  In this work we review the application of a functional method, serving as an
alternative to the Wilsonian Exact Renormalization approach, to stringy bosonic
$\sigma$-models with metric and dilaton backgrounds on a spherical world sheet
[1]. We derive an exact evolution equation for the dilaton with the amplitude
of quantum fluctuations, driven by the kinetic term of the two-dimensional
world-sheet theory. The linear dilaton conformal field theory, corresponding to
a linearly (in cosmic Einstein-frame time) expanding Universe, appears as a
trivial fixed point of this equation. With the help of conformal-invariance
conditions, we find a logarithmic dilaton as another, exact and non trivial,
fixed-point solution. Cosmological implications of our solutions are briefly
discussed, in particular the transition (exit) from the expanding Universe of
the linear dilaton to the Minkowski vacuum, corresponding to the non-trivial
fixed point of our generalised flow. This novel renormalization-group method
may therefore offer new insights into exact properties of string theories of
physical significance.
