On Small Deviations of Exponential Functional of Brownian Motion With Applications

Abstract

We provide the asymptotic bounds on the log-Laplace transform of an exponential functional $A$ of a Brownian motion. Using excursion theory, in the case of $A$ taken at random time, we discover the nice explicit formula of its Laplace transform. We apply the new formula to provide a connection between $A$ and excursions of a two-dimensional Brownian motion from a hyperplane. Finally, we apply the new results to the problem of small deviations of Gaussian multiplicative chaos on Euclidean balls.

Keywords

log-Laplace transform geometric Brownian motion order of convergence at infinity two-dimensional Brownian motion excursions from hyperplane GMC

1. Preliminaries and Main Results

Exponential functionals of Brownian motion play an important role in many modern probabilistic models. The probability distributions of integral exponential functionals of geometric Brownian motion (IEF) and their properties have been the subject of thorough investigations over the last few years (see, e.g., Donati-Martin, Ghomrasni, & Yor, 2001; Matsumoto & Yor, 2001, 2005a, 2005) motivated by, among others, the role of geometric Brownian motion in financial modeling. It is now well known that the distribution of IEF is strictly related to the Hartman–Watson distribution and thus it is technically highly complicated due to the oscillating nature of its density distribution (see Hartman & Watson, 1974; Jakubowski & Wiśniewolski, 2020; Lyasoff, 2016). For this reason, a probabilistic description of this seemingly simple IEF has always been a delicate matter. Since we did not find a description of small deviations of IEF in the literature, we decided to fill this gap in this article.

The one of motivations for this is the fact that the order of convergence of tails of IEF may be helpful in studying the asymptotic properties of Gaussian multiplicative chaos (GMC) or quantum gravity related objects (see Duplantier & Sheffield, 2011; Kahane, 1985; Rhodes & Vargas, 2014; Sheffield, 2007). It is not surprising that additive functionals of Brownian motion are strictly connected to the problem of GMC since in the case of $d = 2$ and Gaussian free field (GFF), the circle average process which describes the average values of GFF on small circles centered around a point is known to be a Brownian motion. GMC, in turn, is an exponential functional of such a process.

Under the small deviation of a positive random variable $η$ , we understand the asymptotic behavior of the log-Laplace transform

\ln (E e^{- e^{x} η})

for large values of

x

. Up to multiplicative constant, by the classical Tauberian arguments, this asymptotic behavior is equivalent to the behavior of

\ln P (η \leq ϵ)

for

ϵ = e^{- x}

close to

0

. In other words, recognizing the asymptotic behavior of the log-Laplace transform at infinity is a way to understand the behavior of distribution of a random variable at

0

1.1. One-Dimensional Perspective

To develop the main formula, we consider a one-dimensional Brownian motion $B$ on the probability space endowed with filtration satisfying usual conditions. For $α \geq 0, β \in R$ , we consider the functional

A_{t}^{α, β} = \int_{0}^{t} e^{α B_{u} - β u} d u, t \geq 0.

(1)

Although much is known about

A_{t}^{α, β}

(see Mansuy & Yor, 2008; Matsumoto & Yor, 2005a, 2005 for a survey in this area), we are able to give some new insight into this now classical theory. Some close to the explicit formula of the Laplace transform of IEF is known, however, it does not give the answer to the question about the asymptotic behavior. For example, if

α = 2

β = 0

, we have by Matsumoto and Yor (2005a, Corollary 3.2)

E e^{- e^{x} A_{t}^{2, 0}} = \sqrt{\frac{2}{π t}} \int_{0}^{\infty} \cos (\sqrt{2} e^{\frac{x}{2}} \sinh (z)) e^{- \frac{z^{2}}{2 t}} d z .

So the first natural motivation is to study the order of convergence of

E e^{- e^{x} A_{t}^{α, β}}

x \to \infty

. To say more about the asymptotic behavior of IEF, we use the excursion theory of Markov processes. In what follows, for simplicity of computations, we will set

β = 0, α = 1

and denote

A_{t} = A_{t}^{1, 0}

. Notice that for

β \geq 0

E e^{- \frac{1}{α^{2}} e^{x} A_{α^{2} t}} \leq E e^{- e^{x} A_{t}^{α, β}} \leq E e^{- \frac{1}{α^{2}} e^{x - β t} A_{α^{2} t}},

so for a fixed

t

, the asymptotic behavior and other properties can be deduced in a general case from

A_{t}

. If

β < 0

we easily obtain analogous estimations. For further consideration, we define

d_{t} = inf {s > t : B_{s} = 0}, t \geq 0.

K_{γ}

and

I_{γ},

we denote the modified Bessel functions with parameter

γ \in R

Theorem 1

Let $β = 0$ and $α = 1$ . Then for every $x \in R$

\ln (E e^{- e^{x} A_{d_{t}}}) = - \sqrt{\frac{t}{π}} e^{\frac{x}{2}} (\frac{K_{1} (2 \sqrt{2 e^{x}})}{K_{0} (2 \sqrt{2 e^{x}})} + \frac{I_{1} (2 \sqrt{2 e^{x}})}{I_{0} (2 \sqrt{2 e^{x}})}) .

Moreover, for

x \to \infty

\ln (E e^{- e^{x} A_{d_{t}}}) = - 2 \sqrt{\frac{t}{π}} e^{\frac{x}{2}},

(2)

in a sense that

e^{- x / 2} \ln (E e^{- e^{x} A_{d_{t}}}) \to - 2 \sqrt{t / π}

x \to \infty

As a corollary of Theorem 1 and estimations on Bessel functions (Laforgia & Natalini, 2010), we get elementary bounds on $\ln (E e^{- e^{x} A_{d_{t}}})$ .

Corollary 2

For every $x$ and $t > 0$ , we have

- \sqrt{\frac{t}{π}} \frac{\sqrt{8 e^{x} + 1} + 2 \sqrt{2 e^{x}} + 1}{2 \sqrt{2}} < \ln (E e^{- e^{x} A_{d_{t}}}) < - \sqrt{\frac{t}{π}} \frac{\sqrt{8 e^{x} + 1} + 2 \sqrt{2 e^{x}} - 1}{2 \sqrt{2}} .

Although the known links between IEF and Bessel processes concern mainly the case of a Bessel process with index $\frac{1}{2}$ , Theorem 1 ties the distribution of $A_{d_{t}}$ to the local time at $0$ of $R$ being a Bessel process with index $0$ ( $BES (0)$ ). Recall that $R$ starting from $x \geq 0$ can be constructed as $R_{t} = \sqrt{B_{t}^{2} + W_{t}^{2}}$ , where $B$ and $W$ are two independent Brownian motions such that $W_{0} = 0$ and $B_{0} = x$ . It is well known that the inverse of local time is a subordinator. Application of Theorem 1 gives its new characterization.

Theorem 3

Let $(L_{t}^{1})_{t \geq 0}$ be the local time at $1$ of a $BES (0)$ $R$ starting from $1$ and $(τ_{t}^{(1)})_{t}$ be the process of inverse local time of $R$ at $1$ . For any $t > 0$ and $s > 0$ , we have

E e^{- λ A_{d_{8 π t^{2}}}} = E_{1} e^{- 4 λ τ_{t}^{(1)}},

(3)

so for

t > 0, s > 0

P (A_{d_{t}} > s) = P (L_{\frac{s}{4}}^{1} < \sqrt{\frac{t}{8 π}}) .

