Asymptotic properties of solutions of the Cauchy problem for the Burgers’ equation with random initial conditions. (English. Ukrainian original) Zbl 0941.60044

Leonenko, N. N.; Parkhomenko, V. V.

Theory Probab. Math. Stat. 53, 89-95 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 80-86 (1995).

Let \(u(x,t),\;x \in R,\;t>0,\) be a solution of the Cauchy problem for Burgers’ equation with weakly dependent random initial conditions. It is proved that under some conditions the finite-dimensional distributions of the process \[ X^t(a)={{t^{1/2+\alpha /4}}\over{\sqrt{L(\sqrt t)}}} u(a\sqrt t,t),\quad a \in R, \;\alpha \in (0,1), \] converge weakly as \(t \to \infty\) to the finite-dimensional distributions of a stationary Gaussian process \(X(a).\) The spectral representation of the process \(X(a)\) is obtained.

Reviewer: A.Ya.Olenko (Kyïv)

MSC:

60F05	Central limit and other weak theorems
70L05	Random vibrations in mechanics of particles and systems
35Q53	KdV equations (Korteweg-de Vries equations)
60G15	Gaussian processes

Keywords:

nonlinear partial differential equations; spectral representation; Gaussian initial conditions; long-range dependence