techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$

Question

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log covariance $R(|t-s|)=E[X(s)X(t)]=\ln(\frac{\delta}{\epsilon \vee |t-s|\wedge \delta})$.

Apparently very little is known about the inverse $T_{a}$ of this measure i.e. $\mu(0,T_{a})=a$. In computing the moments of increments $E[(T_{b}-T_{a})^p]$ for $0<a<b$ we end up with having to understand moments for the shifted $\mu(T_{a},T_{a}+t)$ (analogous to "Slepian process") and thus understanding $e^{X(T_{a}+s)}$ (even the expectation $E[e^{X(T_{a}+s)}]$ is hard to understand).

I list here some references and ideas in estimating $E[\mu(T_{a},T_{a}+t)^{q}]$.

• If X is further an Ito process, then one can use a change of variables to study $e^{X_{T_a}}$ ("First Hitting Time of Integral Diffusions and Applications")
• For example, in the case integrated geometric Brownian motion, one has $e^{B_{T_a}-\frac{1}{2}T_{a}}$ be equal in law to squared Bessel. Even possible to get Laplace transform for the inverse $T_{a}$.
• Another hope is to use Malliavin calculus to study that shifted process. One promising route is "Hitting times for Gaussian processes" where they study $e^{X_{\tau_a}-R(\tau_{a})}$ for $X_{\tau_{a}}=a$, but some of the techniques can be modified to study $e^{X_{T_{a}}}$ instead. That work can be generalized to hitting times of additive functionals $\mu(x)=\int e^U$ i.e. $\mu(T_a)=a$. This is because instead of taking Malliavin derivative of the supremum at lemma 3.2, we can take that of $\mu(x)$.
• A beautiful result would be to bound by the unshifted moment $$E[\mu(T_{a},T_{a}+t)^{q}]\leq c E[\mu(0,t)^{q}],$$

for some uniform bound $c=c_{q}$. Usually such proofs go by Gaussian-interpolation but here $X_{T_a}$ is most likely not a Gaussian anymore. So one hope is to use the generalization of interpolation in Malliavin calculus "Comparison inequalities on Wiener space". The issue here is that one seems to necesarily need Malliavin differentiability of $X_{T_a}$, which is not for similar reasons as here possible.

Answer 1

For any one interested in this type of shifted measure for fields with short-range correlation, we made some progress here in the article "Inverse of the Gaussian multiplicative chaos: Moments".

Basically, start with decomposing the variable $T_{a}$ and take supremum

$$E[\mu(T_{a},T_{a}+t)]\leq \sum_{k}E[\sup_{T\in I_{k}}\mu(T,T+t)\chi\{T_{a}\in I_{k}\}]$$

for some partition intervals $I_{k}=[a_{k},a_{k+1}]$. Next one either uses Holder inequality (or FKG inequality) to separate the two factors or a kind of strong-Markov decoupling that is true for fields with short-range correlation.

The factor $E[\sup_{T\in I_{k}}\mu(T,T+t)]$ can be studied using a chaining argument to reduce the supremum to maximum over finitely many intervals.

The factor $T_{a}\in I_{k}$ can be studied using single point estimates eg. for large $a_k$, we can use $\{T_{a}\geq a_{k+1}\}= \{a\geq \mu(a_{k+1}))\}$ and negative moments of $\mu$.