Stochastic Quantization of the three-dimensional polymer measure via the Dirichlet form method

Three dimensional polymer measureThree dimensional polymer measure

Dedicated to the Memory of Xian Yin Zhou, with great admiration and gratitude for his work, on which the present contribution builds

We prove that there exists a diffusion process whose invariant measure is the three dimensional polymer measure $\nu_\lambda$ for small $\lambda>0$.
We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X.Y. Zhou. For the construction of $\nu _\lambda$ we rely on previous work by J. Westwater, E. Bolthausen and X.Y. Zhou.
Using $\nu _\lambda$, the diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. The closability of the appropriate pre-Dirichlet form which is of gradient type is proven, by using a general closability result in [18]. This result does not require an integration by parts formula  (which does not even hold for the two-dimensional polymer measure $\nu_\lambda$) but requires the quasi-invariance of $\nu_\lambda$ along a basis  of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives have versions which form a continuous process.

AMS 2010 Subject Classification: Primary 81S20, 60J65. Secondary 60J46, 60H30.

Key words: Three-dimensional polymer measure, closability, Dirichlet forms, diffusion processes, quasi-invariance.

§ INTRODUCTION

§.§ Polymer measure

In the literature on polymer physics, random walks and Brownian motions can be regarded as realizations of polymer chains that consist of a huge number of monomers. Moreover, it is natural to think of self-interaction of polymer chains, e.g. an attractive or repulsive effect due to Van der Waals force between monomers, and a repulsive effect due to the physical restriction such that different monomers should not occupy the same point in space.
Edwards' model (see [60]) is a probabilistic model for long polymer chains that takes into account the self-exclusive volume effects.   It is formally given by a probability measure on Wiener space:

\begin{align}\label{eq:polmea}
\nu_\lambda(\dd\omega)=N_\lambda^{-1}\exp\left(-\lambda \int_0^1\int_s^1 \delta_0(\omega_t-\omega_s)\dd s \dd t\right)\nu_0(\dd \omega),
\end{align}

where $\nu_0$ is the Wiener measure and
\begin{align*}
J(\omega):= \int_0^1\int_s^1 \delta_0(\omega_t-\omega_s)\dd s \dd t
\end{align*}
is the self-interaction local time of the $d$-dimensional Brownian motion $\{\omega_t\}_{t\geq 0}$ in time interval $[0,1]$,  $\lambda\in [0,\infty)$ is a coupling constant, and $N_\lambda$ is the normalizing constant to assure that $\nu_\lambda$ is a probability measure. We will give their precise definitions below.

In constructive quantum field theory (CQFT), Symanzik independently introduced a representation of $\phi^4$ theory in terms of a gas of Brownian paths (see [114]). In this representation, a self-intersection local time naturally appear. Later, Brydges, Fröhlich, Sokal, and Spencer established the existence and nontriviality of $\phi^4_2$ and $\phi^4_3$ via a random walk representation (discrete counterpart to (<ref>)), Schwinger-Dyson equation, and skeleton inequality (see [35, 36, 37, 33, 61]).

The intersection properties of Brownian paths have been investigated since
the forties [94]. In [51, 52] Dovorezcky, Erdös, Kakutani, and Taylor  proved that the $d$-dimensional Brownian motion $\omega= \{\omega_t\}_{t\geq 0}$ has no multiple points when $d\geq 4$ almost surely so that $J=0$  and   $\nu_\lambda$ is trivial, whereas it has multiple points for $d\leq 3$ almost surely, that is $\{0\leq s< t\leq 1:\omega_s=\omega_t\}\not=\emptyset$. Hence, we focus on the case for $d\leq 3$. We remark that the critical dimension of multiple points is $d=6$ from the viewpoint of quasi-every [97].  (In connection with the triviality of $\phi^4_4$ theory, intersection local time and Edwards' model in $d=4$ are discussed in [9, 4, 10, 6, 24, 3, 66, 67, 75, 102].)

In order to describe  $J$ and $\nu_\lambda$ more precisely, we start with recalling some facts about the $d$-dimensional polymer measure.  Let $X:=C_0([0,1],\R^d)$ be the set of all continuous paths in $\R^d$ indexed by $[0,1]$ and starting at zero. Let $\mathcal{B}$ be the $\sigma$-algebra generated by all maps $\omega\mapsto \omega_t$, $t\geq 0$, from $X$ to $\R^d$ and let $\nu_0$ denote the Wiener measure on $(X,\mathcal{B})$.

First, we “approximate" $J(\omega)$ in the following way:
For $\e\in (0,1)$, $a\in (0,1)$, $0\leq s\leq t\leq 1$, $0\leq u\leq v\leq 1$, and $\omega\in X$, let
\begin{align*}
J_{s,t;u,v}^{\e,a}:=\int_s^t\dd \sigma \int_{(u\vee (\sigma+\e))\wedge v}^v \dd \tau p_a(\omega_\sigma-\omega_\tau)
\end{align*}

\[
p_a(x) := \frac{1}{(2\pi a)^{d/2}}\exp \left( -\frac{|x|^2}{2a}\right) \quad \text{for $a>0$, $x\in \R^d$}.
\]

Taking the limit of $J_{s,t:u,v}^{\e,a}$ as $a \searrow 0$ formally,
$J^{\e,0}_{s,t;u,v}$ describes the “self-intersection local time" of Brownian paths $\omega[s,t]$ and $\omega[u,v]$ with removal of times near the diagonal
by parameter $\e>0$. The probability measure in (<ref>) is defined by

\begin{align}
\nu_{\lambda}(\dd \omega)=\lim_{\e\searrow 0}\frac{1}{N_\lambda^\e}\exp\left(-\lambda J_{0,1;0,1}^{\e ,0}\right)\nu_0(\dd\omega) \label{eq:nulambda}
\end{align}

if the limit exists.
For $\lambda>0$, the realization of paths under $\nu_\lambda$ likes to avoid self-intersection other than those already present under $\nu_0$.
The difficulty to define $\nu_\lambda$ mathematically rigorously is that it depends on the dimension of the space.

For the case $d=1$, Westwater discussed the $L^1$-convergence of $J_{0,1;0,1}^{0,a}$ as $a\searrow 0$  and  the limit is  given by

\begin{align}
\frac{1}{2}\int_\R \ell_1(x)^2\dd x\label{eq:d=1SelfInLocal}
\end{align}

where $\{\ell_1(x)\}_{x\in\R}$ is the local time of Brownian motion $\omega$ (almost surely defined) up to time $1$, which are continuous and hence (<ref>) makes sense (see [123]). Furthermore, (<ref>) does make sense. This explicit expression allows us to analyze the properties of the long-time behavior of $\omega$  under the polymer measures (see [91, 125]). Westwater has shown that for $T \rightarrow \infty$, the law of $\frac{\omega_T}{T}$ converges under $\nu_{\lambda,T}:=\frac{1}{Z_{\lambda,T}}\exp\left(-\lambda J_{0,T;0,T}^{0,0}\right)\nu_0$ to $\frac{1}{2}(\delta _{\lambda^* \lambda ^{1/3}} + \delta _{-\lambda ^* \lambda ^{1/3}} )$, where $J_{0,T;0,T}^{0,0}$ is defined similarly to $J_{0,1;0,1}^{0,0}$ by replacing $[0,1]$ by $[0,T]$ and   $\lambda^* \sim 1.1$ is a constant. This result is improved by van der Hofstad, den Hollander, König in [117, 118, 120, 119] and Najnudel in [101].

In the case that $d=2$, it is known that $J_{0,1;0,1}^{0,a}$ diverges almost surely as $a\searrow 0$ (see [116, 95, 126]). But, there exists a random variable $Y$ such that $J_{0,1;0,1}^{0,a}-E[J_{0,1;0,1}^{0,a}]$ converges to $ Y$ in $L^2(\nu_0)$ as $a\searrow 0$ (Varadhan's renormalization) and $E[e^{-\lambda  Y}]<\infty$ for all $\lambda \geq 0$ (see [116, 95, 107]). Therefore, (<ref>) is mathematically rigorously defined by $\nu_\lambda(\dd \omega)=\frac{1}{E[e^{-\lambda  Y}]}e^{-\lambda Y}\nu_0(\dd \omega)$.
Moreover, there exists $g_0>0$ such that $E[e^{gY}]<\infty$ for $g\in (-\infty,g_0)$ (see [113, 96]).
Then, Bass and Chen showed that the best constant $g_0$ coincides with $A^{-4}$, where
$A$ is the best constant in the Gagliardo-Nirenberg inequality [28].
Here, we remark that for $\lambda\in (-g_0,\infty)$, $\nu_\lambda$ is absolutely continuous with respect to $\nu_0$. The $k$-multiple intersection local times for the planar Brownian motion are also discussed in [58, 59, 109].

The case $d=3$ is more complicated, because Varadhan's renormalization does not work, since the variance of $J_{0,1;0,1}^{\e,a}$ diverges as $a\searrow 0$ and $\e\searrow 0 $ (see (<ref>) below). However, Westwater constructed the polymer measure $\nu_\lambda$ in [123, 124] and then Bolthausen gave a simpler construction in [46] for small values of the coupling constant $\lambda\geq 0$. In [124] it is shown that the latter approach is still suitable for all positive coupling constants, and in [27] it is proved that the polymer measure given by Bolthausen is actually the same as the one given by Westwater. It is also proved that the three-dimensional polymer measure is singular with respect to the Wiener measure (see [124]). More precisely, $\nu_{\lambda_1}$ and $\nu_{\lambda_2}$ are mutually singular for $0\leq \lambda_1<\lambda_2$. We remark that Bovier, Felder, and Fröhlich proved the tightness of approximated polymer measures in [34].
Also, Gubinelli and Lörinczi discussed the construction of $\nu_\lambda$ (for $d=1$) with the general theory of Gibbs measures via rough path theory in [72].

