croises.• 6.6.2001. Seminar on Stochastic Processes, ETH/Universitat Zurich, Stochastic res-

onance and hysteresis.• 30.11.2000. Talks in Mathematical Physics, ETH Zurich, Slowly time-dependent sys-

tems with additive noise: delay and resonance phenomena.• 10.12.1999. Seminaire de Physique Theorique, EPFL, Atome d’hydrogene en champs

croises.• 11.1999. Physics Colloquia, Georgia Tech, Adiabatic Dynamical Systems and Hys-

teresis.

Other talks at conferences

• 27.5.2015. Journees de Probabilites, Toulouse: Structures de regularite et renormal-isation dEDPS de FitzHugh–Nagumo.

• 27.5.2014. Journees de Probabilites, CIRM, Marseille: Theorie des valeurs extremeset probleme de sortie stochastique.

• 9.6.2009. Journees de Probabilites, Poitiers: Metastabilite dans des EDPS du typeGinzburg–Landau.

• 4.9.2008. Journees de Probabilites, Lille: Metastabilite et loi de Kramers pour lesdiffusions dans des potentiels a selles non-quadratiques.

Curriculum Vitæ – Nils Berglund 15

• 4.6.2007. CCT07, Marseille: Metastability in interacting nonlinear stochastic differ-ential equations.

• 20.9.2006. Journees de Probabilites, CIRM, Marseille: Synchronisation dans les dif-fusions couplees.

• 5.9.2005. Journees de Probabilites, Nancy: Distribution des temps de residence pourla resonance stochastique.

• 13.9.2004. QMath9, Giens: On the stochastic exit problem — irreversible case.• 8.9.2004. Journees de Probabilites, CIRM, Marseille: Le probleme de sortie d’une

diffusion d’un domaine borne par une orbite periodique instable: Au-dela des grandesdeviations.

• 6.7.2002. Let’s face chaos through nonlinear dynamics, Maribor: Stochastic reso-nance: Some new results.

• 5.6.2000. Second Nonlinear Control Network (NCN) Workshop, Paris: Effect of noiseand open loop control on dynamic bifurcations.

• 2.8.1999. Equadiff’99, Berlin: Bifurcations, scaling laws and hysteresis in singularlyperturbed systems.

• 30.6.1999. Let’s face chaos through nonlinear dynamics, Maribor: Dynamic bifurca-tions: hysteresis, scaling laws, and feedback control .

• 15.3.1999. Nonlinear Control Network (NCN) Workshop “Stability and Stabiliza-tion”, Ghent: Effect of noise and open loop control on dynamic bifurcations.

Other participation in conferences

• 3–5.6.2015. Journees en l’honneur de Marc Yor, Paris.• 2–6.2.2015. 7eme Rencontre du GDR Dynamique Quantique, Nantes.• 27–28.1.2015. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 5–7.2.2014. 6eme Rencontre du GDR Dynamique Quantique, Roscoff.• 15–17.7.2013. Stochastics and Real World Models, IGK, Bielefeld.• 22–23.1.2013. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 18–22.6.2012. Journees de Probabilites, Roscoff.• 8–10.2.2012. 4eme Rencontre du GDR Dynamique Quantique, IMT Toulouse.• 24–25.1.2012. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 3–6.8.2011. Limit theorems in Probability, Statistics and Number Theory, Bielefeld.• 20–24.6.2011. Journees de Probabilites, Nancy.• 25–27.11.2010. Quantum dynamics, CIRM, Marseille.• 14–16.10.2010. Workshop on Probabilistic Techniques in Statistical Mechanics, Berlin.• 24–26.3.2010. Second meeting of the GDR “Quantum Dynamics”, Dijon.• 26–27.1.2010. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 1–8.6.2008. Hyperbolic dynamical systems, Erwin Schodinger Insitute, Wien.• 22–23.1.2008. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 5–9.11.2007. Entropy, turbulence and transport, Institut Henri Poincare, Paris.• 9–13.7.2007. ICIAM 07, Zurich.• 23–24.1.2007. Inhomogeneous Random Systems, Institut Henri Poincare, Paris.• 15–19.1.2007. Vth Meeting of the GDRE Mathematics and Quantum Physics, CIRM,

Marseille.• 4–29.7.2005. Mathematical Statistical Physics, Les Houches.• 16–19.3.2005. Workshop random spatial models from physics and biology, Berlin.• 18–23.4.2004. Equilibrium and Dynamics of Spin Glasses, Ascona.• 3–28.3.2003. Statistical Mechanics and Probability Theory, CIRM, Marseille.

