ResearchTalks and ConferencesTeachingCVResearch “He who seeks for methods without having a definite problem in mind seeks for the most part in vain.” (David Hilbert) Scientific Interests stochastic partial differential equations (stochastic) variational calculus singular and degenerate stochastic diffusion equations in Hilbert space nonlocal and local nonlinear stochastic evolution equations linear growth functionals and variational convergence ergodic theory and invariant distributions Dirichlet forms and their geometry topology of information Weighted Sobolev space theory Links Google Scholar, arXiv, Orcid, ResearcherID. Coauthors (in chronological order) Ioana Ciotir Wei Liu Max-K. von Renesse Benjamin Gess Michael Scheutzow Abdelhadi Es-Sarhir Onno van Gaans Patrick Beissner Christian Kuehn List of Publications for Jonas M. Tölle updated January 15, 2019 Erdős number: 3 P. Erdős — C. J. Colbourn — M. Scheutzow — J. M. Tölle Submitted for peer review J. M. Tölle (with C. Kuehn). A gradient flow formulation for the stochastic Amari neural field model. Preprint, submitted (2018), 18 pp., arXiv:1807.02575. Abstract. show abstract We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient-flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures and ergodic properties for the associated Feller semigroups are also discussed. J. M. Tölle. Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions. Preprint, submitted (2018), 25 pp., arXiv:1803.07005. Abstract. show abstract We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the d-dimensional torus with singular p-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with C1-vector field coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula. J. M. Tölle (with P. Beissner). A compact topology for σ-algebra convergence. Preprint, submitted (2017), 23 pp., arXiv:1802.05920. Abstract. show abstract We propose a sequential topology on the space of sub-σ-algebras of a separable probability space (Ω, ℱ, ℙ) by linking conditional expectations on L2 along sequences of sub-σ-algebras. The varying index of measurability is captured by a bundle space construction. As a consequence, we establish the compactness of the space of sub-σ-algebras. The proposed topology preserves independence and is compatible with join and meet operations. Journal Publications J. M. Tölle (with B. Gess). Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. SIAM J. Math. Anal. 48 (2016), no. 6, 4094–4125, http://dx.doi.org/10.1137/15M1049774, preprint available at arXiv:1507.04545. J. M. Tölle (with I. Ciotir). Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise. J. Funct. Anal. 271 (2016), no. 7, 1764–1792, http://dx.doi.org/10.1016/j.jfa.2016.05.013, preprint available at arXiv:1507.02576. J. M. Tölle (with B. Gess). Stability of solutions to stochastic partial differential equations. J. Differential Equations 260 (2016), no. 6, 4973–5025, http://dx.doi.org/10.1016/j.jde.2015.11.039, preprint available at arXiv:1506.01230. J. M. Tölle (with B. Gess). Multi-valued, singular stochastic evolution inclusions. J. Math. Pures Appl. 101 (2014), no. 6, 789–827, http://dx.doi.org/10.1016/j.matpur.2013.10.004, preprint available at arXiv:1112.5672. J. M. Tölle (with A. Es-Sarhir, M. Scheutzow and O. van Gaans). Invariant measures for monotone SPDEs with multiplicative noise term. Appl. Math. Optim. 68 (2013), no. 2, 275–287, http://dx.doi.org/10.1007/s00245-013-9206-4, preprint available at arXiv:0910.0960. J. M. Tölle (with I. Ciotir). Corrigendum to “Convergence of invariant measures for singular stochastic diffusion equations” [Stochastic Process. Appl. 122 (2012) 1998–2017.] Stochastic Process. Appl. 123 (2013), no. 3, 1178–1181, http://dx.doi.org/10.1016/j.spa.2012.10.009, preprint available at arXiv:1211.4404. J. M. Tölle. Uniqueness of weighted Sobolev spaces with weakly differentiable weights. J. Funct. Anal. 263 (2012), no. 10, 3195–3223, http://dx.doi.org/10.1016/j.jfa.2012.08.002, preprint available at arXiv:1110.2888. J. M. Tölle (with I. Ciotir). Convergence of invariant measures for singular stochastic diffusion equations. Stochastic Process. Appl. 122 (2012), no. 4, 1998–2017, http://dx.doi.org/10.1016/j.spa.2011.11.011, preprint available at arXiv:1201.2839. J. M. Tölle (with M.