Research Projects:
- National Science Centre (Poland) grant 2019/33/B/ST1/02494 (2020-2023) [Principal Investigator]
Geometric quantities related to nonlocal equations
- National Science Centre (Poland) grant 2017/27/B/ST1/01339 (2018-2021) [Investigator]
Regularity and asymptotics for multidimensional Markovian evolutions
- National Science Centre (Poland) grant 2016/23/B/ST1/01665 (2017-2021) [Principal Investigator]
Heat kernels: construction and estimates
- National Science Centre (Poland) grant 2015/17/B/ST1/01043 (2016-2019) [Investigator]
Levy and Markov processes on Euclidean and hyperbolic spaces
- National Science Centre (Poland) grant Harmonia 2014/14/M/ST1/00600 (2015-2018) [Investigator]
Analysis of nonlocal operators
- Ministry of Science and Higher Education (Poland) grant N N201 373136 (2009-2012) [Investigator]
Potential theory of a class of Levy processes and their Feynman-Kac semigroups
- Ministry of Science and Higher Education (Poland) grant N N201 4112 33 (2007-2008) [Investigator]
Potential theory of relativistic processes
- Polish State Committee for Scientific Research grant 1 P03A 020 28 (2005-2008) [Investigator]
Harmonic measures and properties of semigroups of stable processes
Preprints:
- W. Cygan, T. Grzywny and J. Lenczewska, Asymptotics and geometric flows for a class of nonlocal curvatures.
arxiv: https://arxiv.org/abs/2308.16660
- T. Grzywny and M. Kwaśnicki,
Liouville's theorems for Lévy operators.
arxiv: https://arxiv.org/abs/2301.08540
- T. Grzywny, T. Jakubowski and G. ¯urek,
Estimates of gradient of L-harmonic functions for nonlocal operators with order α>1.
arxiv: https://arxiv.org/abs/2208.00747
- T. Grzywny and B. Trojan, Subordinated Markov processes: sharp estimates for heat kernels and the
Green functions.
arxiv: https://arxiv.org/abs/2110.01201
- T. Grzywny, First exit times from a bounded interval for Lévy processes
arxiv: http://arxiv.org/abs/1911.05022
Papers:
- W. Cygan and T. Grzywny, Asymptotics of non-local perimeters.
Annali di Matematica Pura ed Applicata 202(6) (2023), 2629-2651
DOI: 10.1007/s10231-023-01332-z
arxiv: https://arxiv.org/abs/2203.12343
- G. Armstrong, K. Bogdan, T. Grzywny, £. Le¿aj, and L. Wang, Yaglom limit for unimodal Lévy processes.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 59(3) (2023), 1688-1721
DOI: 10.1214/22-AIHP1301
arxiv: http://arxiv.org/abs/2110.00873
- K. Bogdan, T. Grzywny, K. Pietruska-Pa³uba and A. Rutkowski, Nonlinear nonlocal Douglas identity
Calculus of Variations and Partial Differential Equations 62(5) (2023), article 151
DOI: 10.1007/s00526-023-02458-x
arxiv: https://arxiv.org/abs/2006.01932
- T. Grzywny and J. Lenczewska, Asymptotic expansion of the nonlocal heat content.
Studia Mathematica 270(3) (2023), 339-359
DOI: 10.4064/sm220831-26-1
arxiv: https://arxiv.org/abs/2202.11662
- T. Grzywny, £. Le¿aj and B. Trojan, Transition densities of subordinators of positive order
Journal of the Institute of Mathematics of Jussieu 22(3) (2023), 1119 - 1179
DOI: 10.1017/S1474748021000360
arxiv: https://arxiv.org/abs/1812.06793
- T. Grzywny, K. Kaleta and P. Sztonyk, Heat kernels of non-local Schrödinger operators with Kato potentials.
