Superprocesses on ultradistributions

References

• J.A. Acebrón, and A. Rodríguez-Rozas, A new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees, J Comput. Phys. 230 (2011), pp. 7891–7909.
   Web of Science ®Google Scholar
• J.A. Acebrón, A. Rodriguez-Rozas, and R. Spigler, Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees, J. Comput. Phys. 228 (2009), pp. 5574–5591.
   Web of Science ®Google Scholar
• J.A. Acebrón, A. Rodríguez-Rozas, and R. Spigler, Efficient parallel solution of nonlinear parabolic partial differential equations by a probabilistic domain decomposition, J. Sci. Comput. 43 (2010), pp. 135–157.
   Web of Science ®Google Scholar
• R.N. Bhattacharya, et al., Majorizing kernels and stochastic cascades with applications to incompressible Navier--Stokes equations, Trans. Amer. Math. Soc. 355 (2003), pp. 5003–5040.
   Web of Science ®Google Scholar
• F. Cipriano, H. Ouerdiane, and R. Vilela, Mendes; Stochastic solution of a KPP-type nonlinear fractional differential equation, Fract. Calc. Appl. Anal. 12 (2009), pp. 47–56.
   Google Scholar
• E.B. Dynkin, Branching exit Markov systems snd superprocesses, Ann. Probab. 29 (2001), pp. 1833–1858.
   Web of Science ®Google Scholar
• E.B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, AMS Colloquium Publications, Providence (RI), 2002.
   Google Scholar
• E.B. Dynkin, Superdiffusions and positive solutions of nonlinear partial differential equations, AMS, Providence (RI), 2004.
   Google Scholar
• P.J. Fitzsimmons, Construction and regularity of measure-valued Markov branching processes, Israel J. Math. 64 (1988), pp. 337–361.
   Web of Science ®Google Scholar
• P.J.Fitzsimmons, On the Martingale problem for measure-valued Markov branching processes, in Seminar on Stochastic Processes 1991, Progress in Probability, Vol. 29, Springer, Boston (MA), 1992.39–51.
   Google Scholar
• E. Floriani, R. Lima, and R. Vilela Mendes, Poisson--Vlasov: Stochastic representation and numerical codes, Eur. Phys. J. D 46 (2008), pp. 295–302. and 407
   Web of Science ®Google Scholar
• E. Floriani, and R. Vilela, Mendes; A stochastic approach to the solution of magnetohydrodynamics equations, J. Comput. Phys. 242 (2013), pp. 777–789.
   Web of Science ®Google Scholar
• M. Hasumi, Note on the n-dimensional tempered ultradistributions, Tôhoku Math. J. 13 (1960), pp. 94–104.
   Google Scholar
• R.F. Hoskins and J. Sousa Pinto, in Theories of generalized functions: Distributions, ultradistributions and other generalized functions, 2nd ed., Woodhead Publication, Philadelphia (PA), 2011.
   Google Scholar
• Y. LeJan, and A.S. Sznitman, Stochastic cascades and 3-dimensional Navier--Stokes equations, Prob. Theory Relat. Fields 109 (1997), pp. 343–366.
   Web of Science ®Google Scholar
• Z. Li, Measure-Valued Branching Markov Processes, Springer, Berlin, 2011.
   Google Scholar
• H.P. McKean, Application of Brownian motion to the equation of Kolmogorov--Petrovskii--Piskunov, Commun. Pure Appl. Math. 28 (1975), pp. 323–331. 29 (1976), pp. 553--554.
   Web of Science ®Google Scholar
• J.S. Oliveira, Sur un produit multiplicatif de ultradistributions, Bolletino del Un. Mat. Ital. 6 (1982), pp. 943–955.
   Google Scholar
• J.C. Orum, Stochastic cascades and 2D Fourier Navier--Stokes equations, in Lectures on multiscale and multiplicative processes. Available at www.maphysto.dk/publications/MPS-LN/2002/11.pdf.
   Google Scholar
• M. Ossiander, A probabilistic representation of solutions of the incompressible Navier-Stokes equations in ℝ3, Prob. Theory Relat. Fields 133 (2005), pp. 267–298.
   Web of Science ®Google Scholar
• J. Sebastião e Silva, Les fonctions analytiques comme ultra-distributions dans le calcul opérationnel [Analytic functions as ultradistributions in operational calculus], Math. Annalen 136 (1958), pp. 58–96.
   Web of Science ®Google Scholar
• J. Sebastião e Silva, Les séries de multipôles des physiciens et la théorie des ultradistributions [Physicists’ multipole series and ultradistribution theory], Math. Annalen 174 (1967), pp. 109–142.
   Web of Science ®Google Scholar
• E.M. Stein, and R. Shakarchi, Princeton Lectures in Analysis IV, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, Princeton (RI), 2011.
   Google Scholar
• R.Vilela Mendes, Stochastic Solutions of Nonlinear PDE’s and an Extension of Superprocesses, in Stochastic and Infinite Dimensional Analysis, Birkhäuser Trends in Mathematics, C.C. Bernido, M.V. Carpio-Bernido, M. Grothaus, T. Kuna, M.J. Oliveira and J.L. Silva, eds., Springer, Switzerland, 2016.243–262.
   Google Scholar
• R. Vilela Mendes, and F. Cipriano, A stochastic representation for the Poisson-Vlasov equation, Commun. Nonlinear Sci. Num. Simul. 13 (2008), pp. 221–226 and 1736.
   Web of Science ®Google Scholar
• R. Vilela Mendes, Stochastic solutions of nonlinear PDE’s: McKean vs. superprocesses, in Chaos, Complexity and TransportX. Leoncini and M. Leonetti, eds., World Scientific, 2012.143–154. Available at arXiv:1111.5504.
   Google Scholar
• R. Vilela, Mendes, Stochastic solutions of some nonlinear partial differential equations, Stochastics 81 (2009), pp. 279–297.
   Google Scholar
• R. Vilela, Mendes; Poisson--Vlasov in a strong magnetic field: A stochastic solution approach, J. Math. Phys. 51 (2010), p. 043101.
   Web of Science ®Google Scholar
• E.C. Waymire, Probability & incompressible Navier--Stokes equations: An overview of some recent developments, Prob. Surveys 2 (2005), pp. 1–32.
   Google Scholar