16 February 23, 2016

• 27–31.5.2002. Aspects Mathematiques des Systemes Aleatoires et de la MecaniqueStatistique, CIRM, Marseille.

• 9–11.7.2001. Workshop on stochastic climate models, Chorin.• 2–7.7.2001. Stochastic analysis, Berlin.• 11–17.3.2001. Stochastics in the sciences, Oberwolfach.• 5–9.7.1999. ICIAM 99, Edinburgh.

Popularisation

• 5.10.2014. Fete de la Science, Orleans: Mathematiques du Quotidien.• 20.6.2014, 27.6.2014, 19.6.2015 and 26.6.2015. Centre Galois, Orleans: Mathematiques

et Biologie.• 22.6.2011, 29.6.2011 and 28.6.2013. Centre Galois, Orleans: Probabilites et Physique

Statistique.

Curriculum Vitæ – Nils Berglund 17

Description of Research Activities

Past Research Activities

Hamiltonian Systems

With Herve Kunz, as well as Alex Hansen, Eivind Hauge and Jarek Piasecki, I haveanalysed dynamical systems of the billiard type. These systems are described by symplecticmaps of the plane, which can display a great variety of behaviours, from integrable tochaotic.

In particular, we have used methods from perturbation theory (Kolmogorov–Arnold–Moser theory, Nekhoroshev theory) to describe the dynamics of billiards in magneticfields [BK96] and in electromagnetic fields [BHHP96, Ber00b].

With Turgay Uzer, I have studied the dynamics of atoms in electromagnetic fields,using methods of Celestial Mechanics (Delaunay variables, averaging) [BU01].

Singular perturbation theory

With Herve Kunz, I have studied slow–fast ordinary differential equations, and systemswith slowly time-dependent parameters. These are described by singularly perturbedequation of the form

εx = f(x, t) , x ∈ R n

where ε is a small parameter.Using methods from analytic and geometric singular perturbation theory [Fen79], we

have analysed in particular the following aspects:

• The dynamics in a neighbourhood of equilibrium branches t 7→ x?(t), f(x?(t), t) = 0.Fenichel’s geometrical theory shows the existence of invariant manifolds of the formx = x(t, ε), x(t, ε) = x?(t) +O(ε). We have shown, in particular, that these manifoldscan be approximated by asymptotic expansion with exponential accuracy.

• The assumptions for the existence of invariant manifolds break down in the vicin-ity of bifurcation points, that is in points (x?(t0), t0) in which the Jacobian matrix∂xf(x?(t0), t0) admits eigenvalues with vanishing real part. One speaks in this case ofa dynamic bifurcation. We have developed a method allowing to determine the scalingbehaviour of solutions in a neighbourhood of these points, in a very general setting,using the bifurcation point’s Newton polygon [BK99, Ber01, Ber98, Ber00d, Ber00a].This method also allows to describe the scaling behaviour of hysteresis cycles [Ber01].

• In a mechanical system, we have proved the existence of chaotic hysteresis, which canbe described by geometrical methods [BK97].

With Klaus Schneider, I have applied these methods to problems of feedback controlof dynamic bifurcations [Ber00c, BS99].

Slow–fast stochastic differential equations

With Barbara Gentz, I have studied the pathwise behaviour of solutions of stochasticdifferential equations with two time scales, of the form

dxt =1

εf(xt, yt, ε)dt+

σ√εF (xt, yt, ε)dWt , x ∈ R n ,

dyt = g(xt, yt, ε)dt+ σ′G(xt, yt, ε)dWt , y ∈ Rm ,

18 February 23, 2016

where ε, σ and σ′ are small parameters. Such equations describe systems with fast andslow variables, perturbed by random noise. They play an important role in applications inPhysics (e.g. hysteresis in ferromagnets, Josephson junctions), in Climatology (models ofthe thermohaline circulation, Ice Ages), and biology (neural dynamics, generic regulatorynetworks).

The monograph [BG06] describes a new, constructive method for the quantitativedescription of the sample-path behaviour of such equations. This approach combinesmethods from singular perturbation theory (Fenichel’s geometric theory [Fen79], asymp-totic expansions) with methods from stochastic analysis (large deviations [FW98], the exitproblem, properties of martingales). A large part of the book is dedicated to applicationsof this approach to concrete problems from Physics, Climatology and Biology.

