On Small Deviations of Exponential Functional of Brownian Motion With Applications

We provide the asymptotic bounds on the log-Laplace transform of an exponential functional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> </mml:math> of a Brownian motion. Using excursion theory, in the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> </mml:math> taken at random time, we discover the nice explicit formula of its Laplace transform. We apply the new formula to provide a connection between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>A</mml:mi> </mml:math> and excursions of a two-dimensional Brownian motion from a hyperplane. Finally, we apply the new results to the problem of small deviations of Gaussian multiplicative chaos on Euclidean balls.