##
**Scale renormalization and random solutions of the Burgers equation.**
*(English)*
Zbl 0624.60071

This paper shows how various limit theorems could be useful to study the asymptotic behaviour (in space) of the solution of the Burgers’ equation
\[
u_ t+uu_ x = \mu u_{xx},\quad \mu >0,
\]
with the initial data \(u_ 0(x)\) a continuous stationary stochastic process. It is well known that provided the growth condition \(\int^{x}_{0}u_ 0(\xi)d\xi =o(x^ 2)\) holds, for which the existence of the mean of \(u_ 0\) is enough, the above equation can be changed into the heat equation with a simple transformation. This allows to establish that u(x,t) is stationary in space for any \(t>0.\)

The main result of the paper is that if \(\sigma^ 2(R)=var(\int^{R}_{0}u_ 0(y)dy)\) diverges as \(R\to \infty\), and the renormalized process \[ \sigma (R)^{-1}\int^{Ry}_{0}\{u(t,x)-Eu_ 0(x)\}dx,\quad 0\leq y\leq 1, \] converges weakly as \(R\to \infty\) for \(t=0\), the same limit is obtained for any time \(t>0\). The nature of the limit depends on the correlation of \(u_ 0\). If this is strongly mixing the corresponding central limit theorem in I. Ibragimov, Teor. Veroyatn. Primen. 20, 134-141 (1975; Zbl 0335.60023), yields the Brownian motion. By using a noncentral limit theorem as in R. Dobrushin and P. Major, Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 27-52 (1979; Zbl 0397.60034), different limiting behaviours are exhibited by \(u_ 0(x)=h(G(x))\), where G is a Gaussian process with suitable long-range dependence.

Finally the extension of the main result to the case \(\mu\downarrow 0\) is discussed and a counterexample is given for which the growth condition does not hold.

The main result of the paper is that if \(\sigma^ 2(R)=var(\int^{R}_{0}u_ 0(y)dy)\) diverges as \(R\to \infty\), and the renormalized process \[ \sigma (R)^{-1}\int^{Ry}_{0}\{u(t,x)-Eu_ 0(x)\}dx,\quad 0\leq y\leq 1, \] converges weakly as \(R\to \infty\) for \(t=0\), the same limit is obtained for any time \(t>0\). The nature of the limit depends on the correlation of \(u_ 0\). If this is strongly mixing the corresponding central limit theorem in I. Ibragimov, Teor. Veroyatn. Primen. 20, 134-141 (1975; Zbl 0335.60023), yields the Brownian motion. By using a noncentral limit theorem as in R. Dobrushin and P. Major, Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 27-52 (1979; Zbl 0397.60034), different limiting behaviours are exhibited by \(u_ 0(x)=h(G(x))\), where G is a Gaussian process with suitable long-range dependence.

Finally the extension of the main result to the case \(\mu\downarrow 0\) is discussed and a counterexample is given for which the growth condition does not hold.

Reviewer: M.Piccioni

### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

60G10 | Stationary stochastic processes |

60F05 | Central limit and other weak theorems |