The self-intersection local times are interesting objects themselves,
and many studies are dedicated to them. For example, the Wiener-Chaos
decomposition of regularizations of the self-intersection local times
are studied in [48, 53, 82, 85]. On the other hand, nondifferentiability of the self-intersection local times in the sense of Meyer-Watanabe
is discussed in [8, 82] in the case that $d=2$. The
self-intersection local times are generalized formally by letting
$J_x(\omega):=\int_0^1\int_0^t \delta _0(\omega_t-\omega_s-x)\dd s\dd
t$ for $x\in\R^d$. Then, it is known that for $x\not =0$,
$J_x(\omega)$ is well-defined as a function for $d=1,2,3$ and as a
generalized function in Sobolev space with negative order for $d\geq
4$ (see [106, 95, 126, 85, 110]).
We remark that for $d\geq 4 $, the self-intersection local time
$J_0(\omega)$ of Brownian motion can be defined as a generalized
functional of Brownian motion in the space of Hida distributions (see
[122, 84]) with some suitable renormalization.

Now, we give a rigorous definition of $\nu_\lambda$ as in [46, 27]: For the rest of this paper, we fix $\lambda\in [0,\infty)$.
We recall the following result by Bolthausen.

There exists a version of $J_{s,t;u,v}^{\e,a}$ which is jointly continuous in all the variables $0\leq s\leq t\leq 1$ and $0\leq u\leq v\leq 1$, and parameters $\e\in (0,1)$ and $a\in (0,1)$.
Moreover, the limit as $a\downarrow 0$ exists and also has these properties.

We can prove in a similar way to <cit.> that for each $\e>0$, $0\leq s\leq t\leq 1$, $0\leq u\leq v\leq 1$, and for any $p\geq 1$, $J_{s,t;u,v}^{\e,a}$ converges in $L^p(\nu_0)$ as $a\searrow 0$, which is not mentioned in the paper [46] explicitly.

From now on, we regard $J_{s,t;u,v}^{\e,a}$ as the version given in Proposition <ref>. In particular, we write $J_{s,t;u,v}^{\e}:=J_{s,t;u,v}^{\e,0}$ and we replace $p_a(\omega_\sigma-\omega_\tau)$ by $\delta_0({\omega_\sigma-\omega_\tau})$ in equations. Also, we denote $J_{0,1;0,1}^{\e,a}$ by $J_{0,1}^{\e,a}$ for $\e>0$ and $a\geq 0$ (see also Corollary <ref>).

For $\e\in (0,1)$, we set

\begin{align}
\kappa_1(\e)=\int_\e^1 p_t(0)\dd t=2(2\pi)^{-\frac{3}{2}}(\e^{-\frac{1}{2}}-1)\label{eq:kappa1a}
\end{align}

\begin{align*}
\kappa_2(\e) &= (2\pi)^{-3}\int_0^1\dd s_1{\int_{s_1}^{1}\dd s_2\int_{s_2}^1\dd s_31_{\{\e\leq s_2\}}1_{\{\e\leq s_3-s_1\}}}\\
&\quad \hspace{4cm} \times (s_1(s_2-s_1)+s_1(s_3-s_2)+(s_3-s_2)(s_2-s_1))^{-\frac{3}{2}}.
\end{align*}

In Section <ref>, we shall prove that there exist constants $K_1,K_2,K_3\in \R$ such that

\begin{align}
&\lim_{\e\searrow 0}\left(E[J_{0,1}^\e]-\kappa_1(\e)\right)=K_1,\label{eq:kappa1}\\
&\lim_{\e\searrow 0}\left(\mathrm{Var}\left(J_{0,1}^\e\right)+\frac{2}{(2\pi)^2}\log \e\right)=K_2,\label{eq:kappa2-2}\\
&\lim_{\e\searrow 0}\left(\kappa_2(\e)+\frac{1}{(2\pi)^2}\log \e\right)=K_3.\label{eq:kappa2}
\end{align}

These imply that $\kappa_1(\e)$ and $2\kappa_2(\e)$ are divergent parts of the expectation and variance of $J_{0,1}^\e$, respectively.

\begin{align*}
&\bar{J}_{s,t}^{\e,\lambda}:=\lambda J_{s,t}^\e-\lambda(t-s)\kappa_1(\e)+\lambda^2(t-s)\kappa_2(\e),
\intertext{and}
\end{align*}

where $E$ is the expectation with respect to $\nu_0$. It was shown in [46] and [27] that the limit $\dis \lim_{\e\searrow 0}\nu_{\e,\lambda}$ exists in the weak sense. This limit is denoted by $\nu_\lambda$, which is a rigorous version of (<ref>). The reader may refer to the lecture notes <cit.> treating Edwards' model.

As mentioned above, $\nu_\lambda$ is singular with respect to $\nu_0$ for $\lambda>0$ for the three-dimensional case, and hence it is hard to analyze path properties under $\nu_\lambda$. In [92], Shigeo Kusuoka proved the $\{\omega_{t}\}_{t\in[0,1]}$ under $\nu_\lambda$ is a Dirichlet process in the sense of Föllmer [64]. In [127, 130, 129, 128, 131], Xianyin Zhou discussed the self-intersection local times, the Hausdorff dimensions of the set of double points, and the other properties of the paths.

Also, the self-intersection local time and the associated polymer measures have been studied for $d$-dimensional fractional Brownian motions with Hurst parameter $H\in (0,1)$ [38, 39, 48, 73, 83, 80, 81, 108]. In [80], it is proved that if $dH<1$, $J_{0,1;0,1}^{0,a}$ converges in $L^2$-sense; and if $dH\in \left[1,\frac{3}{2}\right)$,  Varadhan's renormalization does also work for $J^{0,a}_{0,1;0,1}$. In [73, 39], a construction of the polymer measures for $dH\leq 1$ has been given by analyzing the corresponding self-intersection local times.

We can also consider a similar model in a discrete setting by replacing the Brownian motion by a simple random walk $\{S_n\}_{n=0}^\infty$, that is, the self-intersection local time is defined by $L_N(S)=\sum_{0\leq i<j\leq N}1\{S_i=S_j\}$ and define Gibbs measures by
\begin{align*}
P_N^g(\dd S)=\frac{1}{Z_N^g}\exp\left(-g L_N(S)\right)P(\dd S),\quad \text{for }g\geq 0,
\end{align*}

where $P$ is the law of simple random walk $S$.
This model is called the weakly self-avoiding walk or the Domb-Joyce model, and has been well-studied (see  [54, 93, 99]).
Unlike Brownian motion, the self-intersection local time with respect to the simple random walk can be positive.
Hence, $L_N$ and $P_N^g$ do make sense. As discussed in [93], the interaction in Edwards' model is weaker than the one in the weakly-self avoiding walk. To approximate Edwards' model for $d=2,3$, it is suitable to renormalize $g$ as $g_N=2N^{\frac{d-2}{2}}$ with some time-space rescaling. Actually, Stoll in [113] constructed Edwards' model for $d=2$ via this rescaling and the use of Robinson's nonstandard analysis (see e.g. [6]).
In [2], the same method was discussed for $d=3$.
For $d=4$, it is known that paths of the simple random walk have infinitely many self-intersection points, thus behave in another way with respect to Brownian motion (see [93]).
The limit of the rescaled measures $P_{N}^{g_N}$ may in principle under certain restriction on $\lambda$ yield the polymer measure for $d=4$. In [24], the question of the asymptotics of the probabilities of the event that two independent simple random walks in $\Z^4$ have no intersection points up to time $N$ is discussed.
Symanzik, Nelson and Aizenman pointed out that this quantity is related to the $\phi^4_4$ theory (see [9, 4]). In the view point of self-intersection local times, the first author and Zhou in [25] proved the central limit theorem for the renormalized self-intersection local times $L_N(S)$ for $d\geq 3$ as discussed in [126].

§.§ Stochastic quantization

Parisi and Wu introduced the idea for stochastic quantizations in the literature of quantum fields theory. For a given probability measure $\nu$ on a measurable space $\mathcal{X}$, the stochastic quantization of $\nu$ means a construction of a Markov process that has an invariant distribution $\nu$ (see [105]).
In the case of finite-dimensions Kolmogorov had studied such a problem via Fokker-Plank equations many years before (see [88]).
In the present paper we discuss stochastic quantization of the three-dimensional polymer measure, which is a much more difficult case, because the measure is on the (infinite-dimensional) path space and is defined via renormalization.

In the case of $\phi ^4$ Euclidean quantum field theory, the stochastic quantization has been a very hot topic starting many years ago and it remains so also now.
In [87] stochastic quantization for dimension $2$ given by stochastic partial differential equations (SPDE) with regularized noise is studied.  Stochastic quantization of the two-dimensional $\phi ^4$ theory is
studied by infinite-dimensional Dirichlet forms in [22] (see also [31]) and also by SPDE approach in [55].
On the other hand, stochastic quantization for quantum field models in the three-dimensional case had been remained as a difficult problem for a long time.
However, recently the theories of regularity structures introduced by Hairer in [77] and paracontrolled calculus introduced by
Gubinelli, Imkeller and Perkowski in [71] appeared and these methods enable us  to solve nonlinear SPDEs with singular noise via renormalization.
These methods work very well on stochastic quantization of quantum field theories, and are now developing remarkably.
Stochastic quantization of the three-dimensional $\phi ^4$ theory via SPDEs was first achieved locally in time on torus by Hairer via regularity
structure (see <cit.>), also via paracontrolled calculus locally in time on torus (see [49]), then globally in
time on torus (see [100]), and globally in time on the whole space (see [68]).
Construction of the three-dimensional $\phi ^4$-quantum field measure also has succeeded by the methods of singular SPDEs, e.g., on
torus (see [11]), on the whole space by discrete approximation [69], and on the whole space by continuous approximation (see
Besides, stimulated by the development of the methods of singular SPDEs, many mathematicians study similar but different approaches,
e.g, by renormalization group (see [90]), by a variational method (see [40]), by elliptic stochastic quantization (see
[68, 5]), and by scaling argument (see [57]).
For the details of the history of the stochastic quantization of $\phi ^4$ theory, see the introductions of [11, 12, 69] and
references therein. Here, we recall that, as mentioned in above, in [114] Symanzik
suggested a close relation between the three-dimensional polymer measure and the three-dimensional $\phi ^4$ theory by formal
calculations, and indeed many similarities of the models are known, e.g., the renormalization constants which appeared in both models
have the same asymptotics, both measures are singular with respect to the base measures (the base measure is the free field measure in the case of the
$\phi ^4$ theory, and the Wiener measure in the case of the polymer measure).
Also, the renormalization constants of the one-dimensional KPZ equation have the same asymptotics of those of the three-dimensional
polymer measure. See [76] and [74] for approaches from singular SPDEs.
The relation between the one-dimensional KPZ equation and the $(1+1)$-dimensional directed polymers in random environment (DPRE) had already been
known in the paper [89], which first introduced the KPZ equation. Recently, DPRE has focused on the connection between the KPZ equation
and the stochastic Burgers equation (see [26, 29, 30, 43, 44, 45, 50, 56, 62, 63, 70, 74, 86, 103, 115]).
Furthermore, stochastic quantization techniques for random measures on a path space have been applied to other problems (see [30]).
Besides, the stochastic quantization of exponential interaction (Høegh-Krohn model) has also been studied both from Dirichlet
form theory and singular SPDEs (see [13, 78, 79] and references therein).

