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Unless otherwise noted, all picture and video content on this website: Copyright © 2010–2018 by Jonas Tölle. All rights reserved.

Research

Home 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 partial differential equations
  • linear growth functionals and variational convergence
  • ergodic theory and invariant distributions
  • Dirichlet forms and their geometry
  • Sobolev space theory

Links

Google Scholar, arXiv, Orcid, ResearcherID.

Coauthors

(in chronological order)

List of Publications

for Jonas M. Tölle

updated March 20, 2018
Erdős number: 3
P. Erdős — C. J. Colbourn — M. Scheutzow — J. M. Tölle

Submitted for peer review

  1. 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. 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.

  2. J. M. Tölle (with P. Beissner). A compact topology for σ-algebra convergence. Preprint, submitted (2017), 23 pp., arXiv:1802.05920.

    Abstract. We propose a sequential topology on the collection of sub-σ-algebras included in a separable probability space (Ω, ℱ, ℙ). We prove compactness of the conditional expectations with respect to L2-bounded random variables 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. Finally, a new application to information economics is discussed.

Accepted for publication in proceedings

  1. J. M. Tölle. Estimates for nonlinear stochastic partial differential equations with gradient noise via Dirichlet forms. Stochastic Partial Differential Equations and Related Fields, Festschrift in honor of Michael Röckner, in press, Springer Proceedings in Mathematics & Statistics 229 (2018), 14 pp.

Abstract. We present a priori estimates for nonlinear Stratonovich stochastic partial differential equations on the d-dimensional torus with p-Laplace-type drift with sublinear non-homogeneous nonlinearities and Gaussian gradient Stratonovich noise with C1-vector field coefficients. Assuming a commutator bound, the results are obtained by using resolvent and Dirichlet form methods and an approximative Itô-formula.

Journal Publications

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.

Other works (online)

  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 (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.
  3. 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.
  4. 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.
  5. 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.