Moreover, the Stjeltjes measure of subordinator

(τ_{t}^{(1)})

is given on

(0, \infty)

S (d z) = \frac{1}{π^{2} z} \frac{d z}{J_{0}^{2} (\sqrt{2 z}) + Y_{0}^{2} (\sqrt{2 z})} - 2 \sum_{n = 1}^{\infty} \frac{J_{1} (j_{n})}{J_{- 1} (j_{n})} e^{- \sqrt{2} j_{n}^{2} z} δ_{j_{n}^{2} / 2} (\sqrt{2} d z),

(4)

where

J_{0} (\cdot), Y_{0} (\cdot)

are Bessel functions with parameter

0

δ_{z}

is the Dirac measure at

z

and

(j_{n})_{n \geq 1}

are zeros of

J_{0} (\cdot)

The another application of Theorem 1 gives the bounds on the Laplace transform of IEF at the deterministic time.

Theorem 4

Fix $k \in (0, 1]$ . Let $z_{k}$ be a real number such that

\frac{K_{0} (2 \sqrt{2 e^{x + z_{k}}})}{K_{0} (2 \sqrt{2 e^{x}})} = k .

Then

e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 e^{x}}) I_{0} (2 \sqrt{2 e^{x}})}} \leq E e^{- e^{x} A_{t}} \leq \frac{e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 e^{x}}) I_{0} (2 \sqrt{2 e^{x}})}}}{Φ (\frac{z_{k}}{\sqrt{t}}) (k \land \frac{1}{I_{0} (2 \sqrt{2 e^{x}})})},

for every

t > 0

and

x > 0

. In particular, if

t > 2 π

we have for

x \to \infty

e^{- 2 \sqrt{e^{x}} \sqrt{\frac{t}{π}}} \leq E e^{- e^{x} A_{t}} \leq \frac{1}{\sqrt{π \sqrt{2 e^{x}}}} e^{- 2 \sqrt{e^{x}} (\sqrt{\frac{t}{π}} - \sqrt{2})} .

(5)

Using the explicit form of the Laplace transform of $A_{d_{t}}$ , we express the distribution of subordinator $(τ_{t}^{(1)})$ in terms of the transition density of $BES (0)$ . We refer to the theory of algebra of convolutions of locally integrable functions. In particular, for any locally integrable function $f$ there exists a quasi-inverse element defined by $f^{‡} = - \sum_{i \geq 1} f^{* i}$ , see, for example, Lew (1972).

Theorem 5

Fix $λ > 0$ . Let $T_{λ}$ be an exponential random variable with parameter $λ$ , independent of a $BES (0)$ denoted by $R$ , $R_{0} = 1$ . Then

P (τ_{T_{λ}}^{(1)} \in d r) = - (- λ p)^{‡} (1, 1) (r) d r,

where

p (\cdot, \cdot)

denotes the transition density of

R

1.2. Two-Dimensional Perspective: Excursions From Hyperplane

The explicit formula of Theorem 1 is used to find the distribution of IEF in some specific function space related to two-dimensional Brownian motion. Somehow, this is motivated by thinking about applications of the theory of exponential functionals in other objects with a given correlation structure such as GMC on log-correlated Gaussian field (see Aru, 2020; Berestycki, 2017). For a pair $(B, W)$ of the correlated Brownian motions, we consider the integral functional

I_{α, β} (t) = \int_{0}^{t} e^{α B_{s} + β W_{s}} d s, α, β > 0,

(6)

and we would like to describe the distribution of

I_{α, β}

in terms of the input data

(B, W)

. It is possible to look at

I_{α, β}

as an IEF on the excursion space

U

. Indeed, the canonical process under the measure of excursion is the Markov process with the specified transition density related to the initial process. So, let

(B, Z)

be a two-dimensional Brownian motion with independent coordinates and define

W = ρ B + \sqrt{1 - ρ^{2}} Z

, for

ρ \in (- 1, 1)

. We have

I_{α, β} (t) = \int_{0}^{t} e^{(α + β ρ) B_{s} + β \sqrt{1 - ρ^{2}} Z_{s}} d s .

Case 1. Normalized covariance.

First, we normalize the structure and consider the following functional at a random time $τ$

I_{ρ} = \int_{0}^{τ} e^{ρ B_{s} + \sqrt{1 - ρ^{2}} Z_{s}} d s, ρ \in (- 1, 1) .

Since

W_{s} = ρ B_{s} + \sqrt{1 - ρ^{2}} Z_{s}

s \geq 0,

is a Brownian motion, setting

τ = d_{t} = inf {s > t : W_{s} = 0},

we retrieve from

I_{ρ}

the familiar object

A_{d_{t}}

and we can apply the explicit formula for its Laplace transform (Theorem 1).

Let $U$ be the space of excursions of a two-dimensional Brownian motion from the hyperplane

H = {(x, y) \in R^{2} : ρ x + \sqrt{1 - ρ^{2}} y = 0} .

Define

F_{ρ} (u) = \int_{0}^{R (u)} e^{ρ u_{1} (s) + \sqrt{1 - ρ^{2}} u_{2} (s)} d s,

(7)

for

u = (u_{1}, u_{2}) \in U

, where

R (u)

is the lifetime of

u

. We will find explicitly the distribution of

F_{ρ} (u)

under the measure of excursions. Since

H

is closed, the theory of exit system states that there exists a set of

σ

-finite measures

P^{(x_{1}, x_{2})}

(x_{1}, x_{2}) \in R \times R

, on the two-dimensional path space

U

of excursions and an additive functional

K

such that for all nonnegative predictable processes

Y

and measurable functions

f

U

vanishing on the constant excursion, it holds a generalized version of master formula, called later the exit system connection formula,

E_{0, 0} \sum_{s \in G \cap [0, t]} Y_{s} (f \circ θ_{s}) = E_{0, 0} \int_{0}^{t} Y_{u} P^{(B_{u}, Z_{u})} (f) d K_{u},

(8)

where

G

denotes left-end points of excursions interval (see Blumenthal, 1992, Chapter VII or Burdzy, 1986) and the function

(x_{1}, x_{2}) \mapsto P^{(x_{1}, x_{2})} (f) = \int_{U} f (u) P^{(x_{1}, x_{2})} (d u)

is universally measurable. Our aim is now to describe the function

Q^{(x_{1}, x_{2})} (r) = P^{(x_{1}, x_{2})} (F_{ρ} > r), r > 0.

(9)

Γ (0, t)

, we denote incomplete Gamma function

Γ (0, t) = \int_{t}^{\infty} (1 / r) e^{- r} d r

t > 0

. Now, for

v > 0, t > 0

define

{\hat{Q}}_{ρ} (v, t) = \int_{- \infty}^{\infty} Q^{(\sqrt{1 - ρ^{2}} z, - ρ z)} (v) Γ (0, \frac{z^{2}}{2 t}) d z .

(10)

It turns out that the transform

{\hat{Q}}_{ρ} (s, t)

is finite and expressed in terms of the Lévy measure

μ^{(1)} (d r)

(τ_{t}^{(1)})_{t}

—the inverse of local time at

1

BES (0)

Theorem 6

For almost all $t > 0$ , every $r > 0$ and $ρ \in (- 1, 1)$

{\hat{Q}}_{ρ} (r, t) = \sqrt{\frac{π t}{2}} μ^{(1)} ((4 r, \infty)) .

(11)

Moreover, let

n_{1} (\cdot)

denote the characteristic measure of excursions of

BES (0)

from

1

and let

R

denote the associated lifetime of excursions. Then for almost every

s > 0

and every

r > 0

\int_{(- 2 \sqrt{s}, 2 \sqrt{s})} Q^{(\sqrt{1 - ρ^{2}} z, - ρ z)} (r) d z = \sqrt{s} n_{1} (R > 4 r) .