Our main result in this paper is about the stochastic quantization of the three-dimensional polymer measure via the Dirichlet form method.

In [7], the stochastic quantization of the two-dimensional
polymer measure was constructed using the theory of Dirichlet forms on
infinite-dimensional state spaces. In the present paper the same is done in
the case of the three-dimensional polymer measure, which is technically much
more difficult, since in three-dimensions the polymer measure is not absolutely continuous with respect to the Wiener measure (as opposed to the lower-dimensional cases after Varadhan's renormalization for $d=2$).
In the unpublished work [23] a construction of the Dirichlet form associated with the three-dimensional polymer measure was attempted.
However, the proofs were not complete or contained gaps (e.g. <cit.>).
This work in 1996 remained unfinished due to the sudden tragic departure of Xianyin Zhou in his early age.
The present paper is largely based on the previous work [23], that in turn was possible through deep in sights accumulated by X.Y.Zhou in his deep work (see also e.g. [127, 128, 129, 130, 131]).
In the present paper we present a self-contained full proof of the construction of the Dirichlet form associated with the three-dimensional polymer measure, which guarantees that the statement of the main theorem of the preprint [23] is correct.

Let us remind that the construction of the polymer measure $\nu _\lambda$ in the cases that $d\leq 2$ are somewhat similar in difficulty with the construction of the Euclidean measures of models of quantum field theory in dimension $d\leq 2$, see e.g. the references  given in the introductions of [11, 12, 69].
For the construction of associated Dirichlet forms see [22, 17, 111, 13, 78, 79, 41] for $d\leq 2$.
For the case $d=3$, the construction of the polymer measure and associated Dirichlet form is comparable in difficulty with the construction of the $\phi ^4_3$-measure and and associated Dirichlet form (see [132, 41]).
A direct construction of a stochastic process with the $\phi ^4_3$-measure as an invariant measure has been provided in various recent works [76, 68, 11, 132] and the references in [12].
See below after Remark <ref> for further comments on this.

We formulate the main results of the present paper in the next subsection (Theorems <ref>, <ref> and <ref>).
Let us note that the irreducibility of the Dirichlet form associated to $\nu _\lambda$ also remains as an open problem (whereas it has been solved for the case that $d=2$ in [7], see also [14, 15] for relations between irreducibility and ergodicity of associated process).

When one replaces Brownian motion by a fractional Brownian motions, similar constructions of associated polymer measures and stochastic quantization have been discussed in [38]. In [83, 81], it is proved that the self-intersection local times are Meyer-Watanabe differentiable for $dH<1$ where $H$ is the Hurst index of fractional white noise. In [38], the authors used [104] and Meyer-Watanabe differentiability to obtain a diffusion process whose invariant measure is the fractional Edwards' model for $d$-dimensional fractional Brownian motion. The diffusion is constructed via Dirichlet form techniques in infinite-dimensional analysis, where the closability was shown by an integration by parts formula, and irreducibility of the constructed diffusion follows as in the case of planar Brownian motion (discussed in [7]).

§.§ Main results

Recall that $X:=C_0([0,1],\R^d)$ and $X$ equipped with the supremum-norm is a separable real Banach space.
Let $X'$ be its dual. Let $H\subset X$ be the classical Cameron-Martin space, i.e., $H:=\{h\in X:\text{$h$ is absolutely continuous and $|h|_H^2=\int_0^1|h'(t)|^2\dd t<\infty$}\}$.
Furthermore, let $K:=\{h\in H: \sup_{0\leq t\leq 1}|h'(t)|<\infty\}$.
Then, $(H,\langle \cdot , \cdot \rangle)$ is a real separable Hilbert space which is densely and continuously embedded into $X$, and $K$ is a linear subspace of $(H,\langle \cdot , \cdot \rangle_H)$.
By identifying $H$ with its dual we obtain that $X'$ is densely embedded into $H$.
Hence, $X'\subset H\subset X$, and $\langle \cdot , \cdot \rangle_H$ restricted to $X'\times H$ coincides with the dualization between $X'$ and $X$.
Define the set  of bounded smooth cylinder functions by

\begin{align*}
\mathcal{F}C_b^\infty :=\{f(l_1,\cdots,l_m):m\in \N,f\in C_b^\infty(\R^m),\l_1,\cdots,l_m\in X'\},
\end{align*}

where $C_b^\infty(\R^m)$ denotes the set of all bounded infinitely differentiable functions on $\R^m$ with all bounded continuous partial derivatives.

Let $h\in X$. For $u\in \mathcal{F}C_b^\infty$, $\omega\in X$, define
\begin{align*}
\frac{\partial u}{\partial h}(\omega):=\left.\frac{\dd }{\dd s}u(\omega+sh)\right|_{s=0}
\end{align*}

and let $\nabla u(\omega)$ denote the unique element in $H$ such that
\begin{align*}
\langle \nabla u(\omega), h\rangle_H=\frac{\partial u}{\partial h}(\omega)\quad \text{for all }h\in H.
\end{align*}

Define for $u,v\in \mathcal{F}C_b^\infty$
\begin{align*}
\cE_{\nu_\lambda}(u,v):=\int \langle \nabla u(\omega),\nabla v(\omega)\rangle_H\nu_\lambda(\dd \omega)
\end{align*}

and $\cE_{\nu_\lambda,1}:=\cE_{\nu_\lambda}+(\cdot , \cdot )_{L^2(X;\nu_\lambda)}$.

Now, we can formulate our first result.

* The support of $\nu _\lambda$, $\mathrm{supp}\nu_\lambda$, coincides with $X$ (i.e. $\nu_\lambda(U)>0$ for all non-empty open $U\subset X$). In particular, we can identify each $u\in \mathcal{F}C_b^\infty$ with the corresponding class in $L^2(X;\nu_\lambda)$, and thus $(\cE_{\nu_\lambda},\mathcal{F}C_b^\infty)$ is a symmetric non-negative definite bilinear form on $L^2(X;\nu_\lambda)$.

* The symmetric bilinear form $(\cE_{\nu_\lambda},\mathcal{F}C_b^\infty)$ is closable on $L^2(X;\nu_\lambda)$ and the closure $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$ is  a symmetric Dirichlet form, i.e., a closed non-negative definite symmetric bilinear form such that $u^\#:=(u\vee 0)\wedge 1\in D(\cE_{\nu_\lambda})$ and $\cE_{\nu_\lambda}(u^\#,v^\#)\leq \cE_{\nu_\lambda}(u,v) $ for all $u\in D(\cE_{\nu_\lambda})$.

The closability of $(\cE_{\nu_\lambda},\mathcal{F}C_b^\infty)$ on $L^2(X;\nu_\lambda)$ means that the unique continuous extension $\bar{\iota}$ of the continuous embedding $\iota : \mathcal{F}C_b^\infty \hookrightarrow L^2(X;\nu_\lambda)$, where $\mathcal{F}C_b^\infty$ is equipped with norm $(\cE_{\nu_\lambda,1})^{\frac{1}{2}}$ and $L^2(X;\nu_\lambda)$ with $\| \cdot \| _{L^2(X;\nu_\lambda)}$, to the completion $\overline{\cF C_b^\infty}$ of $\cF C_b^\infty $ with respect to $(\cE_{\nu_\lambda,1})^{\frac{1}{2}}$ is still continuous embedding.
The closure $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$ is then the smallest closed extension of $(\cE_{\nu_\lambda},\cF C_b^\infty)$ on $L^2(X;\nu_\lambda)$. For more details on Dirichlet forms the reader is referred, for example,  to [98, 65, 42, 112], and for the special type of Dirichlet form appearing here, namely so-called classical Dirichlet forms to [20].

Let $(L,D(L))$  be the generator of $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$, i.e., the unique non-positive definite self-adjoint operator on $L^2(X;\nu_\lambda)$ such that
\begin{align}
D(\sqrt{-L})=D(\cE_{\nu_\lambda})\quad \text{and}\quad (\sqrt{-L}u,\sqrt{-L}v)=\cE_{\nu_\lambda}(u,v)\quad \text{for all }u,v\in D(\cE_{\nu_\lambda}).
\end{align}

Let $T_t:=e^{tL},$ $t\geq 0$.
Then, the general theory of Dirichlet form yields the following theorem.

