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Density of integral of exponential of Gaussian bailiwick with log-covariance retort

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Density of integral of exponential of Gaussian bailiwick with log-covariance

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 Borel-Sudakov 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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