On the evolution of distributed systems when there is a fluctuation of the density on the boundary

A dynamical system is considered which is described by a parabolic equation in a circle of length $2\pi$ when acted upon by an undistributed stochastic source with a power $\dot\pi(t)$ (the derivative of Poisson's process):
$$ \frac{\partial W(x,t)}{\partial t}-D^2\frac{\partial^2W(x,t)}{\partial x^2}=\delta(x)\dot\pi(t). $$
The characteristic functional for this system which defines a countable additive measure iii the phase space is constructed. It is proved that almost all $W(x)$ are infinitely differentiable. This measure is not quasi-invariant.