Linear additive functionals of superdiffusions and related nonlinear P.D.E.

Dynkin, E.; Kuznetsov, S.

doi:10.1090/S0002-9947-96-01602-9

Linear additive functionals of
superdiffusions and related nonlinear P.D.E.

Authors: E. B. Dynkin and S. E. Kuznetsov
Journal: Trans. Amer. Math. Soc. 348 (1996), 1959-1987
MSC (1991): Primary 60J60, 35J65; Secondary 60J80, 31C15, 60J25, 60J55, 31C45, 35J60
DOI: https://doi.org/10.1090/S0002-9947-96-01602-9
MathSciNet review: 1348859
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $L$ be a second order elliptic differential operator in a bounded smooth domain $D$ in $\mathbb{R}^{d}$ and let $1<\alpha \le 2$ . We get necessary and sufficient conditions on measures $\eta , \nu$ under which there exists a positive solution of the boundary value problem

$\begin{equation*}\begin{gathered} -Lv+v^{\alpha }=\eta \quad \text{ in } D,\\ v=\nu \quad \text{ on } \partial D. \end{gathered}\tag{*} \end{equation*}$

The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called $(L,\alpha )$ -superdiffusions).

We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain $E$ in $\mathbb{R}^{d}$ , we establish a 1-1 correspondence between a class of such functionals and a class of $L$ -excessive functions $h$ (which we describe in terms of their Martin integral representation). The Laplace transform of $A$ satisfies an integral equation which can be considered as a substitute for (*).

1. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, forthcoming book.
2. P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 185–206 (French, with English summary). MR 743627
3. G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953-54), 131-295. MR 18:295g
4. Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel, Hermann, Paris, 1975 (French). Chapitres I à IV; Édition entièrement refondue; Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV; Actualités Scientifiques et Industrielles, No. 1372. MR 0488194
Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], vol. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres IX à XI, Revised edition, Publications de l’Institut de Mathématiques de l’Université de Strasbourg [Publications of the Mathematical Institute of the University of Strasbourg], XVIII, Hermann, Paris, 1983 (French). Théorie discrète du potential. [Discrete potential theory]; Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], 1410. MR 727641
Claude Dellacherie and Paul-André Meyer, Erratum: Probabilités et potentiel. Chapitres XII–XVI [Hermann, Paris, 1987; MR0898005 (88k:60002)], Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 600 (French). MR 960549, https://doi.org/10.1007/BFb0084158
5. E. B. Dynkin, Functionals of trajectories of Markov stochastic processes, Doklady Akademii Nauk SSSR 104:5 (1955), 691-694. MR 17:501b
6. E. B. Dynkin, Markov processes. Vols. I, II, Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, vol. 122, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. MR 0193671
7. E. B. Dynkin, The exit space of a Markov process, Uspehi Mat. Nauk 24 (1969), no. 4 (148), 89–152 (Russian). MR 0264768
8. E. B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), no. 3, 1185–1262. MR 1235414
9. Eugene B. Dynkin, An introduction to branching measure-valued processes, CRM Monograph Series, vol. 6, American Mathematical Society, Providence, RI, 1994. MR 1280712
10. E. B. Dynkin, Minimal excessive measures and functions, Trans. Amer. Math. Soc. 258 (1980), no. 1, 217–244. MR 554330, https://doi.org/10.1090/S0002-9947-1980-0554330-5
11. E. B. Dynkin, Superprocesses and their linear additive functionals, Trans. Amer. Math. Soc. 314 (1989), no. 1, 255–282. MR 930086, https://doi.org/10.1090/S0002-9947-1989-0930086-7
12. E. B. Dynkin, A probabilistic approach to one class of nonlinear differential equations, Probab. Theory Related Fields 89 (1991), no. 1, 89–115. MR 1109476, https://doi.org/10.1007/BF01225827
13. E. B. Dynkin, Branching particle systems and superprocesses, Ann. Probab. 19 (1991), no. 3, 1157–1194. MR 1112411
14. E. B. Dynkin, Path processes and historical superprocesses, Probab. Theory Related Fields 90 (1991), no. 1, 1–36. MR 1124827, https://doi.org/10.1007/BF01321132
15. E. B. Dynkin, Additive functionals of superdiffusion processes, Random walks, Brownian motion, and interacting particle systems, Progr. Probab., vol. 28, Birkhäuser Boston, Boston, MA, 1991, pp. 269–281. MR 1146452, https://doi.org/10.1007/978-1-4612-0459-6_14
16. E. B. Dynkin, Superdiffusions and parabolic nonlinear differential equations, Ann. Probab. 20 (1992), no. 2, 942–962. MR 1159580
17. E. B. Dynkin and S. E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure & Appl. Math (1996) (to appear).
18. E. B. Dynkin, S. E. Kuznetsov, Solutions of $Lu = u^{\alpha }$ dominated by -harmonic functions, Journale d'Analyse (1996) (to appear).
19. Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
20. Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
21. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
22. Abdelilah Gmira and Laurent Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991), no. 2, 271–324. MR 1136377, https://doi.org/10.1215/S0012-7094-91-06414-8
23. J.-F. Le Gall, The Brownian snake and solutions of $\Delta u=u^{2}$ in a domain, preprint, 1994.
24. Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700
25. Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914