There exists a diffusion process $\mathbb{M}=(\Omega,\cF,(\cF_t)_{t\geq 0},(X_t)_{t\geq0},(P_\omega)_{\omega\in X})$ which is associated with $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$, i.e., for all ($\nu_\lambda$-versions of ) $f\in L^2(X;\nu_\lambda)$ and all $t\geq 0$ the function

\begin{align*}
\omega\mapsto p_tf(\omega):=\int_\Omega f(X_t)\dd P_\omega,\quad \omega\in X,
\end{align*}

is a $\nu_\lambda$-version of $T_tf$. Moreover, $\mathbb{M}$ is conservative and $\nu_\lambda$-symmetric. In particular, $\nu_\lambda$ is an invariant measure for $\mathbb{M}$.

* $\mathbb{M}$ is in fact even properly associated to $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$ in the sense of <cit.>.

* $\Omega$ in Theorem <ref> can, of course, always be chosen to be $C([0,\infty),X)$, since $\mathbb{M}$ has continuous sample paths and is conservative.

* The proof of Theorem <ref> fills gaps in the proof of [23]. In particular, in Section <ref> we adjust Rosen's method to the three-dimensional case, and by applying the estimates obtained in Section <ref> we prove some suitable estimates in Section <ref>, which are not correctly proved in [23].

As already emphasized in [7] the construction of the stochastic quantization of $\nu_\lambda$ by the theory of Dirichlet forms uses $\nu_\lambda$ and is based on some of its properties. It thus does not lead to an independent construction of $\nu_\lambda$ itself. This is similar to early works on the stochastic quantization of measures in Euclidean field theory (such as e.g. $P(\phi)_2$-fields, see for example [87, 18, 19, 22, 17, 1]) where the respective measures have been used in an essential way.
A truly constructive approach of the invariant measures (more precisely by the stochastic quantization equations for Euclidean quantum fields) was not known (for dimension $d=2,3$) before the work initiated in [55] for $d=2$, and [76, 71, 11, 69, 12] for $d=3$ .
For polynomial and exponential type models in $d=2$ even uniqueness results are known (see e.g. [111, 32, 121]).
For $d=3$ $\phi ^4_3$-models results are less compete (see [16, 132, 40, 11, 12]).
To the best of our knowledge, the problem of a constructive approach of $\nu _\lambda$ for polymer measures in $d=2,3$ does not seem to have been discussed from this point of view, i.e. via solutions of SPDEs.
As mentioned in Section <ref>, Theorem <ref> is the first construction of the Dirichlet form with respect to a singular Euclidean quantum field measure.

For the proof of Theorem <ref>, we need the following result, which is of its own interest and is in fact the heart of this paper. For $h\in X$, we define $\tau_h(\omega):=\omega+h$, $\omega\in X$, and let

\begin{align*}
K_0:=\left\{h\in K|\sup_{0\leq t\leq 1}|h''(t)|<\infty\right\}.
\end{align*}

Let $h\in K_0$. Then, $\nu_\lambda$ is $k$-quasi-invariant, i.e., $\nu_\lambda\circ \tau_{sh}^{-1}$ is absolutely continuous with respect  to $\nu_\lambda$ for all $s\in\R$. If $a_{sh}:=\dis \frac{\dd (\nu_{\lambda}\circ \tau_{sh}^{-1})}{\dd \nu_{\lambda}}$, $s\in \R$, then the process $(a_{sh})_{s\in\R}$ has a version which has $\nu_\lambda$-almost every continuous sample paths.

The proof of Theorem <ref> is given in Section <ref> below after the necessary preparations in Section <ref>-<ref>.

Once we prove Theorem <ref>, Theorem <ref> and Theorem <ref> follow from the theory of the Dirichlet forms as in <cit.>.

(i) By Theorem <ref>, $\nu_\lambda$ is $k$-quasi-invariant for each $k\in K_0$. Since  $K_0$  is a linear space and dense in $H$, hence also in $X$, it follows by  <cit.> that $\mathrm{supp}\nu_\lambda=X$.

(ii) Since $K_0$ is dense in $H$, we can find an orthogonal basis $\{h_n:n\in \N\}$ of $H$ in $K_0$. Then, we can find
\begin{align*}
\sum_{n\geq 1}{}_{E'}\langle l,h_n\rangle_{E}^2<\infty\quad \text{for all }l\in E'.
\end{align*}

Suppose that for each $n\in \N$, $(\mathcal{E}_{h_n},\mathcal{F}C_b^\infty)$ on $L^2(X;\nu_\lambda)$ is closable, where
\begin{align*}
\mathcal{E}_{h_n}(u,v)=\int \frac{\partial u}{\partial h_n}\frac{\partial v}{\partial h_n }\dd \nu_\lambda,\quad u,v\in \mathcal{F}C_b^\infty.
\end{align*}

Then, we can find from <cit.> that
for $u,v\in \mathcal{F}C_b^\infty$,

\begin{align}
\mathcal{E}_{\nu_\lambda}(u,v)=\sum_{n=1}^\infty \int \frac{\partial u}{\partial h_n}\frac{\partial v}{\partial h_n }\dd \nu_\lambda,			\label{eq:Dirichletnug}
\end{align}

and from <cit.> or <cit.> that $(\mathcal{E}_{\nu_\lambda},\mathcal{F}C_b^\infty)$ is closable on $L^2(\nu_\lambda)$ and its closure $(\mathcal{E}_{\nu_\lambda},D(\mathcal{E}_{\nu_\lambda}))$  is a symmetric Dirichlet form.
Thus, it suffices to prove the closability of each $(\mathcal{E}_{h_n},\mathcal{F}C_b^\infty)$ on $L^2(X;\nu_\lambda)$.

Since $s\mapsto a_{sh_n}$ is continuous  $\nu_\lambda$-almost everywhere by Theorem <ref> for each $n\in \N$, $h_n$ is admissible <cit.> in the sense <cit.>.    Then, the closability follows from <cit.>.

The proof of Theorem <ref> then follows by the standard machinery of Dirichlet forms on infinite-dimensional state spaces and is exactly the same as that of <cit.>.

Since we have verified the closability of $(\mathcal{E}_{\nu_\lambda},\mathcal{F}C_b^\infty)$ on $L^2(X;\nu_\lambda)$, it follows from <cit.>   that  $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$ is quasi-regular. Then, the existence of a $\nu_{\lambda}$-symmetric  $\mathbb{M}$ is a consequence of <cit.>.

Also, by <cit.>, the quasi-regular Dirichlet form  $(\cE_{\nu_\lambda},D(\cE_{\nu_\lambda}))$ possesses the local property, and hence  sample paths of $\mathbb{M}$ are continuous by <cit.>. The conservativity of $\mathbb{M}$ is obvious since $1\in D(L)$
and $L1= 0$, hence $T_t1=1$ for all $ t\geq  0$. The fact that $\nu_\lambda$ is an invariant measure for $\mathbb{M}$ then follows immediately.

Thus, the rest of this paper will be devoted to proving Theorem <ref>.

In contrast to the two-dimensional case we have not yet succeeded in proving the irreducibility (which is equivalent to ergodicity) of $(\mathcal{E}_{\nu_\lambda},D(\mathcal{E}_{\nu_\lambda}))$ in Theorem <ref>.

§.§ Some observations and organization of the paper

This subsection explains the idea of the proof of Theorem <ref> and the organization of the present paper. As mentioned above, we are able to construct

\begin{align*}
\nu_\lambda(\dd \omega)=E[e^{-\lambda J_{0,1}^0}]^{-1}\exp\left(-\lambda {J_{0,1}^0}\right)\nu(\dd \omega)
\end{align*}

\begin{align*}
J_{0,1}^0=\int_{0}^1\int_s^1 \delta_0(\omega_t-\omega_s)\dd t\dd s
\end{align*}

by approximations.

By using this formal expression we would see from Girsanov's transformation that

\begin{align}
\frac{\dd (\nu_{\lambda}\circ \tau_{uh}^{-1})}{\dd\nu_\lambda}=\exp\left(-\lambda \left(J_{0,1}^{0}(\omega-uh)-J_{0,1}^0(\omega)\right)+u\int_0^1 h_s'\dd \omega_s-\frac{u^2}{2}\int_0^1 \left(h_s'\right)^2\dd s\right)\label{eq:shiftder}
\end{align}

which should corresponds with $a_{uk}$ in Theorem <ref>. Of course, the discussion above is formal and not mathematically rigorous.
First, it is not clear whether the stochastic integral $\int_0^1 h_s'\dd \omega_s$ is well-defined or not in Itô's sense, because we do not know whether  $\omega$ is a semimartingale under $\nu_\lambda$ or not.
However, we can avoid such a problem by interpreting the stochastic integral $\int_0^1 h_s'\dd \omega_s$ as

\begin{align}
\int_0^1 h_s'\dd \omega_s=h_1'\omega_1-\int_0^1 \omega_s \dd h'_s,\label{eq:stointrepre}
\end{align}

for $h\in K_0 $,
where the right-hand side of (<ref>) is well-defined for any continuous function $\omega$. Next, we should make sense of the difference $J_{0,1}^{0}(\omega-uh)-J_{0,1}^0(\omega)$ under $\nu_\lambda$.
In view of the heuristic observation above we provide the proofs of the main theorems step by step in the following sections.

In Section <ref>, we will give an estimate of
\begin{align*}
\end{align*}

for $\ve_0\in (0,1)$ small.
As a corollary, we will find that
\begin{align}
\left\{J_{0,1}^{2^{-\ve_0n},2^{-n}}(\omega-uh)-J_{0,1}^{2^{-\ve_0n},2^{-n}}(\omega)\right\}_{n\geq 1} \label{eq:JomegauhJomega}
\end{align}

is an $L^4(\nu_0)$-Cauchy sequence.
This is the most technical and hard part in the proofs of the
main theorems. For the proof we give an extension of Rosen's method, which is an adjusted
version to the three-dimensional case. Also in Section <ref> we provide some lemmas in Sections
<ref> and <ref>, which guarantee the treatment of Dirac functions and the
differentiability of a key quantity in the parameter of approximation, respectively.

