On the Solution to the Probability Problem for Non-uniformly Distributed System

A dynamical system is considered which is described by a parabolic equation in a circle of length $2l$ when acted upon by an undistributed stochastic force $f(t)$ (white noise)
$$ \frac{\partial W(x,t)}{\partial t}-\frac{\partial^2W(x,t)}{\partial x^2}=\delta(x)f(t). $$
The Green's function for this system (a countable additive measure in the phase space) is constructed. It is proved that almost all $w(x)$ are infinitely differentiable. This measure is not quasi-invariant.