(12)

Remark 7

There are known connections between excursion’s distribution of one-dimensional Brownian motion with $BES (\frac{1}{2})$ –Bessel process with index $\frac{1}{2}$ (Blumenthal, 1992, Chapter III). Furthermore, Burdzy (1986) described the distribution of excursions of $n$ -dimensional Brownian motion from a hyperplane

{x \in R^{n} : x_{1} = 0},

and his descriptions also showed the strict connections of excursion’s distribution with the distribution of

BES (\frac{1}{2})

. Our results show the relations of excursion’s distribution of a two-dimensional Brownian motion from a hyperplane depending on a covariance structure with the Lévy measure of the inverse of local time at

1

a of one-dimensional

BES (0)

Case 2. Unormalized covariance.

In the second case, instead of $W = ρ B + \sqrt{1 - ρ^{2}} Z$ , we consider $\bar{W} = ρ B + Z$ , where $(B, Z)$ is a two-dimensional Brownian motion with independent coordinates. We will study how $ρ$ affects the distributional description of functionals generated by $\bar{W}$ . Let $\bar{τ} = inf {s > t : {\bar{W}}_{s} = 0}$ and define

{\bar{I}}_{ρ} = \int_{0}^{\frac{\bar{τ}}{1 + ρ^{2}}} e^{ρ B_{s} + Z_{s}} d s .

τ_{ρ} = inf {s > t : W_{s} = 0}

, then we have

\bar{τ} = τ_{ρ / \sqrt{1 + ρ^{2}}}

a.s. and we observe that

{\bar{I}}_{ρ} \overset{law}{=} \frac{1}{1 + ρ^{2}} I_{\frac{ρ}{\sqrt{1 + ρ^{2}}}},

for

I

defined in the first case. Thus, for the associated hyperplane

\bar{H} = {(x, y) \in R^{2} : ρ x + y = 0},

we define

{\bar{F}}_{ρ} (u) = \int_{0}^{\frac{R (u)}{1 + ρ^{2}}} e^{ρ u_{1} (s) + u_{2} (s)} d s,

where

R (u)

denotes the lifetime of excursions from

\bar{H}

u \in U

. We have

{\bar{F}}_{ρ} (u) = \frac{1}{1 + ρ^{2}} F_{\frac{ρ}{\sqrt{1 + ρ^{2}}}} (u),

a.s. under the measure of excursions, where

F

is given by (7). Hence, if we define

{\bar{Q}}^{(x_{1}, x_{2})} (r) = P^{(x_{1}, x_{2})} ({\bar{F}}_{ρ} (u) > r), r > 0,

then

{\bar{Q}}^{(x_{1}, x_{2})} (r) = P^{(x_{1}, x_{2})} (F_{\frac{ρ}{\sqrt{1 + ρ^{2}}}} (u) > (1 + ρ^{2}) r) .

Finally, we define

{\hat{\bar{Q}}}_{ρ} (v, t) = \int_{- \infty}^{\infty} {\bar{Q}}^{(\frac{z}{\sqrt{1 + ρ^{2}}}, - \frac{ρ z}{\sqrt{1 + ρ^{2}}})} (v) Γ (0, \frac{z^{2}}{2 t}) d z .

The next theorem is then a straightforward corollary of Theorem 6.

Theorem 8

For almost all $t > 0, r > 0$ and $ρ > 0$

{\hat{\bar{Q}}}_{ρ} (r, t) = \sqrt{\frac{π t}{2}} μ^{(1)} ((4 (1 + ρ^{2}) r, \infty)) .

Moreover, let

n_{1} (\cdot)

denote the characteristic measure of excursions of

BES (0)

from

1

and let

R

denote the associated lifetime of excursions. Then for almost every

s > 0

and

r > 0

\int_{(- 2 \sqrt{s}, 2 \sqrt{s})} {\bar{Q}}^{(\frac{z}{\sqrt{1 + ρ^{2}}}, - \frac{ρ z}{\sqrt{1 + ρ^{2}}})} (r) d z = \sqrt{s} n_{1} (R > 4 (1 + ρ^{2}) r) .

1.3. IEF and Small Deviations of GMC

Let $(h (x))_{x \in A}$ be a centered log-correlated Gaussian field over a bounded domain $A \subset R^{d}$ such that $Cov (h (x), h (z)) = - \ln | x - y | + g (x, y)$ and $g$ is a continuous function (Aru, 2020; Berestycki, 2017; Kahane, 1985; Rhodes & Vargas, 2014). It is well known that for such $h$ we may define a random measure

M (B) = \int_{B} e^{γ h (x) - \frac{γ^{2}}{2} E h^{2} (x)} d x, B \in B (A) .

Notice that the last definition is by no means straightforward. Indeed, since

E h^{2} (x)

cannot be defined point-wise, a limiting procedure has to be applied. The elegant description of this can be found in Berestycki (2017). The random measure

M

is called a GMC. For technical reasons, it is assumed that

γ < \sqrt{2 d}

. In our consideration, we assume that

g

is chosen in such a way that

B (0, 1) = {x \in R^{d} : | x | < 1} \subset A,

and we denote as

M_{1} = M (B (0, 1)), M_{r} = M ({x \in R^{d} : | x | < r}), r \in (0, 1) .

We assume

d \geq 2

and set, for simplicity,

γ = 1

. It is known (and formally proven in case

d = 2

) that

E e^{- e^{x} M_{1}} \leq C_{1} e^{- c_{1} x^{2}},

(13)

for every

x > 0

and for some positive constants

C_{1}, c_{1}

(Aru, 2020). Usually, the random variable

M_{1}

carries the asymptotic property of a log-normal random variable. In fact, as it was recently shown in Lacoin, Rhodes, and Vargas (2018) in case

d = 1

, the last property depends on the value of the spatial mean, and in some cases the rate of convergence in (13) can be faster.

In case $g \equiv T$ , where $T$ is a constant such that $x \mapsto \ln T / | x |$ is positive definite on $B (0, 1)$ , there exist natural bounds on the constants in small deviations of GMC on balls $B (0, r) \subset R^{d}$ for $r \in (0, 1)$ (see Talarczyk-Noble & Wiśniewolski, 2023, Theorem 2.4). We can use these bounds, which will be assumed here under hypothesis S, to provide an explicit upper bound on $c_{1}$ .

Definition 9

We will say that the hypothesis S holds if: (1)

There exist $c_{1} \in (0, \infty)$ and $x_{0} > 0$ such that for every $r \in (0, 1)$ and $x \geq x_{0}$

E e^{- e^{x} M_{r}} \leq e^{- \frac{c_{1}}{1 + 2 c_{1} \ln \frac{1}{r}} x^{2}} .

(14)

(2)

$M_{1}$ has a subscaling property, that is, for all $λ > 0$ , $r \in (0, 1)$ and $Z_{r}$ being an independent of $M_{1}$ centered Gaussian random variable with covariance $\ln 1 / r$ , there exists a constant $C_{2} \in [1, \infty)$ such that

E e^{- λ r^{d} M_{1} e^{Z_{r} - \frac{1}{2} \ln \frac{1}{r}}} \leq C_{2} E e^{- λ M_{r}} .

(15)

The volume of ball $B (0, 1) \subset R^{d}$ is denoted by $| B (0, 1) |_{d}$ .

Theorem 10

Under hypothesis $S$ we have

c_{1} \leq \frac{1}{(\ln α)^{2}} \ln (C_{2} e^{\sqrt{α π^{- 1}} (1 + | B (0, 1) |_{d}})),

(16)

where

α

is the unique solution in

(1, \infty)

of equation

\frac{1 + | B (0, 1) |_{d}}{4 \sqrt{π}} \ln α = \frac{\ln C_{2}}{\sqrt{α}} + \frac{1 + | B (0, 1) |_{d}}{\sqrt{π}} .

(17)

In particular, if

C_{2} = 1

, then

c_{1} \leq \frac{e^{2}}{\sqrt{π}} \frac{1 + | B (0, 1) |_{d}}{16} .