The main steps of our approach are the following:

• A slow manifold of the systems is a set of the form {(x?(y), y) ∈ R n+m : f(x?(y), y) =0, y ∈ D0}. It is stable if all eigenvalues of the Jacobian matrix ∂xf(x?(y), y) havenegative real parts, bounded away from zero, for all y ∈ D0. In this case one constructsa neighbourhood B(h) of the manifold, where hmeasures the size of the neighbourhood,such that an estimate of the form

C−(t, ε) e−κ−h2/2σ2

6 Px?(y0),y0

{τB(h) < t

}6 C+(t, ε) e−κ+h

2/2σ2

holds for the first-exit time τB(h) of paths from B(h). This implies that paths arelikely to spend exponentially long time spans in B(h), as long as h is chosen somewhatlarger than σ.

• In case the slow manifold in unstable, one can show that sample paths typically leavea vicinity of the manifold in a time of order ε|log σ|.

• The slow manifold admits bifurcation points whenever eigenvalues of the Jacobianmatrix ∂xf(x?(y), y) have a vanishing real part at some y ∈ D0. In such cases, theslow manifold can fold back on itself, or split into several manifolds. The analysis ofsuch cases proceeds in two steps. First, one reduces the problem to a lower-dimensionalone by projection on the centre manifold. Next one carries out a local analysis of thereduced dynamics. Using informations on the deterministic dynamics, partitions ofthe time interval, and the strong Markov property, one can describe the behaviour oftypical paths. Depending on the situations, paths choose to follow one of the new slowmanifolds, our jump to a remote manifold.

In addition to the general theory [BG03], we have described in particular the followingphenomena and applications:

• Dynamic bifurcations with noise: Pitchfork bifurcation [BG02d], saddle–node [BG02b]and Hopf bifurcation [BG06, Chapter 5]. A remarkable fact is that the noise canfacilitate transitions between manifolds, so that these can happen earlier as in thedeterministic case.

• Stochastic resonance: [BG02e, BG04, BG05, BG14]. This phenomenon appears whena deterministic periodic perturbation is combined with the noise, yielding almost pe-riodic transitions between stable manifolds, which would be impossible in the absenceof noise. The analysis of the distribution of waiting times between transitions requiredan extension of the classical Wentzell–Freidlin theory [FW98] to systems with periodicorbits.

• Hysteresis: Effect of noise on scaling laws of the area of hysteresis loops [BG02b].• Models from Physics: In particular from Solid State Physics [BG02a].• Climate models: Models of Ice Ages and the Gulf Stream [BG02c].

Curriculum Vitæ – Nils Berglund 19

• Application in Biology: Models of neurons, such as the Hodgkin–Huxley equations, inwhich noise is responsible for the phenomenon of excitability [BG06, Chapter 6].

• Quantum dynamics: The effect of classical noise on certain quantum systems has beenstudied in the PhD thesis of Jean-Philippe Aguilar [AB08].

Effect of noise on spatially extended systems

With Barbara Gentz and Bastien Fernandez I have examined spatially extended systemswith stochastic perturbations, in the framework of the research project “Modelisationstochastique de phenomenes hors equilibre” (supported by the French Ministry of Researchby a contract of 70’000 Euros). Such systems model in particular the influence of differentheat reservoirs, which influence transport properties through physical systems [EPRB99].

In [BFG07a, BFG07b], we examine the dynamics of a chain of bistable coupled units,described by a diffusion on R Z /NZ of the form

dxσi (t) = f(xσi (t))dt+γ

2

[xσi+1(t)− 2xσi (t) + xσi−1(t)

]dt+ σdWi(t) .

Here bistability is created by the form of f , e.g. the choice f(x) = x − x3 causes thesystem to hesitate between values +1 and −1. On the other hand, the coupling term withnearest neighbours compels the units to cooperate, yielding a tendency of local systemsto synchronise. The system has two fundamental states, given by I+ = (1, 1, . . . , 1) andI− = −I+. The noise triggers transitions between these fundamental states, and possiblyother existing metastable states.

With the help of Wentzell–Freidlin theory, we were able to prove the following:

• At weak coupling γ, the system has 2N local (metastable) equilibrium states. Transi-tions between these states are similar to those of an Ising model with Glauber dynam-ics, indeed they occur through the growth of a droplet of one phase within the phasewith opposite sign [dH04].

