On stochastic diffusion equations and stochastic Burgers' equations

Journal of Mathematical Physics

Volumn 37, Issue 1, 1996, Pages 283-307

  Truman, A ^a   Zhao, H Z ^a  

^a SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030547154     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531391     Document Type: Article

References (38)

1. V. P. Maslov, Theorie des Perturbations et Methods Asymptotiques (Dunod, Paris, 1972).
2. A. Truman, "Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation," J. Math. Phys. 18, 2308-2315 (1977).
3. K. D. Elworthy and A. Truman, "The classical limit of quantum mechanics in a curved space background," in Functional Integration: Theory and Applications, edited by J. P. Antoine and E. Tirapegui (Plenum, New York, 1980), pp. 65-87.
4. K. D. Elworthy and A. Truman, "Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation on a Riemannian manifold," J. Math. Phys. 22, 2144-2166 (1981).
5. K. D. Elworthy and A. Truman, "The diffusion equation and classical mechanics: An elementary formula," in Stochastic Processes in Quantum Physics, edited by S. Albeverio et al., Lecture Notes in Physics No. 173 (Springer, Berlin, 1982), pp. 136-146.
6. R. S. R. Varadhan, "Diffusion process in a small time interval," Commun. Pure Appl. Math. XX, 659-685 (1967).
7. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, Berlin, 1984).
8. K. D. Elworthy, A. Truman, and H. Z. Zhao, "Approximate traveling waves for the generalized KPP equations and classical mechanics," Proc. R. Soc. London, Ser. A 446, 529-554 (1994).
9. K. D. Watling, "Formulas for solutions to (possibly degenerate) diffusion equation exhibiting semiclassical asymptotics," in Stochastics and Quantum Mechanics, edited by A. Truman and I. M. Davies (World Scientific, Singapore, 1992), pp. 248-271.
10. A. Truman and H. Z. Zhao, "The stochastic Hamilton Jacobi equation, stochastic heat equation, and Schrödinger equation," in Stochastic Analysis and Applications, edited by A. Truman, I. M. Davies, and K. D. Elworthy (World Scientific, Singapore, 1996), pp. 441-464.
11. M. Kardar, G. Parisi, and Y. C. Zhang, "Dynamical scaling of growing interfaces," Phys. Rev. Lett. 56, 889-892 (1986).
12. V. I. Arnold, Singularities of Caustics and Wave Fronts (Kluwer Academic, Dordrecht, 1990).
13. F. Flandoli, "Some remarks on regularity theory and stochastic flows for parabolic SPDE's" (Preprint di Mathematica n.135, Scuola Normale Superiore di Pisa, 1992).
14. K. D. Elworthy, Stochastic Differential Equations on Manifolds, London Mathematical Society Lecture Notes Series 70 (Cambridge University, Cambridge, 1982).
15. L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 2. Itô Calculus (Wiley, New York, 1987).
16. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Kodansha, Amsterdam, Tokyo, 1981).
17. K. D. Elworthy, "Geometric aspects of diffusions on manifolds," LNM (Springer, Berlin 1988), Vol. 1362, pp. 277-425.
18. H. P. McKean, Stochastic integrals (Academic, New York, 1969).
19. L. Markus and A. Weerasinghe, "Stochastic oscillators," J. Diff. Eqn. 21, 288-314 (1988).
20. S. Albeverio, A. Hilbert, and E. Zehnder, "Hamiltonian systems with stochastic forces: Nonlinear versus linear and a Girsanov formula," Stochastic Stoch. Rep. 39, 159-188 (1992).
21. N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators (Springer-Verlag, Berlin, 1992).
22. J. M. Burgers, "Applications of a model system to illustrate some points of the statistical theory of free turbulence," Proc. Acad. Sci. Amsterdam 43, 2-12 (1940).
23. J. M. Burgers, "A mathematical model illustrating the theory of turbulence," Adv. Appl. Mech. 1, 171-199 (1948).
24. xx," Commun. Pure Appl. Math. 3, 201-230 (1950).
25. L. Bertini, N. Cancrini, and G. Jona-Lasinio, "The stochastic Burgers equation," Commun. Math. Phys. 165, 211-232 (1994).
26. L. Bertini and N. Cancrini, "The stochastic heat equation: Feynman-Kac formula and intermittence," J. Stat. Phys. 78, 1377-1401 (1995).
27. S. Albeverio, S. Molchanov, and D. Surgailis, "Stratified structure of the universe and Burgers equation - A probability approach," Probab. Theory Relat. Fields 100, 457-484 (1994).
28. G. Da Prato, A. Debusche, and R. Temam, "Stochastic Burgers equation" (Preprint di Mathematica n.27, Scuola Normale Superiore di Pisa, 1993).
29. H. Holden, T. Lindstrom, B. Oksendal, J. Uboe, and T. S. Zhang, "The Burgers' equation with a noise force and the stochastic heat equations," Comm. PDE 19, 119-141 (1994).
30. H. Holden, T. Lindstrom, B. Oksendal, J. Uboe, and T. S. Zhang, "The stochastic Wick-type Burgers equation," in Stochastic Partial Differential Equations, edited by A. M. Ethridge, London Mathematical Society Lecture Note Series 216 (Cambridge University, Cambridge, 1995), pp. 141-161.
31. B. Oksendal, "Stochastic partial differential equations and applications to hydrodynamics," in Stochastic Analysis and Applications in Physics, edited by A. I. Cardaso, M. de Faria, J. Potthoff, and L. Streit, Nato ASI Series, Vol. 449 (Kluwer, Dordrecht, 1994). pp. 283-305.
32. T. J. Newman, "Exact results for a model of interface growth," Phys. Rev. E 49, 2525-2527 (1994).
33. A. S. Mishchenko, V. E. Shatolov, and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator (Springer-Verlag, Berlin, 1990).
34. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983).
35. T. Dankel, "Stochastic KdV equations" (preprint).
36. I. Davies and A. Truman, "Laplace asymptotic expansions of conditional Wiener integrals and generalized Mehler kernel formulas," J. Math. Phys. 23, 2059-2070 (1982).
37. W. H. Fleming, Controlled Markov Processes and Viscosity Solution of Nonlinear Evolution Equations (Lezioni Fermiane, Scuola Normale Superiore di Pisa, 1986).
38. W. H. Fleming, "The Cauchy problem for a nonlinear first order partial differential equation," J. Diff. Eqn. 5, 515-530 (1969).