(18)

Remark 11

In case $g \equiv T$ , where $T$ is a constant, we always have $C_{2} = 1$ .

2. Proofs

Proof. Proof of Theorem 1

In the proof, we will use the excursion theory. Recall that $β = 0$ , $α = 1$ by assumption, and

d_{t} = inf {s > t : B_{s} = 0} .

We rewrite

A_{d_{t}}

in terms of excursions of

B

from

0

. Let

L

be the local time of

B

0

. Define

τ_{t -} = inf {s > 0 : L_{s} \geq t}, τ_{t} = inf {s > 0 : L_{s} > t},

(τ_{t})_{t}

is the inverse local time at

0

. Let

U

be the subspace of continuous functions

u : [0, \infty) \to E = R

such that

u (0) = 0

and

0 < R (u) \leq \infty

, where

R (u) = inf {t > 0 : u (t) = 0},

and

u (v) > 0

for

v \in (0, R (u))

. Moreover, we demand that

u (t) = 0

for all

t \geq R (u)

. Let

U_{0}

be the

σ

-algebra generated by the coordinate mappings. Let

δ : [0, \infty) \to E

be a constant function,

δ (t) \equiv 0

. We denote as

U^{δ}

the space

U \cup {δ}

and set

U_{0}^{δ} = σ (U, {δ})

. Finally, on the canonical space, we consider an excursion point process

(e (s))_{s > 0}

and we denote as

n

the associated measure of excursions (Blumenthal, 1992, Chapter III, Revuz & Yor, 2005, Chapter XII). Since

B

has no probability mass at

0

, we have a.s.

A_{d_{t}} = \sum_{τ_{s -} \leq t} \int_{τ_{s -}}^{τ_{s}} e^{B_{u}} d u .

Denote as

σ_{0}

the first hitting time of

0

B

. Clearly

τ_{s} = τ_{s -} + σ_{0} \circ θ_{τ_{s -}}

, where

θ

denotes a shift operator. Thus

A_{d_{t}} = \sum_{τ_{s -} \leq t} (\int_{0}^{σ_{0}} e^{B_{u}} d u) \circ θ_{τ_{s -}} .

Using the immersion of fragmented Brownian paths into the functional space of excursions

(U, U)

, recalling that

R (u)

is the lifetime of an excursion

u \in U

and defining

F : [0, \infty) \times U \to R

F (t, u) = \int_{0}^{R (u)} e^{u (s)} d s,

we conclude that

(\int_{0}^{σ_{0}} e^{B_{u}} d u) \circ θ_{τ_{s -}} = F (τ_{s -}, e (s)) .

The exponential formula for excursions and change of variable formula yields for a measurable (with respect to both variables), nonnegative

H

E e^{- \sum_{0 < τ_{s -} \leq t} H (τ_{s -}, e (s))} = e^{- E \int_{0}^{t} \int_{U} (1 - e^{- H (v, u)}) n (d u) d L_{v}},

where

L

is the local time of

B

(it is a consequence of Revuz & Yor, 2005, Chapter XII, Proposition 1.12, Proposition 2.6). Hence

\begin{aligned} E e^{- e^{x} A_{d_{t}}} & = e^{- E \int_{0}^{t} \int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u) d L_{v}} \\ = e^{- (E L_{t}) \int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u)} . \end{aligned}

For fixed

t

E L_{t} = E | B_{t} |

, so

E e^{- e^{x} A_{d_{t}}} = e^{- \sqrt{\frac{2 t}{π}} \int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u)} .

(19)

To compute the right-hand side (RHS) of (19), we need to find the form of

\int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u)

. It is well known (Revuz & Yor, 2005, Chapter XII, Theorem 4.1) that under

n (\cdot)

the family

(u (s))_{s}

(0, R (u))

forms a strong Markov process with the transition density of a Brownian motion killed at

0

, and the entrance law given on

R ∖ {0}

λ_{s} (d y) = \frac{1}{\sqrt{2 π s^{3}}} | y | e^{- \frac{y^{2}}{2 s}} d y .

Hence, using excursion theory notation, we denote as

\hat{P}

a measure under which a process

({\hat{X}}_{t})_{t > 0}

is Markov with entrance law

(λ_{s})_{s > 0}

and the transition density of a Brownian motion killed at

0

. We have

\begin{aligned} n (\int_{0}^{R (u)} e^{u (s)} d s \in d v) = \hat{P} (\int_{0}^{\infty} e^{{\hat{X}}_{s}} d s \in d v), \\ \int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u) = \hat{P} (1 - e^{- e^{x} \int_{0}^{\infty} e^{{\hat{X}}_{s}} d s}), \end{aligned}

and

\hat{P} (\int_{0}^{\infty} e^{{\hat{X}}_{s}} d s \in d v) = lim_{ϵ \to 0} \int_{R ∖ {0}} λ_{ϵ} (d y) P_{y}^{ϵ} (\int_{ϵ}^{σ_{0}} e^{B_{s}} d s \in d v, σ_{0} > ϵ),

where

B

is a Brownian motion,

σ_{0}

its first hitting time of

0

P_{y}^{ϵ} (B_{ϵ} = y) = 1

. By Markov property, since

σ_{0} = ϵ + σ_{0} \circ θ_{ϵ}

{σ_{0} > ϵ}

, we find that

P_{y}^{ϵ} (\int_{ϵ}^{σ_{0}} e^{B_{s}} d s \in d v, σ_{0} > ϵ) = P_{y} (\int_{0}^{σ_{0}} e^{B_{s}} d s \in d v, σ_{0} > 0) = P_{y} (\int_{0}^{σ_{0}} e^{B_{s}} d s \in d v) .

Thus

\int_{U} (1 - e^{- e^{x} \int_{0}^{R (u)} e^{u (s)} d s}) n (d u) = lim_{ϵ \to 0} \int_{R ∖ {0}} λ_{ϵ} (d y) E_{y} (1 - e^{- e^{x} \int_{0}^{σ_{0}} e^{B_{s}} d s}) .

The explicit formula for the Laplace transform of

\int_{0}^{σ_{0}} e^{B_{s}} d s

has the form

E_{y} e^{- γ \int_{0}^{σ_{0}} e^{B_{s}} d s} = 1_{(- \infty, 0)} (y) \frac{I_{0} (2 \sqrt{2 γ} e^{\frac{y}{2}})}{I_{0} (2 \sqrt{2 γ})} + 1_{(0, \infty)} (y) \frac{K_{0} (2 \sqrt{2 γ} e^{\frac{y}{2}})}{K_{0} (2 \sqrt{2 γ})},

where

γ > 0

(see Borodin & Salminen, 2002, 2.10.3). Furthermore,

\int_{R ∖ {0}} λ_{ϵ} (d y) E_{y} (1 - e^{- e^{x} \int_{0}^{σ_{0}} e^{B_{s}} d s}) = J_{1} (ϵ) + J_{2} (ϵ),

where

\begin{aligned} J_{1} (ϵ) & = \int_{- \infty}^{0} \frac{1}{\sqrt{2 π ϵ^{3}}} (- y) e^{- \frac{y^{2}}{2 ϵ}} (1 - \frac{I_{0} (2 \sqrt{2 e^{x + y}})}{I_{0} (2 \sqrt{2 e^{x}})}) d y, \\ J_{2} (ϵ) & = \int_{0}^{\infty} \frac{1}{\sqrt{2 π ϵ^{3}}} y e^{- \frac{y^{2}}{2 ϵ}} (1 - \frac{K_{0} (2 \sqrt{2 e^{x + y}})}{K_{0} (2 \sqrt{2 e^{x}})}) d y . \end{aligned}

