“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
Coauthors
(in chronological order)
Links
Google Scholar, arXiv, Orcid, ResearcherID.
List of Publications
for Jonas M. Tölle
updated May 5, 2019
Erdős number: 3
P. Erdős — C. J. Colbourn — M. Scheutzow — J. M. Tölle
Submitted for peer review
- 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 abstractWe 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 C^{1}-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 abstractWe propose a sequential topology on the space of sub-σ-algebras of a separable probability space (Ω, ℱ, ℙ) by linking conditional expectations on L^{2} 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.
Accepted for journal publication
- J. M. Tölle (with C. Kuehn). A gradient flow formulation for the stochastic Amari neural field model, 21 pages, 1 figure, Journal of Mathematical Biology, accepted / to appear (2019), arXiv:1807.02575.
Abstract.
show abstractWe 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, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.
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.
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 | |
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 | |
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. | |
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 | |
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 | |
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 |
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 L^{2}, (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.
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 |