-K. von Renesse). On an EVI curve characterization of Hilbert spaces. J. Math. Anal. Appl. 385 (2012), 589–598, http://dx.doi.org/10.1016/j.jmaa.2011.06.080. J. M. Tölle (with W. Liu). Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts. Electr. Comm. Probab. 16 (2011), 447–457, http://ecp.ejpecp.org/article/view/1643, preprint available at arXiv:1109.2437. Published in peer reviewed proceedings J. M. Tölle. Estimates for nonlinear stochastic partial differential equations with gradient noise via Dirichlet forms. In: Eberle A., Grothaus M., Hoh W., Kassmann M., Stannat W., Trutnau G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_14. Other works (online) J. M. Tölle. Convergence of solutions to the p-Laplace evolution equation as p goes to 1. Preprint, http://arxiv.org/abs/1103.0229v2, 2011, 11 pp. J. M. Tölle (with I. Ciotir). Convergence of solutions to the stochastic p-Laplace equation as p goes to 1. BiBoS Preprint 11-01-371, SFB 701 Preprint 11002. http://www.math.uni-bielefeld.de/~bibos/preprints/11-01-371.pdf, 2010, 16 pp. J. M. Tölle. Revision of Variational convergence of nonlinear partial differential operators on varying Banach spaces. http://www.math.uni-bielefeld.de/~bibos/preprints/E10-09-360.pdf, 2010, 250 pp. J. M. Tölle. Variational convergence of nonlinear partial differential operators on varying Banach spaces. Dissertation, Universität Bielefeld, published online on BieSOn, Universitätsbibliothek Bielefeld, URN (NBN): urn:nbn:de:hbz:361-16758, 2010, 250 pp. J. M. Tölle. Convergence of non-symmetric forms with changing reference measures. Diploma thesis, Universität Bielefeld, BiBoS-Preprint E06-09-234, http://www.math.uni-bielefeld.de/~bibos/preprints/E06-09-234.pdf, 2006, 81 pp. Talks and Conferences Co-organization of Workshops We have organized a workshop on “Nonlinear, nonlocal problems and stochastic methods” at Aalto University, Finland, which took place December 7 — 9, 2016. Workshop homepage: NNPSM2016. Workshop poster. Selected past Talks Meeting Location Date Title of Talk Slides 9th International Conference on Stochastic Analysis and Its Applications Bielefeld University September 3 — 7, 2018 Gradient flows for the stochastic Amari neural field model pdf Workshop on Stochastic Systems: their Analysis, Geometry and Perturbation Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China July 10 — 15, 2018 Stochastic nonlinear PDEs with singular drift and gradient noise pdf Stochastic Partial Differential Equations CIRM, Luminy, Marseille, France May 14 — 18, 2018 Gradient flows for the stochastic Amari neural field model 13th German Probability and Statistics Days Uni Freiburg, Freiburg im Breisgau, Germany Feb 27 — March 2, 2018 The set of sub-σ-algebras is a compact metric space. pdf 19th ÖMG Congress and Annual DMV Meeting Paris-Lodron University of Salzburg, Salzburg, Austria Sep 11 — 15, 2017 The p-Laplace evolution equation as p goes to 1 — Toward a general convergence result for parabolic minimizers 19th ÖMG Congress and Annual DMV Meeting Paris-Lodron University of Salzburg, Salzburg, Austria Sep 11 — 15, 2017 Nonlinear SPDE with gradient noise via curvature-dimension conditions Japanese-German Open Conference on Stochastic Analysis 2017 University of Kaiserslautern, Germany Sep 4 — 8, 2017 A compact topology for σ-algebra convergence Oberseminar Mathematische Modellierung und partielle Differentialgleichungen Augsburg University, Augsburg, Germany May 30, 2017 The p-Laplace evolution equation as p goes to 1: Toward a general convergence result for parabolic minimizers pdf Stochastic Partial Differential Equations and Related Fields Bielefeld University, Bielefeld, Germany Oct 10 — 14, 2016 Nonlinear, singular SPDE perturbed by noise acting along infinitesimal motions on domains with symmetries pdf 7th European Congress of Mathematics Technical University of Berlin, Berlin, Germany Jul 18 — 22, 2016 Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise Stochastic Partial Differential Equations and Applications – X Levico Terme, Trento, Italy Apr 30 — May 3, 2016 Nonlinear, singular SPDE with gradient-type Gaussian noise driving infinitesimal vector field actions pdf Teaching Lectures