We have

\begin{aligned} J_{1} (ϵ) & = \int_{0}^{\infty} \frac{y}{\sqrt{2 π ϵ^{3}}} e^{- \frac{y^{2}}{2 ϵ}} (1 - \frac{I_{0} (2 \sqrt{2 e^{x - y}})}{I_{0} (2 \sqrt{2 e^{x}})}) d y \\ = \int_{0}^{\infty} \frac{v}{\sqrt{2 π ϵ}} e^{- \frac{v^{2}}{2}} (1 - \frac{I_{0} (2 \sqrt{2 e^{x - v \sqrt{ϵ}}})}{I_{0} (2 \sqrt{2 e^{x}})}) d v \\ = \frac{1}{I_{0} (2 \sqrt{2 e^{x}})} \int_{0}^{\infty} \frac{v^{2}}{\sqrt{2 π}} e^{- \frac{v^{2}}{2}} \frac{I_{0} (2 \sqrt{2 e^{x}}) - I_{0} (2 \sqrt{2 e^{x - v \sqrt{ϵ}}})}{v \sqrt{ϵ}} d v . \end{aligned}

By integral representation

I_{0} (z) = 1 / π \int_{0}^{π} e^{- z \cos r} d r

we have for any positive

z, w

| I_{0} (z) - I_{0} (w) | \leq c | z - w |,

for a constant

c > 0

, which yields that for some

ϵ_{0} > 0

and all

ϵ \in (0, ϵ_{0})

the function

v \mapsto \frac{I_{0} (2 \sqrt{2 e^{x}}) - I_{0} (2 \sqrt{2 e^{x - v \sqrt{ϵ}}})}{v \sqrt{ϵ}}

has an integrable majorant. Since

lim_{ϵ \to 0} \frac{I_{0} (2 \sqrt{2 e^{x}}) - I_{0} (2 \sqrt{2 e^{x - v \sqrt{ϵ}}})}{v \sqrt{ϵ}} = I_{0}^{'} (2 \sqrt{2 e^{x}}) = \sqrt{2} e^{\frac{x}{2}} I_{1} (2 \sqrt{2 e^{x}}),

we conclude by the dominated convergence theorem that

lim_{ϵ \to 0} J_{1} (ϵ) = \frac{1}{I_{0} (2 \sqrt{2 e^{x}})} \int_{0}^{\infty} \frac{v^{2}}{\sqrt{2 π}} e^{- \frac{v^{2}}{2}} \sqrt{2} e^{\frac{x}{2}} I_{1} (2 \sqrt{2 e^{x}}) d v = \frac{e^{\frac{x}{2}}}{\sqrt{2}} \frac{I_{1} (2 \sqrt{2 e^{x}})}{I_{0} (2 \sqrt{2 e^{x}})} .

In a similar way, we find that

lim_{ϵ \to 0} J_{2} (ϵ) = \frac{e^{\frac{x}{2}}}{\sqrt{2}} \frac{K_{1} (2 \sqrt{2 e^{x}})}{K_{0} (2 \sqrt{2 e^{x}})} .

Plugging obtained limits in (19) yields the first statement, that is,

E e^{- e^{x} A_{d_{t}}} = e^{- \sqrt{\frac{2 t}{π}} \frac{e^{\frac{x}{2}}}{\sqrt{2}} (\frac{K_{1} (2 \sqrt{2 e^{x}})}{K_{0} (2 \sqrt{2 e^{x}})} + \frac{I_{1} (2 \sqrt{2 e^{x}})}{I_{0} (2 \sqrt{2 e^{x}})})} .

Since for large

x

lim_{x \to \infty} \frac{K_{1} (2 \sqrt{2 e^{x}})}{K_{0} (2 \sqrt{2 e^{x}})} + \frac{I_{1} (2 \sqrt{2 e^{x}})}{I_{0} (2 \sqrt{2 e^{x}})} = 2,

we finally get, for

x

large,

E e^{- e^{x} A_{d_{t}}} = e^{- 2 \sqrt{\frac{t}{π}} e^{\frac{x}{2}}} .

The proof is complete.

Proof. Proof of Corollary 2

The estimates are a straightforward consequence of estimates given by Laforgia and Natalini (2010, Theorems 1.1 and 1.2) and Theorem 1.

Proof. Proof of Theorem 3

By Theorem 1 we have for every $λ > 0$

E e^{- λ A_{d_{t}}} = e^{- \sqrt{\frac{t}{π}} \sqrt{λ} (\frac{K_{1} (2 \sqrt{2 λ})}{K_{0} (2 \sqrt{2 λ})} + \frac{I_{1} (2 \sqrt{2 λ})}{I_{0} (2 \sqrt{2 λ})})} .

Since

K_{1} (x) I_{0} (x) + I_{1} (x) K_{0} (x) = \frac{1}{x} .

(20)

(Borodin & Salminen, 2002, App.2,p.4), we have

E e^{- λ A_{d_{t}}} = e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 λ}) I_{0} (2 \sqrt{2 λ})}} .

(21)

On the other hand, using Borodin and Salminen (2002, App.3, p. 27), we find that

\begin{aligned} K_{0} (2 \sqrt{2 λ}) I_{0} (2 \sqrt{2 λ}) & = \int_{0}^{\infty} e^{- 4 λ y} \frac{1}{2 y} e^{- \frac{1}{y}} I_{0} (y^{- 1}) d y \\ = \int_{0}^{\infty} e^{- 4 λ y} p (y, 1, 1) d y = G_{4 λ} (1, 1), \end{aligned}

where

p (y, \cdot, \cdot)

is the transition density with respect to the speed measure of

BES (0)

, which we denote here as

R

, and

G_{λ} (\cdot, \cdot)

denotes the associated Green function. It is known that for

τ^{(1)}

being the process of inverse local time of

R

1

E_{P_{1}} e^{- 4 λ τ_{t}^{(1)}} = \exp (- \frac{t}{G_{4 λ} (1, 1)}),

(22)

where

P_{1} (R_{0} = 1) = 1

. This and (21) give (3), that is,

E e^{- λ A_{d_{8 π t^{2}}}} = E_{P_{1}} e^{- 4 λ τ_{t}^{(1)}}

, which means that

\frac{1}{4} A_{d_{8 π t^{2}}} \overset{d}{=} τ_{t}^{(1)} .

For this reason

P (A_{d_{8 π t^{2}}} > 4 s) = P_{1} (τ_{t}^{(1)} > s) = P_{1} (t > L_{s}^{1}),

where

L^{1}

is under

P_{1}

the process of local time at

1

BES (0)

starting from

1

Finally, to find the Stjeltjes measure of a subordinator $τ_{t}^{(1)}$ , we write

E e^{- λ τ_{t}^{(1)}} = e^{- \frac{λ}{4} A_{d_{8 π t^{2}}}} = e^{- t \sqrt{2 λ} (\frac{K_{1}}{K_{0}} + \frac{I_{1}}{I_{0}}) (\sqrt{2 λ})},

which corresponds to the Stjeltjes measure of a subordinator given by (4) (see Schilling, Song, and Vondracek (2010, Chapter 15, 15.9.116, 15.9.118)) and recall that

K_{- 1} = K_{1}

. This finishes the proof.

Proof. Proof of Theorem 4

We have

A_{d_{t}} = A_{t} + A_{σ_{0}} \circ θ_{t},

so, that

E e^{- e^{x} A_{d_{t}}} \leq E e^{- e^{x} A_{t}} .

Using (21) we get

E e^{- e^{x} A_{t}} \geq e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 λ}) I_{0} (2 \sqrt{2 λ})}} .

By the Markov property

E e^{- e^{x} A_{d_{t}}} = E e^{- e^{x} A_{t}} E_{B_{t}} e^{- e^{x} A_{σ_{0}}} .

