Stochastic solutions of some nonlinear partial differential equations

References

• Acebrón , J.A. and Spigler , R. 2007 . A new probabilistic approach to the domain decomposition method . Lect. Notes Comput. Sci. Eng. , 55 : 475 – 480 .
   Google Scholar
• Acebrón , J.A. , Busico , M.P. , Lanucara , P. and Spigler , R. 2005 . Domain decomposition solution of elliptic boundary-value problems via Monte-Carlo and quasi-Monte-Carlo methods . SIAM J. Sci. Comput. , 27 : 440 – 457 .
   Web of Science ®Google Scholar
• Bass , R.F. 1995 . Probabilistic Techniques in Analysis , New York : Springer .
   Google Scholar
• Bass , R.F. 1998 . Diffusions and Elliptic Operators , New York : Springer .
   Google Scholar
• Bhattacharya , R.N. , Chen , L. , Dobson , S. , Guenther , R.B. , Orum , C. , Ossiander , M. , Thormann , E. and Waymire , E.C. 2003 . Majorizing kernels and stochastic cascades with applications to incompressible Navier–Stokes equations . Trans. Am. Math. Soc. , 355 : 5003 – 5040 .
   Web of Science ®Google Scholar
• Blumenthal , R.M. and Getoor , R.K. 1968 . Markov Processes and Potential Theory , New York : Academic Press .
   Google Scholar
• Champneys , A. , Harris , S. , Toland , J. , Warren , J. and Williams , D. 1995 . Algebra, analysis and probability for a coupled system of reaction-diffusion equations . Phil. Trans. R. Soc. Lond. A , 350 : 69 – 112 .
   Web of Science ®Google Scholar
• Cipriano , F. , Ouerdiane , H. and Vilela Mendes , R. 2009 . Stochastic solution of a KPP-type nonlinear fractional differential equation . Fractional Calculus and Applied Analysis , 12 : 47 – 56 .
   Google Scholar
• Constantin , P. and Iyer , G. 2008 . A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations . Comm. Pure Appl. Math. , 61 : 330 – 345 .
   Web of Science ®Google Scholar
• Courant , R. , Friedrichs , K. and Lewy , H. 1928 . Über die partiellen Differenzengleichungen der mathematischen Physik . Mat. Ann. , 100 : 32 – 74 .
   Web of Science ®Google Scholar
• Dawson , D.A. and Perkins , E. 1991 . Historical processes . Memoirs Am. Mat. Soc. , 93 : 454
   Web of Science ®Google Scholar
• Dynkin , E.B. 1991 . A probabilistic approach to one class of nonlinear differential equations . Probab. Theory Relat. Fields , 89 : 89 – 115 .
   Web of Science ®Google Scholar
• Dynkin , E. 1991 . Branching particle systems and superprocesses . Ann. Probab. , 19 : 1157 – 1194 .
   Web of Science ®Google Scholar
• Dykin , E.B. 1993 . Superprocesses and partial differential equations . Ann. Probab. , 21 : 1185 – 1262 .
   Web of Science ®Google Scholar
• Dynkin , E.B. 2002 . Diffusions, Superdiffusions and Partial Differential Equations , Providence, RI : AMS .
   Google Scholar
• Dynkin , E. 2004 . Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations , Providence, RI : AMS .
   Google Scholar
• Eleutério , S.M. and Vilela Mendes , R. 1987 . Stochastic mechanics of abelian gauge theories . J. Phys. A , 20 : 6411
   Google Scholar
• Floriani , E. , Lima , R. and Vilela Mendes , R. 2008 . Poisson–Vlasov: Stochastic representation and numerical codes . Eur. J. Phys. D , 46 : 295 – 302, 407 .
   Web of Science ®Google Scholar
• Graham , C. and Méléard , S. 2001 . “ Probabilistic tools and Monte-Carlo aproximations for some Boltzmann equations ” . In ESAIM Proceedings , Edited by: Coquel , F. and Cordier , S. vol. 10 , 77 – 126 . Les Ulis : EDP Sciences .
   Google Scholar
• Kolmogorov , A. , Petrovskii , I. and Piskunov , N. 1937 . Étude de l'équation de la diffusion avec croissance de la quantité de la matière et son application a un problème biologique . Moscow Univ. Bull. Math. , 1 : 1 – 25 .
   Google Scholar
• LeJan , Y. and Sznitman , A.S. 1997 . Stochastic cascades and 3-dimensional Navier–Stokes equations . Probab. Theory Relat. Fields , 109 : 343 – 366 .
   Web of Science ®Google Scholar
• Lyne , O.D. 2000 . Travelling waves for a certain first-order coupled PDE system . Electron. J. Probab. , 5 ( 14 ) : 1 – 40 .
   Google Scholar
• Lyne , O.D. and Williams , D. 2001 . Weak solutions for a simple hyperbolic system . Electron. J. Probab. , 6 ( 20 ) : 1 – 21 .
   Google Scholar
• McKean , H.P. 1966 . A class of Markov processes associated with nonlinear parabolic equations . Proc. Nat. Acad. Sci. , 56 : 1907 – 1911 .
   PubMed Web of Science ®Google Scholar
• McKean , H.P. 1967 . An exponential formula for solving Bolzmann's equation for a Maxwellian gas . J. Comb. Theory , 2 : 358 – 382 .
   Google Scholar
• McKean , H.P. 1975 . Application of brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov . Comm. Pure Appl. Math. , 28 : 323 – 331 . 29 (1976) 553–554
   Web of Science ®Google Scholar
• Neveu , J. 1987 . “ Multiplicative martingales for spatial branching processes ” . In Seminar on Stochastic Processes , Progress in Probability and Statistics Vol. 15 , 223 – 242 .
   Google Scholar
• Orum , J.C. 2002 . “ Stochastic cascades and 2D Fourier Navier–Stokes equations ” . In Lectures on Multiscale and Multiplicative Processes www.maphysto.dk/publications/MPS-LN/2002/11.pdf
   Google Scholar
• Ossiander , M. 2005 . A probabilistic representation of solutions of the incompressible Navier–Stokes equations in ℝ3 . Probab. Theory Relat. Fields , 133 : 267 – 298 .
   Web of Science ®Google Scholar
• Seixas , J. and Vilela Mendes , R. 1992 . Large deviation analysis of multiplicity fluctuations . Nucl. Phys. , B383 : 622 – 642 .
   Google Scholar
• Talay , D. and Tubaro , L. 1996 . “ Probabilistic models for nonlinear partial differential equations ” . In Lecture Notes in Mathematics Vol. 1627 ,
   Google Scholar
• Vilela Mendes , R. 1992 . Stochastic processes and the non-perturbative structure of the QCD vacuum . Z. Phys. C , 54 : 273 – 281 .
   Google Scholar
• Vilela Mendes , R. and Cipriano , F. 2008 . A stochastic representation for the Poisson–Vlasov equation . Commun. Nonlinear Sci. Numer. Simul. , 13 : 221 – 226 .
   Web of Science ®Google Scholar
• E. Waymire, Lectures on multiscale and multiplicative processes, www.maphysto.dk/publications/MPS-LN/2002/11.pdf (http://www.maphysto.dk/publications/mps-ln/2002/11.pdf)
   Google Scholar
• Waymire , E.C. 2005 . Probability & incompressible Navier–Stokes equations: An overview of some recent developments . Probab. Surveys , 2 : 1 – 32 .
   Google Scholar