Spontaneously Started Signals with White Noise

We give a complete description of the conditions for equivalency of the distribution of Brownian motion with randomly started drift of the square root type to the Wiener measure. This problem is closely related to the theory of Gaussian multiplicative chaos (GMC). We develop a new elementary method to prove the existence of the density of the corresponding distribution, which, in fact, is related to one-dimensional GMC in the subcritical regime.