On a probability problem for a one-dimensional heat equation

The system (2) for random amplitudes $W_i(t)$ (where $f(t)$ is the derivative of a Wiener process) was considered in [1] in connection with the stochastic heat equation (1) and the two following assertions were obtained:
(a) a solution of the system (2) is a random (normal) element in the Hilbert space $l_2$ for every $t>0$;
(b) almost all solutions $\{W_i(t)\}$ are rapidly decreasing sequences for every $t>0$.
In the present note a simple proof of the assertion (a) is given and the assertion (b) is shown to be wrong.