The Fractional Laplacian: probabilistic structure and extension to Lovy Processes

DATES: 26-30 June 2017 (5 sessions)
TIME: 10:00 - 12:00 (a total of 10 hours)

We will discuss elements of the potential theory of the fractional Laplacian. We will construct from the first principles the semigroup of the operator and the corresponding stochastic process. We will define the Green function and Poisson kernel and show their interconnections. The underlying probabilistic structure allows for natural extensions to Lovy processes and further developments. I plan to discuss some of them with focus on the Lovy systems and on their application. If time allows, I will present applications to Fourier multipliers on Lp obtained via martingale transforms of Lovy processes.

PREREQUISITES: Functional analysis, Partial Differential Equations and Probability, basic knowledge.

REFERENCES
[1] R. Bañuelos and K. Bogdan. Lovy processes and Fourier multipliers. J. Funct. Anal., 250(1):197-213, 2007.
[2] R. Bañuelos, K. Bogdan, and T. Luks. Hardy-Stein identities and square functions for semigroups. J. Lond. Math. Soc. (2), 94(2):462-478, 2016. 

[3] K. Bogdan and T. Byczkowski. Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math., 133(1):53-92, 1999. 

[4] K. Bogdan and T. Byczkowski. Potential theory of Schrödinger operator based on fractional Lapla- cian. Probab. Math. Statist., 20(2, Acta Univ. Wratislav. No. 2256):293-335, 2000. 

[5] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, and Z. Vondraˇ cek. Potential analysis of stable processes and its extensions, volume 1980 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009. Edited by Piotr Graczyk and Andrzej Stos. 

[6] K. Bogdan and B. Dyda. The best constant in a fractional Hardy inequality. Math. Nachr., 284(5- 6):629-638, 2011. 

[7] K. Bogdan, T. Grzywny, and M. Ryznar. Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab., 38(5):1901-1923, 2010. 

[8] K.Bogdan and T.Jakubowski. Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys., 271(1):179-198, 2007.
[9] K. Bogdan, T. Kulczycki, and M. Kwaśnicki. Estimates and structure of α-harmonic functions. Probab. Theory Related Fields, 140(3-4):345-381, 2008. 

[10] K. Bogdan, J. Rosiński, G. Serafin, and Ł. Wojciechowski. Lovy systems and moment formulas for mixed Poisson integrals. ArXiv e-prints, Nov. 2014. 


*Registration is free, but inscription is required before 21st June: So as to inscribe send an e-mail to roldan@bcamath.org. Student grants are available. Please, let us know if you need support for travel and accommodation expenses.

• 2017-06-26-Bcam-Course BOGDAN