The radon measure $\mu$ on [0,1] called GMC (Gaussian multiplicative chaos) satisfies the following:
- $$(1/c)|I|^{a}\leq\mu(I)\leq c|I|^{b},$$
- $$\sup_{x\in [0,1]}\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))^{1-\eta}}\leq K$$
iff $\eta\in [q,1]$ for some $q>0$ (otherwise the ratio is infinite). So it is far from being doubling.
- It can be approximated by flat measures $d\mu_{\varepsilon}:=\rho_{\varepsilon}dx$ i.e.
$$\mu_{\varepsilon}\stackrel{L^{2} ~\&~ a.s.}{\to }\mu$$
convergence in $L^{2}$ and almost surely.
The probabilistic properties:
- It satisfies a scaling law:
$$ \mu[0,t]\stackrel{d}{=}te^{G_{t}-E[G^{2}_{t}]}\mu[0,1] ,$$
where $G_{t}\sim N(0,ln(1/t))$.
- A tail estimate $$P[\mu[0,1]>t]=\frac{c_{1}}{t^{s}}+O(\frac{1}{t^{s}})$$
Q: What is a natural tangent distribution object for such measures? Do we have existence?
Attempts 1) Hochnan proved that for all Radon measures we can define the Quasi-palm distributions. There the approach is to study the "scenery flow"
$$\frac{\mu(e^{-t}A)}{\mu(e^{-t}[0,1])}$$
for $A\subset [0,1]$.
2) A more promising approach is via the tangent distributions of Mörters and Preiss. The average $\phi$-density is
$$\lim_{t\to 0}\frac{1}{\ln(t)}\int^{1}_{t}\frac{\mu(B_{r}(x))}{\phi(r)}\frac{dr}{r}.$$
The $\phi$-tangent distribution is the limit points of
$$\lim_{t\to 0}\frac{1}{\ln(t)}\int^{1}_{t}1_{M}(\frac{\mu(B_{r}(x))}{\phi(r)})\frac{dr}{r},$$
where M is any Borel set of positive radon measures. Here I am experimenting with various gauge functions: a)using the tail estimate b)some of the conjectured densities eg. see.