In Section <ref>, we will give an estimate of roughly speaking, the “difference" between $\mu_{\e,\lambda}(f)$ and $\nu_\lambda(f)$, which will play a key role in controlling $\nu_\lambda(f)$.

In Section <ref>, by combining the results  in Sections <ref> and <ref>, we will prove that (<ref>) is also an $L^p(\nu_\lambda)$-Cauchy sequence for small $\lambda>0$ for each $1\leq  p<4$. We denote by $\rho(-u,k)$ the $L^p(\nu_\lambda)$ limit, which corresponds to the above quantity $J_{0,1}^{0}(\omega-uh)-J_{0,1}^0(\omega)$.

Finally, in Section <ref> we provide the proof of Theorem <ref>, which yields the main theorems as mentioned in Section <ref>.

We remark that some explicit calculations are put in Appendix (Section <ref>).

§ SOME ESTIMATES OF SELF-INTERSECTION LOCAL TIMES IN WIENER SPACE

In this section, we prepare some estimates which are needed for later arguments.
We remark that the proofs in this section are very technical.
We set

\begin{align*}
f_a(x)=(2\pi)^{-3}\exp\left(-\frac{a|x|^2}{2}\right),\qquad \text{for }a>0, x=(x_1,x_2,x_3)\in \R^3.
\end{align*}

Then, the following holds in the weak sense
\begin{align*}
\int_{\R^3} e^{i\langle x,y\rangle}f_a(y)\dd y=\frac{1}{(2\pi a)^\frac{3}{2}}e^{-\frac{|x|^2}{2a}}\to \delta_0(x),\quad a\searrow 0,
\end{align*}

where $\delta_0$ is the Dirac function at point $0\in\R^3$ , and $\lan x,y\ran=x_1y_1+x_2y_2+x_3y_3$ for $x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$ and we denote by $|x|=\sqrt{\lan x,x\ran}$ the Euclidean norm.
We remark that

\begin{align*}
J_{0,1}^{\e,a}(\omega)=\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s\ran}f_a(y)\dd y\dd s\dd t = \int _{T_\e} \frac{1}{(2\pi a )^{\frac 32}} e^{-\frac{|\omega _t -\omega _s|^2}{2a}} \dd s \dd t,
\end{align*}

where $T_\e:=\{(s,t)\in [0,1]^2:t-s\geq \e\}$.
The following lemma has been given already in <cit.>, but we will give in Section <ref> below its proof for self-containedness.

Let $u\in \R$, $h\in K$, and $p\geq 1$ be given. Then, for $\gamma\in \left(0,\frac{1}{4}\right)$, there exists $C$ such that
\begin{align*}
E\left[\left|J_{0,1}^{\e,a}(\cdot+uh)-J_{0,1}^{\e,a'}(\cdot+uh)\right|^p\right]\leq C|a-a'|^{p}a^{-\frac{5}{6}p}+C|a-a'|^{\gamma p} a^{\frac{1-4\gamma}{24}p}\e^{-\frac{3}{4}p}
\end{align*}

for $0<a'<a<1$ and for $\e \in (0,1)$.

As a corollary of Lemma <ref>, we obtain the following.

Let $u\in \R$, $h\in K$, and $p\geq 1$ be given. Then, $\{J_{0,1}^{\e,a}(\omega+uh)\}_{a\in (0,1)}$ has a continuous version (we also denote it by $J_{0,1}^{\e,a}(\omega+uh)$) and has the $L^p(\nu_0)$-limit $J_{0,1}^{\e}(\omega+uh)=\lim_{a\searrow 0}J_{0,1}^{\e,a}(\omega+uh)$.

Moreover, we have

\begin{align}
E\left[\left|J_{0,1}^{\e,a}(\cdot+uh)-J_{0,1}^{\e}(\cdot+uh)\right|^p\right]\leq Ca^{\frac{p}{6}}+C a^{\frac{1+20\gamma}{24}p}\e^{-\frac{3}{4}p}\label{eq:JaJbdd}
\end{align}

for $\gamma \in \left(0,\frac{1}{4}\right)$.

Fix $\gamma\in (0,\frac{1}{4})$.
Then, we can apply the Kolmogorov continuity theorem to $\left\{J_{0,1}^{\e,a}(\omega+uh)\right\}_{a\in (0,1)}$ in view of Lemma <ref> for $p\geq 1$ with $p\gamma>1$. Also, we can see that $ \left\{J_{0,1}^{\e,a}(\omega+uh)\right\}_{a\in (0,1)}$ is an $L^p(\nu_0)$-Cauchy sequence as $a\searrow 0$ which guarantees the existence of the $L^p(\nu_0)$-limit.
Letting $a'\searrow 0$, (<ref>) follows from Fatou's lemma.

We define

\begin{align*}
\end{align*}

\begin{align*}
\widehat{J}_{\e,a}(u_1,u_2,h):=\widetilde{J}_{\e,a}(u_1,h)-\widetilde{J}_{\e,a}(u_2,h),
\end{align*}

for $a,\e\in (0,1)$,  $u,u_1,u_2\in \R$, and  $h\in K$.

For $\ve_0\in \left(0,\frac{1}{21}\right)$, we define

\begin{align}
\e_n&=2^{-\ve_0n} \label{eq:en}
\intertext{and}
a_n&=2^{-n} \label{eq:an}
\end{align}

for $n\geq 1$.

For $\gamma \in (\frac{1}{3},\frac{1}{2})$ and $\ve_0 \in \left(0, \frac{1-2\gamma}{21}\right)$, there exists $\beta>0$ such that the following holds. For any $h\in K$ and any interval $[-M,M]$ with $M>0$, there exists $C>0$ such that

\begin{align*}
&E\left[\left|\widehat{J}_{\e_n,a_n}(u_1,u_2,h)-\widehat{J}_{\e_m,a_m}(u_1,u_2,h)\right|^4\right]\leq C|u_1-u_2|^{4\gamma }2^{-\beta m}
\end{align*}

for all $u_1,u_2\in [-M,M]$, $n\geq m\geq 1$.

We will prove this lemma in Section <ref> below.
Taking $u_2=0$ and combining with Corollary <ref>, we have the following.

$\left\{\widehat{J}_{\e_n,a_n}(u_1,u_2,h)\right\}_{n\geq 1}$, $\left\{ \widetilde{J}_{\e_n,a_n}(u,h)\right\}_{n\geq 1}$ and $\Big\{{J}_{0,1}^{\e_n}(\omega+uh)-J_{0,1}^{\e_n}(\omega)\Big\}_{n\geq 1}$ are $L^4(\nu_0)$-Cauchy sequences for each $u_1,u_2,u\in\R$, and for any $h\in K$.
In particular, $\left\{\widetilde{J}_{\e_n,a_n}(u,h) \right\}_{n\geq 1}$ is bounded in $L^4(\nu_0)$.

In view of Lemma <ref> $\left\{\widehat{J}_{\e_n,a_n}(u_1,u_2,h)\right\}_{n\geq 1}$ is an $L^4(\nu_0)$-Cauchy sequence for each $u_1,u_2\in \R$.

\begin{align*}
\widehat{J}_{\e,a}(u,0,h) =\widetilde{J}_{\e,a}(u,h),
\end{align*}

$\left\{ \widetilde{J}_{\e_n,a_n}(u,h)\right\}_{n\geq 1}$ is also $L^4(\nu_0)$-Cauchy sequence.
Furthermore, from this fact and Corollary <ref>, we also have the assertion for $\Big\{{J}_{0,1}^{\e_n}(\omega+uh)-J_{0,1}^{\e_n}(\omega)\Big\}_{n\geq 1}$.

In <cit.>, a similar statement to Lemma <ref> for another approximation of $J_{0,1}^\e$ and $\eta_{n,\e}$, is discussed.  However, we can find serious errors in its proof.
Indeed, they missed the problem in (Step 4) in the proof of Lemma <ref> in the present paper, which is applied in the proof of Lemma <ref>.
Moreover, they did not give any proof of an analogous result to Proposition <ref> in the present paper. Actually, their statement says that a similar bound to Lemma <ref> holds for any $p\geq 1$, but  we can show it only for $1\leq p\leq  4$. Also, we remark that if one can prove a similar bound for any $p\geq 1$, then the whole argument after Section <ref> in the present paper does work, and hence the stochastic quantization of $\nu_\lambda$ will be achieved for any $\lambda>0$ (i.e. without the restriction we make in this paper that $\lambda$ be small).

§.§ J. Rosen's method

To prove Lemmas <ref> and <ref>, we will modify the technique in <cit.>.
For later use, we give some equations, notations, and some lemmas.

We set $T\subset [0,1]^2\cap \{0\leq s<t\leq 1\}$ and $p\in \mathbb{N}$.  Then, we find that for $g\in L^1(\R^3)$, $h\in K$, and $u\in \R$

\begin{align}
&E\left[\left(\int_{T}\dd s\dd t\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}g(y)\right)^p\right]\notag\\
&=\int_{T^p}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^pe^{iu\lan y_j,h_{\mathbf{s},\mathbf{t}}^{(j)}\ran}g(y_j)\right)e^{-\frac{1}{2}\var{\sum_{j=1}^p \left\lan y_j,\omega_{t_j}-\omega_{s_j}\right\ran }}\label{eq:lpg}
\intertext{and}
&E\left[\left(\int_{T}\dd s\dd t\int_{\R^3}\left(e^{i\lan y,\omega_t+u_1h_t-\omega_s-u_1h_s\ran}-e^{i\lan y,\omega_t+u_2h_t-\omega_s-u_2h_s\ran}\right)g(y)\right)^p\right]\notag\\
&=\int_{T^p}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)g(y_j)\right)%\notag\\
e^{-\frac{1}{2}\var{\sum_{j=1}^p \left\lan y_j,\omega_{t_j}-\omega_{s_j}\right\ran }}\label{eq:lpetan}
\end{align}

where $(\mathbf{s},\mathbf{t})$ denotes an element in $[0,1]^{2p}$ with
$\mathbf{s}=(s_1,\cdots,s_p)$, $\mathbf{t}=(t_1,\cdots,t_p)\in[0,1]^p$ such that $0\leq s_i<t_i\leq 1$ for each $i=1,\cdots,p$, $\mathbf{y}=(y_1,\cdots,y_p)\in \R^{3p}$ with $y_1,\cdots,y_p\in \R^3$, and we define $h_{\mathbf{s},\mathbf{t}}^{(j)}:=h_{t_j}-h_{s_j}$ for $(\mathbf{s},\mathbf{t})$ and $j=1,\dots,p$.

