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Density of integral of exponential of Gaussian bailiwick with logcovariance
Here we make use of the Gaussian bailiwick $Vin N(0,ln(delta/epsilon))$ over $[0,1]$ with covariance
commence{equation}
mathbb{E}[V(x_{1} )V(x_{2} ) ]=ln(frac{delta}{x_{2}x_{1}vee epsilon})=left{commence{matrix}
ln(frac{delta}{epsilon} ) &, x_{2}x_{1}leq epsilon
ln(frac{delta}{x_{2}x_{1}}) &, deltageq x_{2}x_{1}geq epsilon
0&,x_{2}x_{1}>delta
stop{matrix}capable..
stop{equation}
assume the touchstone $mu(x)=int_{0}^{x}e^{barrier{V}(s)}ds$, the place $barrier{V}(s)=V(s)frac{1}{2}E(V^{2}(s))$ (in order that $E[mu(x)]=x$).
Q: We attempt to appraise the likelihood
$$P[aleq mu(1)leq b]$$
for common $0<a<b$ (or aircraft higher the density obligation of $mu$).
Some feedback from the literature

In the paper On the density capabilities of integrals of Gaussian random fields, they hunt the tail $F^{c}(a)=P[aleq mu(1)]$ for sizable $a$. (they too array that the density $f(a)=F'(a)$ exists and has Gaussian tail).

In the papers On the coterie, GMC…, The distribution of Gaussian multiplicative chaos on the unit interval, they handle to compute the density within the illustration $epsilon=0$ above.

In The Integral of Geometric Brownian Motion, he computes the density for $int_{0}^{x}e^{2B_{t}+2mu t}dt$, the place $B_{t}$ is Brownian movement and $muin mathbb{R}$ is a ceaseless.
One workable route to accumulate some estimates is through the use of the BorelSudakov appraise
$$P[sup_{sin [0,h]} V(s)V(0)geq u]lesssim h e^{u^{2}/2ln(delta/epsilon)}$$
and changing the above by
$$P[frac{a}{e^{u_{1}}}leq e^{bar{V}(0)}leq frac{b}{e^{u_{2}}}]+P[sup_{sin [0,1]} V(s)V(0)geq u_{1}]+P[inf_{sin [0,1]} V(s)V(0)geq u_{2}].$$
So when $a,b$ are each sizable, one can purchase some estimates (as instructed in 1 too). Anyhow given the technicalities in 1, we doubt there may be any concrete density, we at the least await to accumulate varied estimates for diminutive and immense $a,b$.
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