• At strong coupling (of the order N2), by contrast, the system synchronises: Only thetwo metastable equilibrium states I± are left. During transitions between these twometastable states, the elements of the chain keep the same position in their respectivelocal potential with high probability.

• The most interesting apect is the transition between these two regimes, which involvesa sequence of symmetry-breaking bifurcations. The description of these bifurcationsinvolves the so-called equivariant bifurcation theory, which takes the system’s sym-metries into account. In particular in the limit of large chain length N , we were ableto determine the bifurcation diagram with the help of the theory of symplectic maps,allowing for a precise determination of the time scales and the geometry of transitionpaths.

In all these regimes, Wentzell–Freidlin theory yields the expected transition time be-tween metastable states on the level of large deviations. Results by Bovier and cowork-ers [BEGK04, BGK05] allow to go beyond Wentzell–Freidlin exponential asymptotics, andto determine prefactors of metastable transition times (by proving the so-called Kramersformula, see for instance [Ber13]).

20 February 23, 2016

Current and planned research activities

Metastability

We continue our work on metastability in systems of coupled stochastic differential equa-tions. As the particular system described above depends on parameters, it undergoesbifurcations whenever the determinant of the Hessian of the potential vanishes. Theclassical Kramers formula fails in these cases, since it would predict vanishing or diverg-ing transition times. We have extended the Kramers formula to situations with bifur-cations [BG10, BG09a], by taking into account the effect of higher-order terms of thepotential at critical points.

Another extension of the Kramers formula is to the infinite-dimensional case of stochas-tic partial differential equations. In [BG13], we have obtained sharp estimates on expectedtransition times for a class of one-dimensional parabolic equations of the form

dut(x) =[∆ut(x)− U ′(ut(x))

]dt+

√2εdW (t, x) .

This result uses spectral Galerkin approximations of the system, and required a control ofcapacities with error bounds uniform in the dimension.

Another pecularity of the above models is that they are invariant under a group ofsymmetries, so that the classical Kramers formula for expected transition times does notapply. The PhD thesis of Sebastien Dutercq addresses the description of these systems,in particular by deriving a corrected Kramers formula taking the symmetries into ac-count [BD15]. These results have been applied in particular to a variant of the systemstudied in [BFG07a, BFG07b], in which the sum of coordinates is constrained to remainequal to zero. For weak coupling, one then obtains a Kawasaki-type dynamics, with anexplicit expression for the spectral gap [BD16].

Neuroscience

Several models for action-potential generation in neurons, such as the Hodgkin–Huxley,FitzHugh–Nagumo and Morris–Lecar equations involve systems of slow–fast differentialequations. For appropriate parameter values, these systems display the phenomenon ofexcitability: They admit a stable equilibrium point, which, however, is such that smalldeterministic or random perturbations cause the system to make a big excursion in phasespace, corresponding to a spike. An important question is to determine the statistics of in-terspike intervals. In certain, effectively one-dimensional cases, the interspike distributionis close to an exponential one [BG09b]. In multidimensional cases, however, one numeri-cally observes different distributions, involving for instance clusters of spikes [MVE08]. Amajor difficulty in the mathematical analysis of these equations stems from the fact thatthey are far from being reversible.

Building on methods in [BG06], we have started developing a mathematical descrip-tion of the effect of noise on excitable systems. This work is part of the ANR projectMANDy, Mathematical Analysis of Neuronal Dynamics (principal investigator: MicheleThieullen, University Paris 6). The PhD thesis of Damien Landon addresses the prob-lem of characterising the interspike distribution for the FitzHugh–Nagumo model [Lan12].Using properties of substochastic Markov chains and quasistationary distributions, wehave been able to describe this distribution in the regime where the excitable stationarypoint is a focus. In particular, we have proved that the distribution is asymptoticallygeometric [BL12].

Curriculum Vitæ – Nils Berglund 21

An interesting feature of three-dimensional models is that they can produce mixed-mode oscillations, which are patterns alternating large- and small-amplitude oscillations(see [DGK+12] for a review). In collaboration with Barbara Gentz and Christian Kuehn(Vienna), we have started a systematic study of the effect of noise on mixed-mode oscil-lations in these systems. First results in [BGK12] quantify the effect of noise on maskingsmall-amplitude oscillations, and on decreasing the number of small oscillations betweenspikes. This description has been extended to the global dynamics, using an analysis ofPoincare sections for the stochastic system [BGK15].