Let

f_{x} (z) = E_{z} e^{- e^{x} A_{σ_{0}}}, z \in R .

By Borodin and Salminen (2002, 2.10.3)

f_{x} (z) = 1_{(- \infty, 0]} (z) \frac{I_{0} (2 \sqrt{2 e^{x + z}})}{I_{0} (2 \sqrt{2 e^{x}})} + 1_{(0, \infty)} (z) \frac{K_{0} (2 \sqrt{2 e^{x + z}})}{K_{0} (2 \sqrt{2 e^{x}})},

which is strictly positive, attains maximum value at

0

, vanishes at infinity and

f (- \infty) = 1 / I_{0} (2 \sqrt{2 e^{x}})

. For any fixed

k \in (0, 1]

there exists a unique

z_{k} > 0

such that

\frac{K_{0} (2 \sqrt{2 e^{x + z_{k}}})}{K_{0} (2 \sqrt{2 e^{x}})} = k > 0.

We have

E e^{- e^{x} A_{t}} f_{x} (B_{t}) \geq (k \land \frac{1}{I_{0} (2 \sqrt{2 e^{x}})}) E e^{- e^{x} A_{t}} 1_{{B_{t} \leq z_{k}}} .

Applying Fortuin–Kasteleyn–Ginibre inequality we further get

E e^{- e^{x} A_{t}} 1_{{B_{t} \leq z_{k}}} \geq E e^{- e^{x} A_{t}} P (B_{t} \leq z_{k}) = Φ (\frac{z_{k}}{\sqrt{t}}) E e^{- e^{x} A_{t}},

where

Φ (\cdot)

denotes the cumulative distribution function of

N (0, 1)

. For this reason

E e^{- e^{x} A_{d_{t}}} \geq Φ (\frac{z_{k}}{\sqrt{t}}) (k \land \frac{1}{I_{0} (2 \sqrt{2 e^{x}})}) E e^{- e^{x} A_{t}} .

(23)

Hence and using again (21), we obtain

E e^{- e^{x} A_{t}} \leq \frac{e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 e^{x}}) I_{0} (2 \sqrt{2 e^{x}})}}}{Φ (\frac{z_{k}}{\sqrt{t}}) (k \land \frac{1}{I_{0} (2 \sqrt{2 e^{x}})})} .

This yields the first assertion.

To prove the second assertion we take $t > 2 π$ , $k = 1$ , and $z_{k} = 0$ . Since

I_{0} (z) K_{0} (z) = \frac{1}{2 z} and I_{0} (z) = \frac{1}{\sqrt{2 π z}} e^{z} as z \to \infty,

we obtain, using the first assertion and taking

x \to \infty

e^{- 4 \sqrt{2 e^{x}} \sqrt{\frac{t}{8 π}}} \leq E e^{- e^{x} A_{t}} \leq 2 e^{- 4 \sqrt{2 e^{x}} \sqrt{\frac{t}{8 π}}} \frac{1}{\sqrt{2 π 2 \sqrt{2 e^{x}}}} e^{2 \sqrt{2 e^{x}}},

which gives (5) after elementary algebra.

Proof. Proof of Theorem 5

Recall that $A = A^{1, 0}$ . Using (3) and (22), we find that

E e^{- \frac{γ}{4} A_{d_{t}}} = E e^{- γ^{2} τ_{\sqrt{t / (8 π)}}^{(1)}} = e^{- \frac{\sqrt{\frac{t}{8 π}}}{G_{γ} (1, 1)}} .

It is known that if

T_{γ}

is an exponential random variable with parameter

γ

, independent of

L^{1}

γ

, then

L_{T_{γ}}^{1}

is an exponential random variable with parameter

1 / G_{γ} (1, 1)

. Hence

E e^{- \frac{γ}{4} A_{d_{t}}} = P (L_{T_{γ}}^{1} > \sqrt{\frac{t}{8 π}}),

which we rewrite as

κ_{γ} (s) := P (L_{T_{γ}}^{1} > s) = E e^{- \frac{γ}{4} A_{d_{8 π s^{2}}}} .

(24)

For this reason, for any integer

m > 0

m \int_{0}^{\infty} s^{m - 1} κ_{γ} (s) d s = E {(L_{T_{γ}}^{1})}^{m} = γ \int_{0}^{\infty} e^{- r γ} E {(L_{r}^{1})}^{m} d r .

(25)

It is known (Jakubowski & Wiśniewolski, 2024, Theorem 2.7) that for an integer

m

E {(L_{r}^{1})}^{m} = m! \int_{0}^{r} p^{* m} (1, 1) (r) d r,

so, from(25),

\begin{aligned} \int_{0}^{\infty} (- λ)^{m - 1} \frac{s^{m - 1}}{(m - 1)!} κ_{γ} (s) d s & = - \frac{γ}{λ} \int_{0}^{\infty} e^{- r γ} \int_{0}^{r} (- λ p)^{* m} (1, 1) (v) d v d r \\ = - \frac{1}{λ} \int_{0}^{\infty} e^{- r γ} (- λ p)^{* m} (1, 1) (r) d r, \end{aligned}

for

λ > 0

. Summing up, we get

\int_{0}^{\infty} e^{- λ s} κ_{γ} (s) d s = - \frac{1}{λ} \int_{0}^{\infty} e^{- r γ} (- λ p)^{‡} (1, 1) (r) d r,

where

f^{‡}

denotes the quasi-inverse element of a locally integrable function

f

. Therefore, from (24) we conclude that

\int_{0}^{\infty} e^{- λ s} \int_{(0, \infty)} e^{- γ r} P (\frac{1}{4} A_{d_{8 π s^{2}}} \in d r) d s = - \frac{1}{λ} \int_{0}^{\infty} e^{- r γ} (- λ p)^{‡} (1, 1) (r) d r,

so by Fubini’s theorem and the Laplace transform argument

\int_{0}^{\infty} e^{- λ s} P (\frac{1}{4} A_{d_{8 π s^{2}}} \in d r) d s = - \frac{1}{λ} (- λ p)^{‡} (1, 1) (r) d r,

which means

P (\frac{1}{4} A_{d_{8 π {T_{λ}}^{2}}} \in d r) = - (- λ p)^{‡} (1, 1) (r) d r .

Applying (3) finishes the proof.

Proof. Proof of Theorem 6

Let $(B, Z)$ be a two-dimensional Brownian motion. Observe that, by construction, the excursions of a one-dimensional Brownian motion are determined by its zeros, so we are in the situation when there are infinitely many excursions and the theory works perfectly, transforming the process of excursions into a Poisson point process. Since $H$ corresponds to zeros of a one-dimensional Brownian motion $W = ρ B + \sqrt{1 - ρ^{2}} Z$ , we conclude that the additive functional $K$ has the form $K_{u} = L_{u}^{(0)} (W)$ , where $L_{u}^{(0)} (W)$ is the process of the local time of $W$ at $0$ . Now, if $A_{d_{t}} = \int_{0}^{d_{t}} e^{W_{s}} d s$ , then the exponential master formula for excursion yields

E e^{- λ A_{d_{t}}} = \exp {- E \int_{0}^{t} P^{(B_{u}, Z_{u})} (1 - e^{- λ F_{ρ}}) d L_{u}^{(0)} (W)} .

