Spontaneously started signals with white noise

P. A. Yaskov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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.

Keywords: Brownian motion, Girsanov theorem, Cameron–Martin theorem, Gaussian multiplicative chaos.

Received: 16.08.2024

DOI: 10.4213/tvp5741

 English version: Theory of Probability and its Applications, 2025, 69:4, 565–578

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