Time-dependent gradient perturbations of fractional Laplacian

Journal of Evolution Equations

Volumn 10, Issue 2, 2010, Pages 319-339

  Jakubowski, Tomasz ^a,b   Szczypkowski, Karol ^b  

^a UNIVERSITÉ D'ANGERS   (France)

^b WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955845394     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-009-0051-5     Document Type: Article

References (39)

1. Aronson D. G. (1968) Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22: 607-694.
2. Berg C., Forst G. (1975) Potential theory on locally compact abelian groups. Springer-Verlag, New York.
3. Bogdan K. (2000) Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243(2): 326-337.
4. Bogdan K., Byczkowski T. (1999) Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1): 53-92.
5. Bogdan, K., and Byczkowski, T. Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20, 2, Acta Univ. Wratislav. No. 2256 (2000), 293-335.
6. Bogdan K., Hansen W., Jakubowski T. (2008) Time-dependent Schrödinger perturbations of transition densities. Studia Math. 189(3): 235-254.
7. Bogdan K., Jakubowski T. (2007) Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys. 271(1): 179-198.
8. Bogdan K., Stós A., Sztonyk P. (2003) Harnack inequality for stable processes on d-sets. Studia Math. 158(2): 163-198.
9. Brandolese L., Karch G. (2008) Far field asymptotics of solutions to convection equation with anomalous diffusion. J. Evol. Equ. 8(2): 307-326.
10. Carmona R., Masters W. C., Simon B. (1990) Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91(1): 117-142.
11. Chen Z.-Q., Song R. (1997) Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150(1): 204-239.
12. Chen Z.-Q., Song R. (2003) Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1): 262-281.
13. Chung, K. L., and Zhao, Z. X. From Brownian motion to Schrödinger's equation, vol. 312 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1995.
14. Cranston M., Fabes E., Zhao Z. (1988) Conditional gauge and potential theory for the Schrödinger operator. Trans. Amer. Math. Soc. 307(1): 171-194.
15. Droniou J., Imbert C. (2006) Fractal first-order partial differential equations. Arch. Ration. Mech. Anal. 182(2): 299-331.
16. Grzywny T., Ryznar M. (2007) Estimates of Green functions for some perturbations of fractional Laplacian. Illinois J. Math. 51(4): 1409-1438.
17. Hansen W. (2005) Uniform boundary Harnack principle and generalized triangle property. J. Funct. Anal. 226(2): 452-484.
18. Jacob, N. Pseudo differential operators and Markov processes. Vol. I. Imperial College Press, London, 2001. Fourier analysis and semigroups.
19. Jacob, N. Pseudo differential operators and Markov processes. Vol. II. Imperial College Press, London, 2002. Generators and their potential theory.
20. Jacob, N. Pseudo differential operators and Markov processes. Vol. III. Imperial College Press, London, 2005. Markov processes and applications.
21. Jakubowski, T. The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22, 2, Acta Univ. Wratislav. No. 2470 (2002), 419-441.
22. Jakubowski, T. The estimates of the mean first exit time from a ball for the α-stable Ornstein-Uhlenbeck processes. Stochastic Process. Appl. 117, 10 (2007), 1540-1560.
23. Jakubowski T. (2008) On Harnack inequality for α-stable Ornstein-Uhlenbeck processes. Math. Z. 258(3): 609-628.
24. Jakubowski T. (2009) On combinatorics of Schrödinger perturbations. Potential Anal. 31(1): 45-55.
25. Karch G., Woyczyński W. A. (2008) Fractal Hamilton-Jacobi-KPZ equations. Trans. Amer. Math. Soc. 360(5): 2423-2442.
26. Kim P., Lee Y.-R. (2007) Generalized 3G theorem and application to relativistic stable process on non-smooth open sets. J. Funct. Anal. 246(1): 113-143.
27. Kondratiev, V., Liskevich, V., Sobol, Z., and Us, O. Estimates of heat kernels for a class of second-order elliptic operators with applications to semi-linear inequalities in exterior domains. J. London Math. Soc. (2) 69, 1 (2004), 107-127.
28. Liskevich, V., and Semenov, Y. Estimates for fundamental solutions of second-order parabolic equations. J. London Math. Soc. (2) 62, 2 (2000), 521-543.
29. Liskevich V., Zhang Q. S. (2004) Extra regularity for parabolic equations with drift terms. Manuscripta Math. 113(2): 191-209.
30. Ryznar M. (2002) Estimates of Green function for relativistic α-stable process. Potential Anal. 17(1): 1-23.
31. Song R. (1993) Probabilistic approach to the Dirichlet problem of perturbed stable processes. Probab. Theory Related Fields 95(3): 371-389.
32. Song R. (1995) Feynman-Kac semigroup with discontinuous additive functionals. J. Theoret. Probab. 8(4): 727-762.
33. Takeda M. (2002) Conditional gaugeability and subcriticality of generalized Schrödinger operators. J. Funct. Anal. 191(2): 343-376.
34. Takeda M. (2006) Gaugeability for Feynman-Kac functionals with applications to symmetric α-stable processes. Proc. Amer. Math. Soc. 134(9): 2729-2738 (electronic).
35. Takeda M., Uemura T. (2004) Subcriticality and gaugeability for symmetric α-stable processes. Forum Math. 16(4): 505-517.
36. Yosida K. (1995) Functional analysis. Classics in Mathematics. Springer-Verlag, Berlin.
37. Zhang Q. (1996) A Harnack inequality for the equation. Manuscripta Math. 89(1): 61-77.
38. Zhang Q. S. (1997) Gaussian bounds for the fundamental solutions of. Manuscripta Math. 93(3): 381-390.
39. Zhang Q. S. (2004) A strong regularity result for parabolic equations. Comm. Math. Phys. 244(2): 245-260.