We compute the RHS of the last equality. Since

L_{u}^{(0)} (W)

increases on zeros of

W = ρ B + \sqrt{1 - ρ^{2}} Z

we have

\begin{aligned} E \int_{0}^{t} P^{(B_{u}, Z_{u})} & (1 - e^{- λ F_{ρ}}) d L_{u}^{(0)} (W) \\ = E \int_{0}^{t} P^{(B_{u}, - \frac{ρ}{\sqrt{1 - ρ^{2}}} B_{u})} (1 - e^{- λ F_{ρ}}) d L_{u}^{(0)} (W) \\ = E \int_{0}^{t} \int_{0}^{\infty} (1 - e^{- λ r}) P^{(B_{u}, - \frac{ρ}{\sqrt{1 - ρ^{2}}} B_{u})} (F_{ρ} \in d r) d L_{u}^{(0)} (W), \end{aligned}

which, by Fubini’s theorem, is equal to

J = \int_{0}^{\infty} (1 - e^{- λ r}) E \int_{0}^{t} P^{(B_{u}, - \frac{ρ}{\sqrt{1 - ρ^{2}}} B_{u})} (F_{ρ} \in d r) d L_{u}^{(0)} (W) .

We observe that if

ϕ

is a nonnegative Borel function, then

E \int_{0}^{t} ϕ (B_{u}) d L_{u}^{(0)} (W) = \int_{0}^{t} \int_{- \infty}^{\infty} \frac{1}{2 π u} ϕ (\sqrt{1 - ρ^{2}} z) e^{- \frac{z^{2}}{2 u}} d z d u .

Indeed, since

(B_{u}, W_{u})_{u} \overset{d}{=} (W_{u}, B_{u})_{u},

we have

\begin{aligned} E \int_{0}^{t} ϕ (B_{u}) d L_{u}^{(0)} (W) & = E \int_{0}^{t} ϕ (W_{u}) d L_{u}^{(0)} (B) = E \int_{0}^{t} E (ϕ (W_{u}) | F_{u}^{B}) d L_{u}^{(0)} (B) \\ = E \int_{0}^{t} E (ϕ (ρ B_{u} + \sqrt{1 - ρ^{2}} Z_{u}) | F_{u}^{B}) d L_{u}^{(0)} (B) \\ = E \int_{0}^{t} \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 π u}} ϕ (ρ B_{u} + \sqrt{1 - ρ^{2}} z) e^{- \frac{z^{2}}{2 u}} d z d L_{u}^{(0)} (B) \\ = E \int_{0}^{t} \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 π u}} ϕ (\sqrt{1 - ρ^{2}} z) e^{- \frac{z^{2}}{2 u}} d z d L_{u}^{(0)} (B) \\ = \int_{0}^{t} \int_{- \infty}^{\infty} \frac{1}{2 π u} ϕ (\sqrt{1 - ρ^{2}} z) e^{- \frac{z^{2}}{2 u}} d z d u, \end{aligned}

where in the last equality we used the known fact

d E L_{u} = 1 / \sqrt{2 π u} d u

. Having this, denoting

ϕ_{ρ} (x, d r) = P^{(x, - \frac{ρ}{\sqrt{1 - ρ^{2}}} x)} (F_{ρ} \in d r),

we rewrite

J

J = \int_{0}^{\infty} (1 - e^{- λ r}) \int_{0}^{t} \int_{- \infty}^{\infty} \frac{1}{2 π u} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, d r) e^{- \frac{z^{2}}{2 u}} d z d u,

which, by Fubini’s theorem, is equal to

\begin{aligned} J & = \int_{0}^{\infty} (1 - e^{- λ r}) \int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, d r) (\int_{0}^{t} \frac{1}{2 π u} e^{- \frac{z^{2}}{2 u}} d u) d z \\ = \frac{1}{2 π} \int_{0}^{\infty} (1 - e^{- λ r}) \int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, d r) Γ (0, \frac{z^{2}}{2 t}) d z, \end{aligned}

where

Γ (0, \cdot)

denotes the incomplete Gamma function. Hence

E e^{- λ A_{d_{t}}} = \exp {- \frac{1}{2 π} \int_{0}^{\infty} (1 - e^{- λ r}) \int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, d r) Γ (0, \frac{z^{2}}{2 t}) d z},

so using (3) and (22) we get

\frac{1}{2 π} \int_{0}^{\infty} (1 - e^{- λ r}) \int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, d r) Γ (0, \frac{z^{2}}{2 t}) d z = \frac{1}{2} \sqrt{\frac{t}{2 π}} \frac{1}{G_{4 λ} (1, 1)},

(26)

where

G_{λ} (\cdot, \cdot)

denotes the Green function of

BES (0)

. Let

Φ_{ρ} (t, d r) = \int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, \frac{1}{4} d r) Γ (0, \frac{z^{2}}{2 t}) d z .

Using (26) and Salminen, Vallois, and Yor (2007, eq.(4)) we obtain in a sense of equality

t

-almost everywhere

Φ_{ρ} (t, d r) = \sqrt{\frac{π t}{2}} μ^{(1)} (d r),

(27)

where

μ^{(1)} (d r)

is the Lévy measure of

(τ_{t}^{(1)})_{t}

—the inverse of local time at

1

BES (0)

. Integrating out (27) on

(r, \infty)

we conclude that

\sqrt{\frac{π t}{2}} μ^{(1)} ((r, \infty)) = {\hat{Q}}_{ρ} (r / 4, t) .

This proves equality (11) and that

{\hat{Q}}_{ρ} (r, t)

is finite for almost every

t

r > 0

and

ρ \in (- 1, 1)

To prove (12) we apply the exponential version of the master formula for excursions of $BES (0)$ from $1$ . It is well known that $BES (0)$ is recurrent and the theory applies exactly in the same manner as in Revuz and Yor (2005, Chapter XII, Proposition 2.7). Thus, we get

E e^{- λ τ_{t}^{(1)}} = e^{- t \int_{0}^{\infty} (1 - e^{- λ s}) n_{1} (R \in d s)} .

Taking into account definition of

Φ_{ρ}

, (27), Salminen et al. (2007, equation (4)), using

\int_{- \infty}^{\infty} ϕ_{ρ} (\sqrt{1 - ρ^{2}} z, \frac{1}{4} d r) Γ (0, \frac{z^{2}}{2 t}) d z = \int_{0}^{\infty} \frac{1}{s} e^{- \frac{s}{t}} (\int_{(- \sqrt{2 s}, \sqrt{2 s})} ϕ_{ρ} (\sqrt{1 - ρ^{2}} v, \frac{1}{4} d r) d v) d s,

which is obtained by changing the order of integration, we can conclude (12) by the Laplace transform argument. This finishes the proof.

Proof. Proof of Theorem 10

Let $B$ be a Brownian motion independent of $M_{1}$ . Consider $A_{d_{t}}$ , $t > 0$ , where $A = A^{1, 0}$ . For $λ > 0$ we have

E e^{- λ M_{1} A_{d_{t}}} \leq E e^{- λ M_{1} \int_{0}^{t} e^{B_{u} - \frac{1}{2} u} d u} \leq \frac{1}{t} \int_{0}^{t} E e^{- λ M_{1} t e^{B_{u} - \frac{1}{2} u}} d u .

Using the subscaling property over

B (0, 1) \subset R^{d}

, i.e. (15), we conclude that

E e^{- λ M_{1} A_{d_{t}}} \leq \frac{C_{2}}{t} \int_{0}^{t} E e^{- λ t e^{u d} M_{e^{- u}}} d u \leq \frac{C_{2}}{t} \int_{0}^{t} E e^{- λ t M_{e^{- u}}} d u = J (t, λ) .

Next, using Theorem 1 and identity (21), we get for all

t > 0

and

λ > 0

E e^{- λ M_{1} A_{d_{t}}} = E e^{- \frac{\sqrt{\frac{t}{8 π}}}{K_{0} (2 \sqrt{2 λ M_{1}}) I_{0} (2 \sqrt{2 λ M_{1}})}} .