Since $g\in L^1(\R^3)$, we can use Fubini's theorem so that

\begin{align*}
&E\left[\left(\int_{T}\dd s\dd t\int_{\R^3}\dd ye^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}g(y)\right)^p\right]\\
&=\int_{T^p}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^pe^{\lan y_j,u(h_{t_j}-h_{s_j}) \ran }g(y_j)\right)E\left[\prod_{j=1}^p e^{i\lan y_j,\omega_{t_j}-\omega_{s_j}\ran}\right]\notag
\end{align*}

and that

\begin{align*}
&E\left[\left(\int_{T}\dd s\dd t\int_{\R^3}\dd y\left(e^{i\lan y,\omega_t+u_1h_t-\omega_s-u_1h_s\ran}-e^{i\lan y,\omega_t+u_2h_t-\omega_s-u_2h_s\ran}\right)g(y)\right)^p\right]\notag\\
&=\int_{T^p}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1\left(h_{t_j}-h_{s_j}\right)\right\ran}-e^{i\left\lan y_j,u_2\left(h_{t_j}-h_{s_j}\right)\right\ran}\right)g(y_j)\right)E\left[\prod_{j=1}^p e^{i\lan y_j,\omega_{t_j}-\omega_{s_j}\ran}\right].\notag
\end{align*}

The statement holds from the fact $\lan y_j,\omega_{t_j}-\omega_{s_j}\ran$ is a Gaussian random variable for any $\mathbf{y}$ and $\mathbf{s},\mathbf{t}$.

Next, we focus on $ \textrm{Var}\left(\sum_{j=1}^p \lan y_j,\omega_{t_j}-\omega_{s_j}\ran\right)$ for $p\geq 1$ and $\mathbf{s},\mathbf{t}$.
As in the proof of <cit.>, we denote the pair $(s_j,t_j)$ by $S^j$ and set $(z_1,\cdots,z_{2p}):=(s_1,t_1,s_2,t_2,\cdots,s_p,t_p)$.  Then, we split the domain of the integral in $\mathbf{s},\mathbf{t}$ on the right-hand side of (<ref>) into $T^p\cap \Delta_p(\pi)$ for some permutation $\pi$ of $\{1,\cdots,2p\}$, where

\begin{align}
\Delta_p(\pi):=\{(z_1,\cdots,z_{2p}):z_{\pi(1)}<\cdots<z_{\pi(2p)}\},\label{eq:labelingpermu}
\end{align}

Let us define a partition $\{R_i\}_{i=1}^{2p-1}$ of the interval $[z_{\pi(1)},z_{\pi(2p)}]$ by

\begin{align}
R_i:=[z_{\pi(i)},z_{\pi(i+1)}],\quad 1\leq i\leq 2p-1,\label{eq:Ri}
\end{align}

and let $W(R_i):=\omega_{z_{\pi(i+1)}}-\omega_{z_{\pi (i)}}$. Then it holds that

\begin{align*}
\omega_{t_j}-\omega_{s_j}=\sum_{i; R_i\subset [s_j,t_j]}W(R_i).
\end{align*}

Since $W(R_i)$, $i=1,2,\cdots,2p-1$, are independent, we have
\begin{align}
\textrm{Var}\left(\sum_{j=1}^p \lan y_j,\omega_{t_j}-\omega_{s_j}\ran\right)&=\textrm{Var}\left(\sum_{j=1}^p\left\lan	y_j,\sum_{R_i\subset [s_j,t_j]}W(R_i)	\right\ran\right)\notag\\
\end{align}

where $\overline{y}^i=\dis \sum_{j;[s_j,t_j]\supset R_i}y_j$ and $|R_j|$ denotes the length of $R_j$.
For convenience, we let $R_0:= [0,z_{\pi (1)}]$, $R_{2p}:= [z_{\pi(2p)}, 1]$, $\overline{y}^0:=0$ and $\overline{y}^{2p}:=0$.

We define

\begin{align}\label{eq:frdef}
f(\ell):= \min \left\{ i| \ R_i \subset [s_\ell ,t_\ell] \right\} , \quad r(\ell):= \max \left\{ i| \ R_i \subset [s_\ell ,t_\ell] \right\} +1
\end{align}

Then, it holds that

\begin{align*}
[s_\ell,t_\ell]=R_{f(\ell)}\cup R_{f(\ell)+1}\cup \cdots\cup R_{r(\ell)-1},
\end{align*}

and hence

\begin{align}
y_\ell=\overline{y}^{f(\ell)}-\overline{y}^{f(\ell)-1}, \quad -y_\ell=\overline{y}^{r(\ell)}-\overline{y}^{r(\ell)-1}.\label{eq:ybary}
\end{align}

We have the following three cases:

*  $\dis \bigcup_{j\in \{1,\dots,p\}}(s_j,t_j)$ is connected.

*  $\dis \bigcup_{j\in \{1,\dots,p\}}(s_j,t_j)$ is disconnected and there exists  $j\in \{1,\dots,p\}$ such that $(s_j,t_j)\cap \bigcup_{i\not= j}[s_i,t_i]=\emptyset$.

*  $\dis \bigcup_{j\in \{1,\dots,p\}}(s_j,t_j)$ is disconnected and for any $j\in \{1,\dots,p\}$, $(s_j,t_j)\cap \bigcup_{i\not= j}[s_i,t_i]\not=\emptyset$.

We can find that for each case, $\left\{\overline{y}^j\right\}_{j=0}^{2p}$ has the following property.

* In the case of <ref>, there exists $j,k\in \{1,\dots,p\}$ such that $r(j)-1=r(k)$ and $\overline{y}^{r(k)}=y_j$. This is true for the largest $t_j$ and the second-largest $t_k$. We denote by $j_{\max}$ and $k_{2-\max}$ the $j$ and $k$ such that $t_j=\max\{t_\ell\}$ and $t_k=\max\{t_j:j\in\{1,\dots,p\}\backslash \{j_{\max}\}\}$.

* In the case of <ref>, there exists $j\in \{1,\dots,p\}$ such that

* $\overline{y}^{f(j)-1}=\overline{y}^{r(j)}=0$.

* $f(j)=r(j)-1$ and  $\overline{y}^{f(j)}=y_j$.

* $\overline{y}^k$ ($k\not= f(j)$) does not contain $y_j$.

Then, we have the following properties, which are obtained in [106].

For $p\in \N$, $S_1,\dots,S_p\in \{(s,t)\in [0,1]^2:t-s>0\}$, $\mathbf{y}\in \R^{3p}$, we set $\{f(\ell)\}_{\ell=1,\dots,p}$, $\{r(\ell)\}_{\ell=1,\dots,p}$, $\{\overline{y}^j\}_{j=0}^{2p}$ as above. Then,
$\overline{y}^{f(1)},\dots,\overline{y}^{f(p)}$ is a nonsingular set of coordinates for $\R^{3p}$.

Let $a\in (0,1)$. For $p\in \N$ with $p\geq 2$, $S_1,\dots,S_p\in \{(s,t)\in [0,1]^2:t-s\geq a\}$, $\mathbf{y}\in \R^{3p}$, we set $\{f(\ell)\}_{\ell=1,\dots,p}$, $\{r(\ell)\}_{\ell=1,\dots,p}$, $\{\overline{y}^j\}_{j=0}^{2p}$ as above. Then, for each $j$, $1\leq j\leq p$, there exists $k\in \{f(j),\dots,r(j)-1\}$ such that $|R_{k}|\geq \frac{a}{2p}$. We denote by $d(j)\in \{f(j),\cdots,r(j)-1\}$ the smallest integer in such $k$.  Also, we can find that

*  $\displaystyle \mathrm{rank}\left(\{\overline{y}^{r(1)},\dots,\overline{y}^{r(p)},\overline{y}^{d(1)},\dots,\overline{y}^{d(p)}\}\right)= 3p$.

*  Moreover, if $S_1,\dots,S_p$ satisfies <ref>, then
\begin{align*}
&\mathrm{rank}\left(\{\overline{y}^{r(1)},\dots,\overline{y}^{r(p)}\}\right)\geq 3.
\end{align*}

Lemma <ref><ref> was given in <cit.> and we can also find the proof of Lemma <ref><ref> there. Also, we remark that there might be the possibility that $R_{d(i)}=R_{d(j)}$ for $i\not=j$.

Changing variables $\widetilde{t}_j=1-s_j$, $\widetilde{s}_j=1-t_j$ in (<ref>), it is rewritten as
\begin{align*}
\int_{T^p}\dd \widetilde{\mathbf{s}}\dd \widetilde{\mathbf{t}}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\widetilde{\mathbf{s}},\widetilde{\mathbf{t}}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\widetilde{\mathbf{s}},\widetilde{\mathbf{t}}}^{(j)}\right\ran}\right)g(y_j)\right)
e^{-\frac{1}{2}\var{\sum_{j=1}^p \left\lan y_j,\omega_{1-\widetilde{s}_j}-\omega_{1-\widetilde{t}_j}\right\ran }}.
\end{align*}

Then, we can introduce  $\{\widetilde{R}_j\}_{j=1}^{2p-1}$, $\left\{\widetilde{\overline{y}}^j\right\}_{j=0}^{2p}$, and $\{\widetilde{f}(\ell), \widetilde{r}(\ell),\widetilde{d}(\ell)\}_{\ell=1}^p$ in a similar manner to the above.

