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Unless otherwise noted, all picture and video content on this website: Copyright © 2010–2020 by Jonas Tölle. All rights reserved.

Overview

Research

“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
  • Models from biology, ecology and neuroscience

Coauthors

(in chronological order)

Links

Google Scholar, arXiv, Orcid, ResearcherID.

List of Publications

for Jonas M. Tölle

updated March 28, 2020
Erdős number: 3
P. Erdős — C. J. Colbourn — M. Scheutzow — J. M. Tölle

Submitted for peer review

  1. J. M. Tölle (with M. Hinz and L. Viitasaari). Variability of paths and differential equations with BV-coefficients. Preprint, submitted (2020), 68 pp., arXiv:2003.11698.

    Abstract.

    show abstract

    We define compositions φ(X) of Hölder paths X in ℝn and functions of bounded variation φ under a relative condition involving the path and the gradient measure of φ. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions φ(X) with respect to a given Hölder path Y. These results are then used, together with Doss’ transform, to obtain existence and, in a certain sense, uniqueness results for differential equations in ℝn driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.

  2. J. M. Tölle (with E. Hausenblas). The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem. Preprint, submitted (2019), 27 pp., arXiv:1912.00996.

    Abstract.

    show abstract

    We investigate the existence of a pair of nonnegative solutions to the stochastic system of advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. The proof of existence is based upon a stochastic version of the Schauder-Tychonoff fixed point theorem, which is also proved here.

  3. J. M. Tölle (with P. Beissner). A compact topology for σ-algebra convergence. Preprint, submitted (2018), 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

  1. J. M. Tölle. Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions. Stochastic Processes and their Applications, 130 (2020), no. 5, 3220–3248, https://doi.org/10.1016/j.spa.2019.09.011, preprint available at arXiv:1803.07005.
  2. J. M. Tölle (with C. Kuehn). A gradient flow formulation for the stochastic Amari neural field model. Journal of Mathematical Biology, 79 (2019), no. 4, 1227–1252 https://doi.org/10.1007/s00285-019-01393-w, preprint available at arXiv:1807.02575.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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, https://doi.org/10.1214/ECP.v16-1643, preprint available at arXiv:1109.2437.

Published in peer reviewed proceedings

  1. 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

  1. 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.
  2. 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.

Theses

  1. J. M. Tölle. Stochastic partial differential equations with singular drift. Habilitation thesis, Universität Augsburg, 2019, 232 pp.
  2. 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.
  3. 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

Future Invited Talks

Meeting Location Date Title of Talk Abstract
8th Austrian Stochastic Days 2020 Leoben, Austria September 10 — 11, 2020 TBA

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
SIAM Conference on Analysis of Partial Differential Equations (PD19) La Quinta, CA, U.S.A. December 11 — 14, 2019 Gradient flows for the stochastic Amari neural field model pdf, video
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” 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

2020 Habilitation in Mathematics (Title of Docent), Habilitation thesis: “Stochastic partial differential equations with singular drift” Universität Augsburg
2019 – now Postdoc at the Geometric Analysis and Partial Differential Equations Research Unit University of Helsinki
2017 – 2019 Research Associate 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, Supervisor: Prof. Michael Röckner (Genealogy), Dissertation: Summary & Download Universität Bielefeld
2007 Stay at the Chinese Academy of Sciences (CAS) 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 (Supervisor: Prof. Michael Röckner) Universität Bielefeld
2006 Graduate School of Mathematics Purdue University
2002 – 2006 Study of Mathematics and Computer Science Universität Bielefeld
2002 Pre-Diploma in Mathematics Universität Bielefeld
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).
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).
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