Abstract
We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.
Highlights
The theory of multiplicative chaos was created by Kahane [20, 21] in the 1980’s in order to obtain a continuous counterpart of the multiplicative cascades, which were proposed by Mandelbrot in early 1970’s as a model for turbulence
During the last 10 years there has been a new wave of interest on multiplicative chaos, due to e.g. its important connections to Stochastic Loewner Evolution [3, 29, 15], quantum field theories and quantum gravity [18, 13, 14, 24, 6, 23], models in finance and turbulence [25, Section 5], and the statistical behaviour of the Riemann zeta function over the critical line [16, 27]
Assuming that the Xn are nice approximations of the field X as explained above, Kahane’s theory yields that in case β ∈ (0, 1) the convergence μn w→∗ μβ takes place almost surely and the obtained chaos μβ is non-trivial. It is an example of subcritical Gaussian chaos, and, as we shall soon recall in more detail, in this normalisation β = 1 appears as a critical value
Summary
The theory of multiplicative chaos was created by Kahane [20, 21] in the 1980’s in order to obtain a continuous counterpart of the multiplicative cascades, which were proposed by Mandelbrot in early 1970’s as a model for turbulence. Assuming that the Xn are nice approximations of the field X as explained above, Kahane’s theory yields that in case β ∈ (0, 1) the convergence μn w→∗ μβ takes place almost surely and the obtained chaos μβ is non-trivial. It is an example of subcritical Gaussian chaos, and, as we shall soon recall in more detail, in this normalisation β = 1 appears as a critical value. An important issue is to understand when the obtained chaos measure is independent of the choice of the approximating fields Xn. As mentioned before, Kahane’s seminal work contained some results in this direction.
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