We can see that $\widetilde{\overline{y}}^{\widetilde{f}(j)}=\overline{y}^{r(j)-1}$ and $\widetilde{\overline{y}}^{\widetilde{r}(j)-1}=\overline{y}^{f(j)}$ for each $j=1,\dots,p$.

Throughout this section, we use these notations as long as they are not confusing.

§.§ Proof of Lemma <ref>

Now we give the proof of Lemma <ref>.
We will split the proof into four steps.

(Step 1)
It is easy to see that
\begin{align*}
&= \left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)\dd y\dd s\dd t\right|\\
&\leq \left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)1\{|y|\leq a^{-\frac{1}{6}}\}\dd y\dd s\dd t\right|\\
&\quad +\left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)1\{|y|>a^{-\frac{1}{6}}\}\dd y\dd s\dd t\right|.
\end{align*}

Since $1-e^{-x}\leq x$ for $x\geq 0$, we have

\begin{align*}
& \left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)1\{|y|\leq a^{-\frac{1}{6}}\}\dd y\dd s\dd t\right|
\leq c(a-a')a^{-\frac{5}{6}}
\end{align*}

for some constant $c>0$. Hence,  it is sufficient to estimate
\begin{align*}
\left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)1\{|y|>a^{-\frac{1}{6}}\}\dd y\dd s\dd t\right|
\end{align*}

for $p\in 2\N$.

(Step 2)
We can use Proposition <ref> by setting $g(y)=f_{a'}(y)1\{|y|>a^{-\frac{1}{6}}\}$ and $T=T_{\e}$, and changing $y_{\frac{p}{2}+1},\dots, y_p\mapsto -y_{\frac{p}{2}+1},\dots,-y_p$. For our convenience, we keep the notation $\{y_j\}_{j=1,\dots,p}$.

We remark that  for $\gamma \in (0,1)$, there exists $C>0$ such that
\begin{align*}
1-e^{-x}\leq Cx^\gamma
\end{align*}

for $x\geq 0$.
Combining this with (<ref>), (<ref>), and (<ref>), we have
\begin{align*}
&E\left[\left|\int_{T_\e}\int_{\R^3}e^{i\lan y,\omega_t-\omega_s+u(h_t-h_s)\ran}\left(1-e^{-\frac{(a-a')|y|^2}{2}}\right)f_{a'}(y)1\{|y|>a^{-\frac{1}{6}}\}\dd y\dd s\dd t\right|^p\right]\\
&\leq |a-a'|^{\gamma p}\int_{T^p_\e}\dd \mathbf{s}\dd \mathbf{t}\int_{Y(a^{-\frac{1}{6}})}\dd \mathbf{y}\prod_{j=1}^p |y_j|^{2\gamma}\exp\left(-\frac{1}{2}\var{\sum_{j=1}^p \left\lan y_j,\omega_{t_j}-\omega_{s_j}\right\ran }\right)\\
&= |a-a'|^{\gamma p}\sum_{\pi}\int_{T_{\e}^p\cap \Delta_p(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{Y(a^{-\frac{1}{6}})}\dd \mathbf{y}\prod_{j=1}^p |y_j|^{2\gamma}e^{-\frac{1}{2}\sum_{i=1}^{2p-1}|R_i|\left|\overline{y}^i\right|^2},
\end{align*}

where we write  $Y(t)=\bigcap_{j=1}^p\{|y_j|\geq t\}$ for $t\geq 0$.

(Step 3)
Using (<ref>), we get
\begin{align*}
|y_\ell|\leq \left(|\overline{y}^{f(\ell)}|+|\overline{y}^{f(\ell)-1}|\right)^\frac{1}{2}\left(|\overline{y}^{r(\ell)}|+|\overline{y}^{r(\ell)-1}|\right)^\frac{1}{2}.
\end{align*}

Since each  $\overline{y}^i$ is equal to either  $\overline{y}^{f(\ell)}$ or  $\overline{y}^{r(\ell)}$ for a unique $\ell$, and either  $\overline{y}^{f(q)-1}$ or  $\overline{y}^{r(q)-1}$ for a unique $q$, we have
\begin{align*}
\prod_{i=1}^p |y_i|&\leq \prod_{\ell=1}^p \left(|\overline{y}^{f(\ell)}|+|\overline{y}^{f(\ell)-1}|\right)^\frac{1}{2}\left(|\overline{y}^{r(\ell)}|+|\overline{y}^{r(\ell)-1}|\right)^\frac{1}{2}\\
&\leq \prod_{i=1}^{2p-1} (1+|\overline{y}^i|^2)^\frac{1}{2},
\end{align*}

by using (<ref>) and $(a+b)^{\frac{1}{2}}\leq (1+a^2)^\frac{1}{4}(1+b^2)^\frac{1}{4}$.

We now change the variable: $J:(S^1,\cdots,S^p)\mapsto (z_{\pi(1)},|R_1|,\cdots,|R_{2p-1}|)$.
Then, the  Jacobian matrix of $J$ has $\{0,1,-1\}$-entries and is invertible almost everywhere. Hence, we can find  $1\leq |DJ(\mathbf{s},\mathbf{t})|\leq 2^{2p}$.

These properties yield that $|DJ(\mathbf{s},\mathbf{t})|=1$.

We may divide $T_{\e}^p\cap \Delta_p(\pi)$  by the indices $\mathbf{d}=(d(1),\cdots,d(p))$, $T_{\e}^p(\pi,\mathbf{d})$. We remark that $J$ can be written as  the invertible linear transformation of $(S^1,\cdots,S^p)$ on $T_{\e}^p(\pi,\mathbf{d})$. Then, we have
\begin{align}
&\int_{T_{\e}^p(\pi,\mathbf{d})}\dd \mathbf{s}\dd \mathbf{t}\int _{Y(a^{-\frac{1}{6}})}\dd \mathbf{y}\exp\left(-\frac{1}{2}\sum_{i=1}^{2p-1}|\overline{y}^i|^2|R_i|\right)\prod_{j=1}^{p}|y_j|^{2\gamma} \notag\\
&\leq C\int_{J(T_{\e}^p(\pi,\mathbf{d}))}\dd z_{\pi(1)}\prod_{i=1}^{2p-1}\dd |R_{i}|\int _{Y(a^{-\frac{1}{6}})}\dd \mathbf{y}\exp\left(-\frac{1}{2}\sum_{i=1}^{2p-1}|\overline{y}^i|^2|R_i|\right)\prod_{j=1}^{2p-1}\left(1+|\overline{y}^j|^2\right)^\gamma \label{eq:bdd1}
\end{align}

The integral in  $\dd |R_{i}|$ is dominated as follows: If $i=d(k)$ for some $k\in \{1,\dots,p\}$, then there is a constant $C>0$ such that
\begin{align*}
\int_{\frac{\e}{2p}}^1 \exp\left({-\frac{|\overline{y}^{d(i)}|^2t}{2}}\right)\dd t\leq \frac{C}{1+|\overline{y}^{d(i)}|^2}\exp\left({-\frac{\e|\overline{y}^{d(i)}|^2}{4p}}\right) \leq C \exp\left({-\frac{\e|\overline{y}^{d(i)}|^2}{4p}}\right).
\end{align*}

If $i\not=d(k)$ for any $k\in \{1,\dots,p\}$, then we have
\begin{align*}
\int_0^1 \exp\left({-\frac{|y|^2 t}{2}}\right)\dd t\leq \frac{C}{1+|y|^2},\quad y\in \R^3.
\end{align*}

From these, we can see that the right-hand side of (<ref>) can be bounded from above by
\begin{align}
C\int_{Y(a^{-\frac{1}{6}})}\dd \mathbf{y}\prod_{i=1}^{p}\left(1+|\overline{y}^{f(i)}|^2\right)^{-1+\gamma}\left(1+|\overline{y}^{r(i)}|^2\right)^{-1+\gamma}e^{-\frac{\e|\overline{y}^{d(i)}|^2}{4p^2}},\label{eq:bdd2}
\end{align}

where $p^2$ of the denominator in  the power of $e$ is put to remove the effect of overlaps of $d(i)$.

(Step 4)
Now, we remark that on $Y(a^{-\frac{1}{6}})$ there are $1\leq i_1<\dots<i_p\leq 2p$ such that $|\overline{y}^{i_j}|\geq \frac{a^{-\frac{1}{6}}}{2}$ for $j=1,\dots,p$. Indeed, for each $i=1,\dots,2p-1$, there exists $k(i)\in \{1,\dots,p\}$ such that $y_{k(i)}=\overline{y}^{i+1}-\overline{y}^i$ or $-\overline{y}^{i+1}+\overline{y}^i$, and hence either $|\overline{y}^{2i-1}|\geq \frac{a^{-\frac{1}{6}}}{2}$ or $|\overline{y}^{2i}|\geq \frac{a^{-\frac{1}{6}}}{2}$ on $Y(a^{-\frac{1}{6}})$.

\begin{align*}
A_{(\pi,\mathbf{d})} &:=\left\{ (y_1,\dots,y_p)\in \R^{3p}: \textrm{There exist $1\leq j_1<\dots<j_\frac{p}{2}\leq p$} \phantom{\frac{a^{-\frac{1}{6}}}{2}}\right.\\
&\quad \hspace{4cm} \left. \textrm{such that $\left|\overline{y}^{f(j_k)}\right|\geq \frac{a^{-\frac{1}{6}}}{2}$, $k=1,\dots,\frac{p}{2}$}\right\}.
\end{align*}