(28)

We use point 1 of hypothesis S, that is, we suppose that (14) holds. Applying (14) for sufficiently large

λ

, we obtain

J (t, λ) \leq \frac{1}{t} \int_{0}^{t} C_{2} e^{- (\ln λ t)^{2} \frac{c_{1}}{1 + 2 c_{1} u}} d u \leq C_{2} e^{- (\ln λ t)^{2} \frac{c_{1}}{1 + 2 c_{1} t}},

(29)

for all

t

. Set

t = λ^{- 1} α

, where

α > 1

. Using (28) and the following asymptotic property of

K_{0}, I_{0}

lim_{x \to \infty} x K_{0} (x) I_{0} (x) = \frac{1}{2},

(30)

we find, applying the dominated convergence theorem, that

lim_{λ \to \infty} E \exp (- λ M_{1} A_{d_{λ^{- 1} α}}) = E \exp (- 2 \sqrt{\frac{α M_{1}}{π}}) .

Joining the last observation with obtained estimation on

J (t, λ)

, that is (29), we conclude for large

λ

and our choice of

t

that

E e^{- 2 \sqrt{\frac{M_{1} α}{π}}} \leq C_{2} e^{- (\ln α)^{2} \frac{c_{1}}{1 + 2 c_{1} λ^{- 1} α}} .

Taking

λ \to \infty

, we obtain

E e^{- 2 \sqrt{\frac{M_{1} α}{π}}} \leq C_{2} e^{- c_{1} (\ln α)^{2}},

which means that for every

α > 1

c_{1} \leq \frac{1}{(\ln α)^{2}} \ln \frac{C_{2}}{E e^{- 2 \sqrt{α π^{- 1}} \sqrt{M_{1}}}} .

Furthermore, by Jensen’s inequality,

E e^{- 2 \sqrt{α π^{- 1}} \sqrt{M_{1}}} \geq e^{- 2 \sqrt{α π^{- 1}} E \sqrt{M_{1}}} \geq e^{- \sqrt{α π^{- 1}} (1 + E M_{1})} .

However,

E M_{1} = | B (0, 1) |_{d}

(see, e.g., Berestycki, 2017), so

E e^{- 2 \sqrt{α π^{- 1}} \sqrt{M_{1}}} \geq e^{- \sqrt{α π^{- 1}} (1 + | B (0, 1) |)} .

As a result, we obtain

c_{1} \leq \frac{1}{(\ln α)^{2}} \ln (C_{2} e^{\sqrt{α π^{- 1}} (1 + | B (0, 1) |}) =: f (α),

for every

α > 1

. By the direct analysis we find that

inf_{α > 1} f (α) = f (α^{*}),

where

α^{*}

is the unique solution of equation (17). Hence (16) holds. Finally, if

C_{2} = 1

then

α^{*} = e^{4}

and (18) follows by straightforward computations. This finishes the proof.

Footnotes

Statements and Declarations

Ethical Approval and Consent to Participate

The authors declare ethical approval and consent to participate.

Consent for publication

The authors declare consent for publication.

Human and Animal Ethics

Not applicable.

Authors’ Contributions

The text is written by Jacek Jakubowski and Maciej Wiśniewolski.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported partially by University of Warsaw Grants IDUB-622-239/2022 and IDUB-POB3-D110-003/2022.

Declaration of Conflicting Interests

The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: We declare that the authors have no potential conflicts and competing interests or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Availability of Supporting Data

Not applicable.

References

Aru

(2020). Gaussian multiplicative chaos through the lens of the 2D Gaussian free field. Markov Processes and Related Fields, 26(1), 17–56.

Berestycki

(2017). An elementary approach to Gaussian multiplicative chaos. Electronic Communications in Probability, 22, 1–12. https://doi.org/10.1214/17-ECP58

Blumenthal

(1992). Excursions of Markov processes. Birkhäuser.

Borodin

Salminen

(2002). Handbook of Brownian motion-facts and formulae (2nd ed). Birkhäuser.

Burdzy

(1986). Brownian excursions from hyperplanes and smooth surfaces. Transactions of the AMS, 295(1), 35–57. https://doi.org/10.2307/2000144

Donati-Martin

Ghomrasni

Yor

(2001). On certain Markov processes attached to exponential functionals of Brownian motion: Application to Asian options. Revista Matematica Iberoamericana, 17(1), 179–193. https://doi.org/10.4171/RMI/292

Duplantier

Sheffield

(2011). Liouville quantum gravity and KPZ. Inventiones Mathematicae, 185(2), 333–393. https://doi.org/10.1007/s00222-010-0308-1

Hartman

Watson

(1974). Normal distribution functions on spheres and the modified Bessel functions. Annals of Probability, 5, 582–585.

Jakubowski

Wiśniewolski

(2020). Another look at the Hartman–Watson distributions. Potential Analysis, 53, 1269–1297. https://doi.org/10.1007/s11118-019-09806-7

10.

Jakubowski

Wiśniewolski

(2024). On bivariate distributions of the local time of Itô-McKean diffusions. Bernoulli, 30, 227–251. https://doi.org/10.3150/23-BEJ1595

11.

Kahane

(1985). Sur le chaos multiplicatif. Annales des Sciences Mathématiques du Québec, 9(2), 105–150.

12.

Lacoin

Rhodes

Vargas

(2022). Path integral for quantum Mabuchi K-energy. Duke Mathematical Journal, 171(3), 483–545. https://doi.org/10.48550/arXiv.1807.01758

13.

Laforgia

Natalini

(2010). Some inequalities for modified Bessel functions. Journal of Inequalities and Applications, 2010, 10. https://doi.org/10.1155/2010/253035

14.

Lew

(1972). On linear Volterra integral equations of convolution type. Proceedings of the American Mathematical Society, 35, 450–455.

15.

Lyasoff

(2016). Another look at the integral of exponential Brownian motion and the pricing of Asian options. Finance and Stochastics, 20, 1061–1096. https://doi.org/10.1007/s00780-016-0307-1

16.

Mansuy

Yor

(2008). Aspects of Brownian motion (Universitext). Springer.

17.

Matsumoto

Yor

(2001). An analogue of Pitman’s

2 M - X

theorem for exponential Wiener functionals, part II: The role of the generalized Inverse Gaussian laws. Nagoya Mathematical Journal, 162, 65–86. https://doi.org/10.1017/S0027763000007455

18.

Matsumoto

Yor

(2005a). Exponential functionals of Brownian motion, I: Probability laws at fixed time. Probability Surveys, 2, 312–347. https://doi.org/10.1214/154957805100000159

19.

Matsumoto

Yor

(2005b). Exponential functionals of Brownian motion, II: Probability laws at fixed time. Probability Surveys, 2, 348–384.

20.

Revuz

Yor

(2005). Continous martingales and Brownian motion (3rd ed). Springer-Verlag.

21.

Rhodes

Vargas

(2014). Gaussian multiplicative chaos and applications: A review. Probability Surveys, 11, 315–392. https://doi.org/10.1214/13-PS218

22.

Salminen

Vallois

Yor

(2007). On excursion theory of linear diffusions. Japanese Journal of Mathematics, 2, 97–127. https://doi.org/10.1007/s11537-007-0662-y

23.

Schilling

Song

Vondracek

(2010). Bernstein functions: Theory and applications. de Gruyter.

24.

Sheffield

(2007). Gaussian free fields for mathematicians. Probability Theory and Related Fields, 139, 521–541. https://doi.org/10.1007/s00440-006-0050-1

25.

Talarczyk-Noble

Wiśniewolski

2025, On Small Deviations of Gaussian Multiplicative Chaos with A Strictly Logarithmic Covariance on Euclidean Ball. Potential Analysis. https://doi.org/10.1007/s11118-025-10197-1