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The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
In this article we study imaginary Gaussian multiplicative chaos-namely a family of random generalized functions which can formally be written as e(iX(x)), where X is a log-correlated real-valued Gaussian field on R-d, that is, it has a logarithmic singula ...
In this article, we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. Th ...
We show that the imaginary multiplicative chaos exp(i beta Gamma) determines the gradient of the underlying field Gamma for all log-correlated Gaussian fields with covariance of the form -log|x-y|+g(x,y) with mild regularity conditions on g, for all d >= 2 ...