Maximum principles for nonlocal parabolic Waldenfels operators
Bulletin of Mathematical Sciences - Trang 1-31 - 2018
Tóm tắt
As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with
$$\alpha $$
-stable Lévy processes are presented to illustrate the maximum principles.
Tài liệu tham khảo
Albeverio, S., Rüdiger, B., Wu, J.-L.: Invariant measures and symmetry property of Lévy type operators. Potential Anal. 13(2), 147–168 (2000)
Albeverio, S., Rüdiger, B., Wu, J.-L.: Analytic and probabilistic aspects of Lévy processes and fields in quantum theory. In: Barndorff-Nielsen, O., Mikosch, T., Resnick, S. (eds.) Lévy Processes: Theory and Applications, pp. 187–224. Birkhäuser, Boston (2001)
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)
Bony, J.-M., Courrège, P., Priouret, P.: Semi-groupes de feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Annales de l’institut Fourier 18(2), 369–521 (1968)
Ciomaga, A.: On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17(7/8), 635–671 (2012)
Courrège, P.: Sur la forme intégro-différentielle des opérateurs de \(C^\infty _k\) dans \(C\) satisfaisant au principe du maximum. Séminaire Brelot-Choquet-Deny. Théorie du Potentiel 10(1), 1–38 (1965)
Coville, J.: Remarks on the strong maximum principle for nonlocal operators. Electron. J. Differ. Equ. 2008(66), 1–10 (2008)
Duan, J.: An Introduction to Stochastic Dynamics. Cambridge University Press, New York (2015)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, vol. 19 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (2010)
Garroni, M.G., Menaldi, J.L.: Green Functions for Second Order Parabolic Integro-Differential Problems, vol. 275 of Chapman & Hall/CRC Research Notes in Mathematics Series. Chapman & Hall/CRC, London (1992)
Jacob, N.: Pseudo differential operators & Markov Processes: Markov Processes and Applications, vol. 1, 2, 3. Imperial College Press, London (2001, 2002, 2005)
Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications, 2nd edn. Springer, New York (2006)
Kolokoltsov, V.N.: Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80(03), 725–768 (2000)
Kolokoltsov, V.N.: Markov Processes, Semigroups, and Generators, vol. 38 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (2011)
Komatsu, T.: Markov processes associated with certain integro-differential. Osaka J. Math. 10, 271–303 (1973)
Komatsu, T.: Pseudo-differential operators and Markov processes. J. Math. Soc. Jpn. 36(3), 387–418 (1984)
Komatsu, T.: Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms. Osaka J. Math. 25(3), 697–728 (1988)
Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4), 833–860 (1995)
Kuwae, K.: Maximum principles for subharmonic functions via local semi-Dirichlet forms. Can. J. Math. 60(4), 822–874 (2008)
Liao, M.: The Dirichlet problem of a discontinuous Markov process. Acta Math. Sin. 5(1), 9–15 (1989)
Øksendal, B.K., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin (2007)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984)
Qiao, H., Kan, X., Duan, J.: Escape probability for stochastic dynamical systems with jumps. Proc. Math. Stat. 34, 195–216 (2013)
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, vol. 13 of Texts in Applied Mathematics, 2nd edn. Springer, New York (2004)
Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions, 2nd edn. Cambridge University Press, Cambridge (1999)
Schertzer, D., Larchevêque, M., Duan, J., Yanovsky, V.V., Lovejoy, S.: Fractional Fokker–Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J. Math. Phys. 42(1), 200–212 (2001)
Shlesinger, M.F., Zaslavsky, G.M., Frisch, U.: Lévy Flights and Related Topics in Physics, vol. 450 of Lecture Notes in Physics. Springer, Berlin (1995)
Stroock, D.W.: Diffusion processes associated with Lévy generators. Probab. Theory Relat. Fields 32(3), 209–244 (1975)
Stroock, D.W.: Markov Processes from K. Itô’s Perspective, vol. 1555 of Annals of Mathematics Studies. Princeton University Press, Princeton (2003)
Taira, K.: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin (2004)
Truman, A., Wu, J.-L.: On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. J. Funct. Anal. 238(2), 612–635 (2006)
Waldenfels, W.v.: Positive Halbgruppen auf einem \(n\)-dimensionalen Torus. Archiv der Mathematik 15(1), 191–203 (1964)