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Ultradistribution Spaces: Superprocesses and Nonlinear Differential Problems

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From Particle Systems to Partial Differential Equations (PSPDE 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 209))

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Abstract

From branching particle systems one obtains, in the scaling limit, measure-value processes called superprocesses. In addition to providing models for evolving populations, superprocesses provide probabilistic representations of the solutions of nonlinear partial differential equations (PDE’s). However, the class of PDE’s that can be handled by measure-valued superprocesses is rather limited. This suggests an extension of the configuration space of superprocesses to ultradistribution-valued processes which have a wider range of applications in the solution of PDE’s. The relevance of the superprocess representation of PDE’s to deal with nonlinear singular problems is also discussed.

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References

  1. Hoskins, R.F., Sousa Pinto, J.: Theories of Generalized Functions: Distributions, Ultradistributions and Other Generalized Functions, 2nd edn. Woodhead Publ, Philadelphia (2011)

    MATH  Google Scholar 

  2. Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6, 351–407 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Roumieu, C.: Sur quelques extensions de la notion de distribution. Ann. Sci. École Norm. Sup. Sér. 3(77), 41–121 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  4. Pilipović, S.: Beurling-Gevrey tempered ultradistributions as boundary values. Portugaliae Math. 48, 483–504 (1991)

    MathSciNet  MATH  Google Scholar 

  5. Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Debrouwere, A., Vindas, J.: On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth. arXiv:1608.07859

  7. Sebastião e Silva, J.: Les fonctions analytiques comme ultra-distributions dans le calcul opérationnel. Math. Annalen 136, 58–96 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sebastião e Silva, J.: Les séries de multip ôles des physiciens et la théorie des ultradistributions. Math. Annalen 174, 109–142 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sebastião e Silva, J.: Integrals and order of distributions. Inst. Gulbekian de Ciencia, Lisboa 35 (1964)

    Google Scholar 

  10. Estrada, R.: The Cesàro behavior of distributions. Proc. R. Soc. Lond. A 454, 2425–2443 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hasumi, M.: Note on the n-dimensional tempered ultradistributions. Tôhoku Math. J. 13, 94–104 (1960)

    Article  MATH  Google Scholar 

  12. Dynkin, E.B.: Diffusions, Superdiffusions and Partial Differential Equations. AMS Colloquium Pubs, Providence (2002)

    Book  MATH  Google Scholar 

  13. Dynkin, E.B.: Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. AMS, Providence (2004)

    Book  MATH  Google Scholar 

  14. Li, Z.: Measure-Valued Branching Markov Processes. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  15. Stein, E.M., Shakarchi, R.: Princeton Lectures in Analysis IV, Functional Analysis: Introduction to Further Topics in Analysis. Princeton Universiyt Press, Princeton (2011)

    MATH  Google Scholar 

  16. Vilela Mendes, R.: Stochastic solutions of nonlinear PDE’s: McKean versus superprocesses. In: Leoncini, X., Leonetti, M. (eds.) Chaos, Complexity and Transport, pp. 143–154. World Scientific (2012)

    Google Scholar 

  17. Vilela Mendes, R.: Stochastic solutions of nonlinear pde’s and an extension of superprocesses. In: Stochastic and Infinite dimensional Analysis, Trends in Math., pp. 243–261. Birkhäuser (2016)

    Google Scholar 

  18. Vilela Mendes, R.: Superprocesses on ultradistributions. Stochastics (2016). https://doi.org/10.1080/17442508.2016.1269768

  19. Dynkin, E.B.: Branching exit Markov systems snd superprocesses. Ann. Probab. 29, 1833–1858 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Isr. J. Math. 64, 337–361 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fitzsimmons, P.J.: On the Martingale problem for measure-valued Markov branching processes. In: Seminar on Stochastic Processes 1991, Progress in Probability, vol. 29, pp. 39–51. Springer, Boston (1992)

    Google Scholar 

  22. Schwartz, L.: Sur l’impossibilité de la multiplication des distributions. C.R. Acad. Sci. Paris 239, 847–848 (1954)

    MathSciNet  MATH  Google Scholar 

  23. Li, C.K.: A review on the products of distributions. In: Kenan Tas et al. (ed.) Mathematical Methods in Engineering. Springer (2007)

    Google Scholar 

  24. Keyfitz, B.L., Kranzer, H.C.: A strictly hyperbolic system of conservation laws admitting singular shocks. In: Nonlinear Evolution Equations that Change Type, pp. 107–125. Springer (1990)

    Google Scholar 

  25. Le Floch, Ph.: Shock waves for nonlinear hyperbolic systems in nonconservative form. Inst. for Math. and Applications - Univ. Minnesotta, preprint , vol. 593 (1989)

    Google Scholar 

  26. Lafon, F., Oberguggenberger, M.: Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case. J. Math. Anal. Appl. 160, 93–106 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Langlais, M.: Generalized functions solutions of monotone and semilinear parabolic equations. Monatshefte Math. 110, 117–136 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mirkov, R., Pilipovic, S., Selesi, D.: Generalized stochastic processes in algebras of generalized functions. J. Math. Anal. Appl. 353, 260–270 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oberguggenberger, M.: Generalized functions and stochastic processes. Prog. Probab. 36, 215–229 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions Mathematics and its Applications, vol. 537. Kluwer (2001)

    Google Scholar 

  31. Steinbauer, R., Vickers, J.A.: The use of generalized functions and distributions in general relativity. Class. Q. Grav. 23, R91–R114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Colombeau, J.F.: New Generalized Functions and Multiplication of Distributions. North-Holland-Elsevier (1984)

    Google Scholar 

  33. Colombeau, J.F.: Nonlinear generalized functions: their origin, some developments and recent advances. São Paulo J. Math. Sci. 7, 201–239 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Aragona, J., Juriaans, S.O., Colombeau, J.F.: Locally convex topological algebras of generalized functions: compactness and nuclearity in a nonlinear context. Trans. Am. Math. Soc. 367, 5399–5414 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mikusinski, J.: Sur la méthode de gé neralisation de M. Laurent Schwartz sur la convergence faible. Fundam. Math. 35, 235–239 (1948)

    Article  MATH  Google Scholar 

  36. Debrouwere, A., Vernaeve, H., Vindas, J.: Optimal embeddings of ultradistributions into differential algebras. arXiv:1601.03512

  37. Vilela Mendes, R.: Stochastic solutions of some nonlinear partial differential equations. Stochastics 81, 279–297 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Vilela Mendes, R.: Poisson-Vlasov in a strong magnetic field: A stochastic solution approach. J. Math. Phys. 51, 043101 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Centro de Matemática e Aplicações Fundamentais, University of Lisbon, Lisbon, Portugal

    R. Vilela Mendes

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  1. R. Vilela Mendes
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Correspondence to R. Vilela Mendes .

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Editors and Affiliations

  1. Center for Mathematics, Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa, Portugal

    Patrícia Gonçalves

  2. Centre of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

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Vilela Mendes, R. (2017). Ultradistribution Spaces: Superprocesses and Nonlinear Differential Problems. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations. PSPDE 2015. Springer Proceedings in Mathematics & Statistics, vol 209. Springer, Cham. https://doi.org/10.1007/978-3-319-66839-0_12

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