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The Burgers equation with a noisy force and the stochastic heat equation. (English) Zbl 0804.35158
As is well known, the classical Burgers equation can be reduced to the heat equation by means of the Cole-Hopf transformation. The authors extend this technique to tackle the Burgers equation with space-time white noise. Since the equation is nonlinear, a difficulty arises concerned with multiplication of distributions. This is overcome by interpreting the nonlinear term as Wick product.

MSC:
35R60 PDEs with randomness, stochastic partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:
[1] Burgers, J. M. 1974. ”The Nonlinear Diffusion Equation.”. Reidel. · Zbl 0302.60048
[2] Forsyth, A.R. 1906. ”Theory of Differential Equations, Part IV - Partial Differential Equations”. Vol. VI, Cambridge University Press.
[3] Gurbatov, S., Malakhov, A. and Saichev, A. 1991. ”Nonlinear random waves and turbulence in nondispersive media: waves, rays and particles.”. Manchester University Press. · Zbl 0860.76002
[4] Gjessing, H., Holden, H., Lindstrøm, T. Øksendal, B., Ubøe, J. and Zhang, T.S. ”New Trends in Probability Theory and Mathematical Statistics.”. Edited by: Melnikov, A. Moscow: TVP Press. The Wick product. To appear · Zbl 0820.60048
[5] Gelfand, I. M. and Vilenkin, N. Y. 1964. ”Generalized Functions”. Applications of Harmonic Analysis. Academic Press. English translation
[6] Hida, T. 1980. ”Brownian Motion”. Springer-Verlag. · Zbl 0423.60063
[7] Hida T., Forthcoming book
[8] Holden H., Stochastic boundary value problems. A white noise functional approach · Zbl 0792.60055
[9] Ito, K. 1951. ”Multiple Wiener integral”. Vol. 3, 157–169. Japan: J. Math. Soc. · Zbl 0044.12202
[10] Kardar, M., Parisi, G. and Zhang, Y.C. 1986. ”Dynamic scaling of growing interfaces.”. Vol. 56, 889–892. Phys. Rev. Lett. · Zbl 1101.82329
[11] Krug, J. and Spohn, H. 1991. ”Kinetic roughening of growing surfaces. In: Solids Far From Equilibrium: Growth, Morphology, and Defects.”. Edited by: Godreche, C. 479–582. Cambridge University Press.
[12] Kuznetsov, N.N. and Rozhdestvenskii, B.L. 1961. ”The solution of Cauchy’s problem for a system of quasi-linear equations in many independent variables.”. Vol. 1, 241–248. Comp. Phys. Math. Phys. · Zbl 0139.10804
[13] Lindstrøm, T. øksendal, B. and Ubøe: J. 1991. ”Stochastic differential equations involving positive noise.”. Edited by: Barlow, M. and Bingham, N. 261–303. Stochastic Analysis. Cambridge Univ. Press. · Zbl 0783.60055
[14] Lindstrøm, T. Øksendal and, B. and Ubøe, J. 1992. ”Wick multiplication and Ito-Skorohod stochastic differential equations.”. 183–206. Cambridge University Press. Ideas and Methods in Mathematical Analysis, Stochastics and Applications. · Zbl 0760.60057
[15] Lindstrøm, T. Øksendal, B. and Ubøe: J. 1991. ”Stochastic modelling of fluid flow in porous media.”. Edited by: Chen, S. and Yong, J. 156–172.
[16] Nualart, D. and Zakai, M. 1989. ”Generalized Brownian functionals and the solution to a stochastic partial differential equation.”. Vol. 84, 279–296. J. Functional Analysis. · Zbl 0682.60046
[17] Potthoff, J. and Streit: L. 1991. ”A characterization of Hida distributions.”. Vol. 101, 212–229. J. Funct. Anal. · Zbl 0826.46035
[18] Sinai, Ya. G. 1991. ”Two results concerning asymptotic behavior of solutions of the Burgers equation with force.”. Vol. 64, 1–12. J. Stat. Phys. · Zbl 0978.35500
[19] DOI: 10.1007/BFb0074920 · doi:10.1007/BFb0074920
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