Research interests

My work focuses on investigating how probabilistic techniques and functional inequalities can be applied and extended to gain a better understanding of the connections between analytic, geometric and probabilistic aspects of stochastic processes in highly porous and non-smooth settings.
Examples of this kind of spaces include the Sierpinski gasket (hand-made picture), the Sierpinski carpet, the Vicsek set, diamond fractals, general polygaskets and so-called fractional spaces.

A full list of my publications is available on arXiv.

Fall 2020 marked the start of the TAMU Fractals Research Team at Texas A&M. Check out our website, our work may draw your interest!

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Preprints

• Korevaar-Schoen p-energies and their $\Gamma$-limits on Cheeger spaces. (with F. Baudoin) arXiV.
• BV functions and Besov spaces associated with Dirichlet spaces (with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev) arXiV.
• Canonical diffusions on the pattern spaces of aperiodic Delone sets (with M. Hinz, R. Treviño and A. Teplyaev) arXiV.

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Publications

• Dirichlet forms on metric measure spaces as Mosco limits of Korevaa-Schoen energies (with F. Baudoin)
  to appear in Annali della Scuola Normale Superiore di Pisa (2024) arXiV.
• Yet another heat semigroup characterization of BV functions on Riemannian manifolds (with F. Baudoin).
  Annales de la Faculte des Sciences de Toulouse Annales de la Faculte des Sciences de Toulouse (6) Vol. 32, no. 3, 577–606 (2023) arXiV.
• Oscillations of BV measures on nested fractals (with F. Baudoin)
  Journal of Fractal Analysis 9, no. 3-4, 373-396 (2022) arXiV.
• Minimal gap in the spectrum of the Sierpinski gasket.
  International Mathematics Research Notices, no. 23, Pages 18874–18894 (2022) arXiV.
• Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates (with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev)
  Cal. Var. and PDE's 60, no. 5, Paper No. 170 (2021) arXiV.
• Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces (with F. Baudoin).
  Journal of Mathematical Analysis and Applications 497, no. 2, 124899 (2021) arXiv.
• Heat kernel analysis on diamond fractals.
  Stochastic Processes and their Applications Vol. 131, 51-72 (2021) arXiv.
• Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev)
  Cal. Var. and PDE's No.3, Vol.59, Paper No.103, 32 pp (2020) arXiV.
• Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities (with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev).
  J. of Functional Analysis Nr. 11, Vol. 278, 108459 (2020) arXiV.
• Analysis on hybrid fractals (with Y. Chen, H. Gu, R.S. Strichartz and Z. Zhou).
  Comm. Pure App. Anal. Nr. 1, Vol. 19: 47-84 (2020) arXiV.
• Explicit formulas for heat kernels on diamond fractals
  Comm. Math. Phys. Nr. 3, Vol. 364, 1305-1326 (2018) arXiv.
• Completely symmetric resistance forms on the stretched Sierpinski gasket (with U. Freiberg and J. Kigami)
  J. of Fractal Geometry, Nr. 3, Vol. 5: 227-277 (2018) arXiv.
• Entropy-based inhomogeneity detection in fiber materials (with E. Spodarev)
  Methodology and Computing in Applied Probability, 1-17 (2017) arXiv.
• Power dissipation in fractal Feynman-Sierpinski AC circuits
  Journal of Mathematical Physics, Nr. 7, Vol. 58: 073503 (2017) arXiv.
• Estimation of entropy for Poisson marked point processes (with E. Spodarev)
  Advances in Applied Probability, Nr. 1, Vol. 49: 258-278 (2017) arXiv.
• Weyl asymptotics for Hanoi atrractors (with U. Freiberg)
  Forum Mathematicum 29, no. 5, 1003–1022, 2017 arXiv.
• The limit theorem for maximum of partial sums of exchangeable random variables (with A. Rakitko)
  Statistics & Probability Letters, Vol. 119: 357-362 (2016) arXiv.
• Energy and Laplacian on Hanoi-type fractal quantum graphs (with D. Kelleher and A. Teplyaev)
  Journal of Physics A: Mathematical and Theoretical, Nr. 4, Vol.49: 1501-1533 (2016) arXiv.
• Hanoi attractors and the Sierpinski gasket (with U. Freiberg)
  Special issue of Int. J. Math. Modelling and Nonlinear Optimization on Fractals, Fractal-based Methods and Applications, Nr. 4, Vol. 3: 251-265, 2012

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Research with students

Fall 2020 marked the start of the TAMU Fractals Research Team at Texas A&M!

Publications
Analysis on hybrid fractals (with Y. Chen, H. Gu, R.S. Strichartz and Z. Zhou). Comm. Pure App. Anal. Nr. 1, Vol. 19: 47-84, 2020. arXiV.

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