The thesis of Manon Baudel is concerned with a formalisation of the theory of randomPoincare maps, and in particular with the spectral theory of these processes.

Regularity structures

With Christian Kuehn (Vienna), I have recently started applying Martin Hairer’s theoryof regularity structures [Hai14] to coupled SPDE–ODE systems of the form

∂tu = ∆u+ f(u, v) + ξ ,

∂tv = g(u, v) ,

where u and v are functions of time t > 0 and x ∈ Td, the torus in dimension d ∈ {2, 3},and ξ denotes space-time white noise. For a cubic nonlinearity f and a linear g, we wereable to show [BK15] local existence of solutions for this highly singular system, by adaptingthe theory in [Hai14]. This includes the case of the FitzHugh–Nagumo equation describinga large population of diffusively coupled neurons. It is of course desirable to obtain morequantitative results, for instance on the appearance of spiral waves in this kind of system.

22 February 23, 2016

Bibliography

[AB08] Jean-Philippe Aguilar and Nils Berglund, The effect of classical noise on a quantumtwo-level system, Journal of Mathematical Physics 49 (2008), 102102 (23 pages).

[BD15] Nils Berglund and Sebastien Dutercq, The Eyring–Kramers law for Markovian jumpprocesses with symmetries, Journal of Theoretical Probability Online First (2015),1–40.

[BD16] , Interface dynamics of a metastable mass-conserving spatially extended diffu-sion, Journal of Statistical Physics 162 (2016), 334–370.

[BEGK04] Anton Bovier, Michael Eckhoff, Veronique Gayrard, and Markus Klein, Metastabilityin reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J.Eur. Math. Soc. (JEMS) 6 (2004), no. 4, 399–424.

[Ber98] Nils Berglund, On the reduction of adiabatic dynamical systems near equilibrium curves,Preprint mp-arc/98-574. Proceedings of the international workshop Celestial Mechan-ics, Separatrix Splitting, Diffusion, Aussois, France, 1998.

[Ber00a] , Bifurcations, scaling laws and hysteresis in singularly perturbed systems, In-ternational Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci.Publishing, River Edge, NJ, 2000, pp. 79–81.

[Ber00b] , Classical billiards in a magnetic field and a potential, Nonlinear Phenom.Complex Syst. 3 (2000), no. 1, 61–70.

[Ber00c] , Control of dynamic Hopf bifurcations, Nonlinearity 13 (2000), no. 1, 225–248.

[Ber00d] , Dynamic bifurcations: hysteresis, scaling laws and feedback control, Prog.Theor. Phys. Suppl. 139 (2000), 325–336.

[Ber01] N. Berglund, Perturbation theory of dynamical systems, ETH Zuerich lecture notes,2001.

[Ber13] Nils Berglund, Kramers’ law: Validity, derivations and generalisations, Markov Pro-cess. Related Fields 19 (2013), no. 3, 459–490.

[BFG07a] Nils Berglund, Bastien Fernandez, and Barbara Gentz, Metastability in interactingnonlinear stochastic differential equations: I. From weak coupling to synchronization,Nonlinearity 20 (2007), no. 11, 2551–2581.

[BFG07b] , Metastability in interacting nonlinear stochastic differential equations II:Large-N behaviour, Nonlinearity 20 (2007), no. 11, 2583–2614.

[BG02a] Nils Berglund and Barbara Gentz, Beyond the Fokker–Planck equation: Pathwise con-trol of noisy bistable systems, J. Phys. A 35 (2002), no. 9, 2057–2091.

[BG02b] , The effect of additive noise on dynamical hysteresis, Nonlinearity 15 (2002),no. 3, 605–632.

[BG02c] , Metastability in simple climate models: Pathwise analysis of slowly drivenLangevin equations, Stoch. Dyn. 2 (2002), 327–356.

23

24 BIBLIOGRAPHY

[BG02d] , Pathwise description of dynamic pitchfork bifurcations with additive noise,Probab. Theory Related Fields 122 (2002), no. 3, 341–388.

[BG02e] , A sample-paths approach to noise-induced synchronization: Stochastic reso-nance in a double-well potential, Ann. Appl. Probab. 12 (2002), 1419–1470.

[BG03] , Geometric singular perturbation theory for stochastic differential equations,J. Differential Equations 191 (2003), 1–54.

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