“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
Coauthors
(in chronological order)
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 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 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 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.
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 |