Strona główna Dydaktyka Konsultacje Research


NCN 2016/23/B/ST1/01665

Heat kernels: construction and estimates

Grant no.: NCN 2016/23/B/ST1/01665
Funding agencies: National Science Centre (NCN), Poland
Funding scheme: OPUS
Funding period: 9.08.2017-8.02.2021 (42 months)
Budget: 471 300 PLN
Title in Polish: Jądra ciepła: konstrukcja i oszacowania

Research team:

Principal Investigator:

Tomasz Grzywny (WUST, Wrocław, PL)

Investigators:

Description:

We will study heat kernels of Markov processes with discontinuous paths. Equivalently, we discuss a class of pseudo-differential operators with non-zero non-local term. We will

  • construct and estimate the corresponding operator semigroups on the whole Euclidean space,
  • obtain estimates and asymptotics for semigroups on domains,
  • find asymptotics behaviour and estimates for heat kernels on discrete structures.

    • In particular we will study heat kernels of non-local operators with non-symmetric Levy kernels, Dunkl operators and subordinators. We will consider processes with non absolutely continuous Levy measures in domains and we will obtain asymptotic behaviour of the Dirichlet heat kernels near the boundary of considered domains and find asymptotics for the spectral heat content. We will also study random walks on discrete structures for instance on affine buildings and topological crystals.

    Publications:

    Peer-reviewed articles in JCR-listed journals

    1. T. Grzywny, H. Park and R. Song, Spectral heat content for Lévy processes
      Mathematische Nachrichten, 292 (4) (2019), 805-825
      DOI: 10.1002/mana.201800035
      arxiv: https://arxiv.org/abs/1705.09463
    2. T. Grzywny and K. Szczypkowski, Heat kernels of non-symmetric Lévy-type operators
      J. Differential Equations, 267 (10) (2019), 6004 - 6064
      DOI: 10.1016/j.jde.2019.06.013
      arxiv: https://arxiv.org/abs/1804.01313
    3. T. Grzywny and K. Szczypkowski, Lévy processes: concentration function and heat kernel estimates
      Bernoulli 26(4) (2020), 3191-3223
      DOI: 10.3150/20-BEJ1220
      arxiv: https://arxiv.org/abs/1907.00778
    4. T. Grzywny and K. Szczypkowski, Estimates of heat kernels of non-symmetric Lévy processes
      Forum Mathematicum 33(5) (2021), 1207-1236
      DOI: 10.1515/forum-2020-0364
      arxiv: https://arxiv.org/abs/1710.07793
    5. D. Kinzebulatov, Y. Semenov, and K. Szczypkowski. Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights
      J. Lond. Math. Soc. (online)
      DOI: 10.1112/jlms.12486
      arxiv: https://arxiv.org/abs/1904.07363

    Forthcoming publications and submitted papers

    1. T. Grzywny, Ł. Leżaj and B. Trojan, Transition densities of subordinators
      to appear in Journal of the Institute of Mathematics of Jussieu
      arxiv: https://arxiv.org/abs/1812.06793
    2. Ł. Leżaj, Transition densities of spectrally positive Lévy processes
      to appear in Lithuanian Mathematical Journal
      arxiv: https://arxiv.org/pdf/2006.16398
    3. G. Armnstrong, K. Bogdan, T. Grzywny, Ł. Leżaj, and L.Wang, Yaglom limit for unimodal Lévy processes.
      arxiv: http://arxiv.org/abs/2110.00873
    4. T. Grzywny and B. Trojan, Subordinated Markov processes: sharp estimates for heat kernels and the Green functions.
      arxiv: https://arxiv.org/abs/2110.01201
    5. B. Rémy, B. Trojan, Martin compactifications of affine buildings
      arxiv: https://arxiv.org/abs/2105.14807
    6. K. Szczypkowski, Regularity of fundamental solutions for Lévy-type operators
      arxiv: https://arxiv.org/abs/2003.09884
    7. K. Szczypkowski, Fundamental solution for super-critical non-symmetric Lévy-type operators
      arxiv: https://arxiv.org/abs/1807.04257
    8. B. Trojan, Asymptotic behaviour of heat kernels and Green functions on affine buildings