Journal of Differential Equations 340 (2022), 273-308
DOI: 10.1016/j.jde.2022.08.038
arxiv: https://arxiv.org/abs/2204.04239
- 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
- T. Grzywny, £. Le¿aj and M. Miśta, Hitting probabilities for Lévy processes on the real line
ALEA, Lat. Am. J. Probab. Math. Stat. 18 (2021), 727-760
DOI: 10.30757/ALEA.v18-27
open access
- T. Grzywny, M. Kassmann and £. Le¿aj, Remarks on the nonlocal Dirichlet problem
Potential Analysis 54(1) (2021), 119-151
DOI: 10.1007/s11118-019-09820-9
open access
- 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
- K. Bogdan, T. Grzywny, K. Pietruska-Pa³uba and A. Rutkowski, Extension theorem for nonlocal operators
Journal de Mathematiques Pures et Appliquees 137 (2020), 33-69
DOI: 10.1016/j.matpur.2019.09.005
arxiv: https://arxiv.org/abs/1710.05880
- T. Grzywny, T. Jakubowski and G. ¯urek, Green function for gradient perturbation of unimodal Lévy processes in the real line
Bulletin of the Malaysian Mathematical Sciences Society 43(2) (2020), 1223-1251
DOI: 10.1007/s40840-019-00738-4
arxiv: https://arxiv.org/abs/1802.01450
- T. Grzywny, K.-Y. Kim and P. Kim, Estimates of Dirichlet heat kernel for symmetric Markov processes,
Stoch. Pr. Appl. 130(1) (2020), 431-470
DOI: 10.1016/j.spa.2019.03.017
arxiv: http://arxiv.org/abs/1512.02717
- T. Grzywny, M. Ryznar and B. Trojan, Asymptotic behaviour and estimates of slowly varying convolution semigroups,
Int Math Res Notices 2019(23) (2019), 7193-7258
DOI: 10.1093/imrn/rnx324
arxiv: http://arxiv.org/abs/1606.04178
- 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
- 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
- K. Bogdan, T. Grzywny, T. Jakubowski and D. Pilarczyk, Fractional Laplacian with Hardy potential
Communications in PDE, 44 (1) (2019), 20-50
DOI: 10.1080/03605302.2018.1539102
arxiv: https://arxiv.org/abs/1710.08378
- W. Cygan and T. Grzywny, A note on the generalized heat content for Lévy processes
Bulletin of the Korean Mathematical Society 55 (5) (2018), 1463-1481
DOI: 10.4134/BKMS.b170835
open access
- T. Grzywny, M. Kwaśnicki, Potential kernels, probabilities of hitting a ball, harmonic
functions and the boundary Harnack inequality for unimodal Lévy processes,
Stoch. Pr. Appl. 128(1) (2018), 1-38
DOI: 10.1016/j.spa.2017.04.004
arxiv: https://arxiv.org/abs/1611.10304
- T. Grzywny and K. Szczypkowski, Kato classes for Lévy processes,
Potential Anal. 47(3) (2017), 245-276
DOI: 10.1007/s11118-017-9614-1
open access
- W. Cygan, T. Grzywny and B. Trojan, Asymptotic behavior of densities of unimodal convolution semigroups,
Trans. Amer. Math. Soc. 369(8) (2017), 5623-5644
DOI: 10.1090/tran/6830
arxiv: https://arxiv.org/abs/1504.08358
- T. Grzywny, T. Jakubowski and G. ¯urek, Green function for gradient perturbation of unimodal Lévy processes,
Prob.
Math. Stat. 37(1) (2017), 119-143
DOI: 10.19195/0208-4147.37.1.5
arxiv: https://arxiv.org/abs/1505.07700
- T. Grzywny and M. Ryznar, Hitting times of points and intervals for symmetric Lévy processes,
Potential Anal. 46(4) (2017), 739-777
DOI: 10.1007/s11118-016-9600-z
open access
- W. Cygan and T. Grzywny, Heat content for convolution semigroups,
J. Math. Anal. Appl. 446(2) (2017), 1393-1414
DOI: 10.1016/j.jmaa.2016.09.051
arxiv: https://arxiv.org/abs/1606.09168
- K. Bogdan, T. Grzywny, M. Ryznar, Barriers, exit time and survival probability for unimodal Lévy processes,
Prob. Th. Related Fields 162 (2015), 155-198
- K. Bogdan, T. Grzywny, M. Ryznar, Dirichlet heat kernel for unimodal Lévy processes,
Stoch. Pr. Appl. 124
(2014), 3612-3650
- K. Bogdan, T. Grzywny, M. Ryznar, Density and tails of unimodal convolution semigroups,
Journal of Functional Analysis 266(6) (2014), 3543-3571
- T. Grzywny, On Harnack Inequality and Hölder Regularity for Isotropic
Unimodal Lévy Processes,
Potential
Anal. 41
(2014), 1-29
- T. Grzywny and M. Ryznar, Potential theory of one-dimensional geometric stable processes,
Colloq. Math. 129(1) (2012), 7-40
- K. Bogdan, T. Grzywny, M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions,
Annals of Probability 38(5) (2010), 1901-1923
- K. Bogdan and T. Grzywny, Heat kernel of fractional Laplacian in cones,
Colloq. Math. 118 (2010), 365-377
- T. Grzywny and M. Ryznar, Two-sided optimal bounds for Green functions of half-spaces
for relativistic α-stable process,
Potential
Anal. 28
(2008), 201-239
- T. Grzywny, Intrinsic ultracontractivity for Lévy processes,
Prob.
Math. Stat. 28
(2008), 91-106
- T. Grzywny and M. Ryznar, Estimates of Green functions for some perturbations of
fractional Laplacian,
Illinois J. Math. 51 (2007),
1409-1438