Course Institution Semester Seminar on “Stochastic Differential Equations” (prospective) Augsburg University 2019 Functional Analysis Ulm University 2018/19 Analysis III Ulm University 2018/19 Functional Analysis Augsburg University 2018 Seminar on “Convex Sets and Convex Functions” (together with Lisa Beck) Augsburg University 2018 Preparatory Course “Brückenkurs” in Mathematics Augsburg University 2017/18 First course in probability and statistics Aalto University 2016/17 Stochastic Analysis University of Hamburg 2015/16 Stand-in lectures Institution Semester Functional Analysis Bielefeld University 2015 Analysis II Bielefeld University 2014 Probability Theory I Bielefeld University 2011 Homework and exams Homework and written exam coordinator Institution Semester Functional Analysis Augsburg University 2018 Ordinary Differential Equations Augsburg University 2017/18 Measure and Integration Theory Bielefeld University 2014/15 Analysis II Bielefeld University 2014 Analysis I Bielefeld University 2013/14 Analysis III for engineers Technical University of Berlin 2012 Analysis I for engineers Technical University of Berlin 2011/12 Exercise classes / teaching assistance Institution Semester Ordinary Differential Equations Augsburg University 2017/18 Measure and Integration Theory Bielefeld University 2014/15 Analysis II Bielefeld University 2014 Analysis I Bielefeld University 2013/14 Analysis III for engineers Technical University of Berlin 2012 Analysis I for engineers Technical University of Berlin 2011/12 MA 303 (Differential Equations and PDEs) Purdue University 2006 Probability Theory I Bielefeld University 2005/06 Measure Theory and Elements of Functional Analysis Bielefeld University 2005 Mathematics for Computer Scientists Bielefeld University 2004/05 Stochastics A Bielefeld University 2003/04 Analysis I Bielefeld University 2003 Linear Algebra I Bielefeld University 2002/03 First examiner in oral exams Graduate level Functional Analysis Analysis III Stochastic Analysis Second examiner in oral exams Graduate level Actuarial Mathematics Mathematics for Finance I Probability Theory I-III Undergraduate level Analysis I–II Analysis C Analysis I-III for engineers Integral Transforms and Differential Equations for engineers Measure and Integration Theory Statistics Master and Diploma students Nelli Schmelzer, Stochastische nichtlineare Schrödingergleichungen: Existenz und Eindeutigkeit von milden Lösungen in L2, (First advisor: Prof. Dr. Michael Röckner), 2015, Bielefeld University. Fadwa Mihad, Extrapolation stationärer stochastischer Prozesse, (First advisor: Lecturer Dr. Henrik Winkler), 2011, Technical University of Berlin. CV 2017 – now Postdoc at the Nonlinear Analysis Chair Universität Augsburg 2018 – 2019 Visiting Professor at the Institute of Analysis Universität Ulm 2016 – 2017 Postdoc at the Nonlinear Partial Differential Equations (NPDE) Group Aalto University 2015 – 2016 Visiting Lecturer at the Center of Mathematical Statistics and Stochastic Processes Universität Hamburg 2014 – 2015 Postdoc at the Stochastic Analysis Group Universität Bielefeld 2011 – 2013 Postdoc at the DFG Research Group 718 “Analysis and Stochastics in Complex Physical Systems” TU Berlin 2010 Postdoc at the DFG Collaborative Research Center 701 “Spectral Structures and Topological Methods in Mathematics” Universität Bielefeld 2010 PhD in Mathematics, Advisor: Prof. Michael Röckner (Genealogy), Dissertation: Summary & Download 2007 Stay at the Chinese Academy of Sciences (CAS) Beijing 2006 – 2010 PhD Studies in Mathematics at the DFG International Research Training Group 1132 “Stochastics and Real World Models” Universität Bielefeld 2006 Diploma in Mathematics (Advisor: Prof. Michael Röckner) 2006 Graduate School of Mathematics Purdue University 2002 – 2006 Study of Mathematics and Computer Science Universität Bielefeld 2002 Pre-Diploma in Mathematics 2000 – 2002 Study of Mathematics, Philosophy and Computer Science Universität Bielefeld Jonas Tölle during an artistic performance involving mathematics (in particular, Hilbert space theory), at the group exhibition "No Lightning Strikes the Nettle" taking place at the showroom Hilbertraum, Berlin-Neukölln (picture taken by Tiina Rajamäki).