  The inviscid Burgers equation with random and spatially smooth forcing is
considered in the limit when the size of the system tends to infinity. For the
one-dimensional problem, it is shown both theoretically and numerically that
many of the features of the space-periodic case carry over to infinite domains
as intermediate time asymptotics. In particular, for large time $T$ we
introduce the concept of $T$-global shocks replacing the notion of main shock
which was considered earlier in the periodic case (1997, E et al., Phys. Rev.
Lett. 78, 1904). In the case of spatially extended systems these objects are no
anymore global. They can be defined only for a given time scale and their
spatial density behaves as $\rho(T) \sim T^{-2/3}$ for large $T$. The
probability density function $p(A)$ of the age $A$ of shocks behaves
asymptotically as $A^{-5/3}$. We also suggest a simple statistical model for
the dynamics and interaction of shocks and discuss an analogy with the problem
of distribution of instability islands for a simple first-order stochastic
differential equation.