By the Cauchy-Schwarz inequality, the right-hand side of (<ref>) is bounded from above by
\begin{align*}
&C\left(\int_{A_{(\pi,\mathbf{d})}} \dd \mathbf{y}\prod_{i=1}^p \left(1+|\overline{y}^{f(i)}|^2\right)^{-2+2\gamma}\right)^\frac{1}{2}\left(\int_{A_{(\pi,\mathbf{d})}} \dd \mathbf{y}\prod_{i=1}^p \left(1+|\overline{y}^{r(i)}|^2\right)^{-2+2\gamma}e^{-\frac{\e|\overline{y}^{d(i)}|^2}{2p^2}}\right)^\frac{1}{2}\\
&+C\left(\int_{A^c_{(\pi,\mathbf{d})}} \dd \mathbf{y}\prod_{i=1}^p \left(1+|\overline{y}^{f(i)}|^2\right)^{-2+2\gamma}\right)^\frac{1}{2} \\
&\quad \hspace{3cm} \times \left(\int_{A^c_{(\pi,\mathbf{d})}\cap Y(a^{-\frac{1}{6}})} \dd \mathbf{y}\prod_{i=1}^p \left(1+|\overline{y}^{r(i)}|^2\right)^{-2+2\gamma}e^{-\frac{\e|\overline{y}^{d(i)}|^2}{2p^2}}\right)^\frac{1}{2}\\
\end{align*}

Lemmas <ref> and <ref> allow us to see that $\{\overline{y}^{f(1)},\cdots,{\overline{y}^{f(p)}}\}$ is a (nonsingular) set of coordinates for $\R^{3p}$ and that we can choose a (nonsingular) set of coordinates (denoted by $(\overline{y}^{k_1},\cdots,\overline{y}^{k_p})$) for $\R^{3p}$ from the set

\begin{align*}
\overline{y}^{r(1)},\cdots,\overline{y}^{r(p)},\overline{y}^{d(1)},\cdots,\overline{y}^{d(p)}.
\end{align*}

Also, it is easy to see that each element in Jacobian matrices is a constant.
Therefore, we have for $\gamma\in (0,\frac{1}{4})$

\begin{align*}
F_{(\pi,\mathbf{d},1)}\leq \left(\int_{\R^3}(1+|y|^2)^{-2+2\gamma} dy\right)^{\frac{p}{2}}\left(\int_{|y|\geq \frac{a^{-\frac{1}{6}}}{2}} (1+|y|^2)^{-2+2\gamma} dy \right)^{\frac{p}{2}}\leq Ca^{\frac{(1-4\gamma)p}{12}}.
\end{align*}

Also, it is easy to see that

\begin{align*}
R_{(\pi,\mathbf{d},1)}\leq C\e^{-\frac{3}{2}p} , \quad F_{(\pi,\mathbf{d},2)}\leq C.
\end{align*}

Hence, the proof of Lemma <ref> is completed, once Proposition <ref> below is proved.

We have
\begin{align*}
R_{(\pi,\mathbf{d},2)}\leq Ca^{\frac{1-4\gamma}{12}p}\e^{-\frac{3}{2}p}.
\end{align*}

First, we remark that
\begin{align*}
A_{(\pi,\mathbf{d})}^c\cap Y(a^{-\frac{1}{6}})
&\subset \bigcup_{1\leq i_1<\dots<i_\frac{p}{2}\leq p}\left\{(y_1,\dots,y_p)\in \R^{3p}:\left| \overline{y}^{r(i_k)}\right|\geq \frac{a^{-\frac{1}{6}}}{2}, k=1, \dots \frac{p}{2} \right\} \\
&=: \bigcup_{\mathbf{i}}B_{\mathbf{i}},
\end{align*}

where we write $\mathbf{i}=\{i_1,\dots,i_\frac{p}{2}\}$ with $1\leq i_1<\dots<i_\frac{p}{2}\leq p$.
Fix a nonsingular coordinate $\overline{y}^{r(j_1)},\dots,\overline{y}^{r(j_k)},\overline{y}^{d(m_1)},\dots,\overline{y}^{d(m_{p-k})}$.

Let $A_1:=\{i_1,\dots,i_{\frac{p}{2}}\}$ and $A_2:=\{j_1,\dots,j_k\}$. Also, let $a_1:=\sharp (A_1\cap A_2)$ and $a_2:=\sharp (A_1\cap A_2^c)$. Then,
\begin{align}
&\int_{B_{\mathbf{i}}} \dd \mathbf{y}\prod_{i=1}^p \left(1+|\overline{y}^{r(i)}|^2\right)^{-2+2\gamma}e^{-\frac{\e|\overline{y}^{d(i)}|^2}{4p^2}}\notag\\
&\leq \int_{\R^{3(p-k)}}\dd \overline{y}^{d(m_1)}\dots \dd \overline{y}^{d(m_{p-k})}\prod_{l=1}^{p-k} e^{-\frac{\e|\overline{y}^{d(m_l)}|^2}{4p^2}}\notag\\
&\times \int_{\R^{3k}}\dd \overline{y}^{r(j_1)}\dots \dd \overline{y}^{r(j_k)}\prod_{i\in A_1\cup A_2} \left(1+|\overline{y}^{r(i)}|^2\right)^{-2+2\gamma}.\label{eq:integralppp}
\end{align}

\begin{align*}
&\int_{\R^3}\dd y e^{-\frac{\e |y|^2}{4p^2}}\leq Cp^{3}\e^{-\frac{3}{2}}\\
&\int_{\R^3}\dd y (1+|y|^2)^{-2+2\gamma}\leq C\\
&\int_{|y|\geq \frac{a^{-\frac{1}{6}}}{2}}\dd y (1+|y|^2)^{-2+2\gamma}\leq C\left(\frac{a^{-\frac{1}{6}}}{2}\right)^{-4+4\gamma+3}\\
&(1+|y|^2)^{-2+2\gamma}1\{|y|\geq \frac{a^{-\frac{1}{6}}}{2}\}\leq C\left(\frac{a^{-\frac{1}{6}}}{2}\right)^{(-4+4\gamma)},
\end{align*}

(<ref>) is dominated by
\begin{align*}
\end{align*}

Since $a_1+a_2=\sharp A_1=\frac{p}{2}$ and $a_1\leq  \frac{p}{2}$, we have
\begin{align*}
(-4+4\gamma+3)a_1+(-4+4\gamma)a_2=(-4+4\gamma)\frac{p}{2}+3a_1\leq (-1+4\gamma)\frac{p}{2}.
\end{align*}

\begin{align*}
R_{(\pi,\mathbf{d},2)}\leq C\e^{-\frac{3}{2}p}a^{\frac{1-4\gamma}{12}p}.
\end{align*}

By Proposition <ref> the proof of Lemma <ref> is finished.

§.§ Proof of Lemma <ref>

Lemma <ref> follows from Propositions <ref> and <ref> below.

Let $\gamma\in (0,\frac{1}{2})$,  and $h\in K$ be given.
For any given $p\geq 1$, there is a constant $C\in (0,\infty)$ such that

\begin{align}\label{eq:pmom}
\nonumber &\leq C|u_1-u_2|^{p}a_\ell^{\frac{p}{7}}+C|u_1-u_2|^{\gamma p}a_\ell^{\frac{(1-2\gamma )p}{28}}\e_m^{-\frac{3}{4}p}
\end{align}

for all $u_1,u_2\in \R$, $k\geq \ell\geq 1$.

Proposition <ref> holds for any $p\geq 1$, while Lemma <ref> will be proved only for the fourth moment.

The proof is similar to the one of Lemma <ref>.
It is sufficient to consider the case $k>\ell$.
The constant $C\in (0,\infty)$ given in this proof is independent of $n,m$, $u_1,u_2$, which can be different from line to line.

First, we note that
\begin{align*}
&=\int_{T_{\e_m}}\dd s\dd t\int_{\R^3}\dd y\left(e^{i\lan y,\omega_t-\omega_s+u_1(h_t-h_s)\ran}-e^{i\lan y,\omega_t-\omega_s+u_2(h_t-h_s)\ran}\right)\left(1-e^{-\frac{a_\ell-a_{k}}{2}|y|^2}\right)f_{a_{k}}(y).
\end{align*}

A similar argument to (Step 1) in the proof of Lemma <ref>, we can see that
\begin{align*}
&\leq C|u_1-u_2||a_{k}-a_\ell| a_\ell^{-\frac{6}{7}}\\
&\quad + C\left|\int_{T_{\e_m}}\dd s\dd t\int_{|y|>a_\ell^{-\frac{1}{7}}}\dd y\left(e^{i\lan y,\omega_t-\omega_s+u_1(h_t-h_s)\ran}-e^{i\lan y,\omega_t-\omega_s+u_2(h_t-h_s)\ran}\right) \right. \\
&\quad \hspace{8cm} \left. \times \left(1-e^{-\frac{a_\ell-a_{k}}{2}|y|^2}\right)f_{a_k}(y)\right|\\
&=: C|u_1-u_2||a_{k}-a_\ell| a_\ell^{-\frac{6}{7}}+|E_{k,\ell}| .
\end{align*}

We use Proposition <ref> by setting

\begin{align*}
g(y)=g_{k,\ell}(y):=\left(1-e^{-\frac{a_{\ell} -a_k}{2}|y|^2}\right)f_{a_k}(y) 1\{|y|\geq a_\ell^{-\frac{1}{7}}\}
\end{align*}

and $T=T_{\e_m}$, and changing $y_{\frac{p}{2}+1},\dots, y_p\mapsto -y_{\frac{p}{2}+1},\dots,-y_p$. For our convenience, we will keep the notation $\{y_j\}_{j=1,\dots,p}$.
By using (<ref>), (<ref>), and (<ref>), we have

\begin{align}
&=\int_{T_{\e_m}^p}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)g_{k.\ell}(y_j)\right) e^{-\frac{1}{2}\sum_{j=1}^{2p-1}|R_i|\left|\overline{y}^i\right|^2}\notag\\
&=\sum_{\pi}\int_{T_{\e_m}^p\cap \Delta_p(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}%\notag\\
\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)g_{k,\ell}(y_j)\right)%\notag\\
e^{-\frac{1}{2}\sum_{j=1}^{2p-1}|R_i|\left|\overline{y}^i\right|^2}\label{eq:etanmp} .
\end{align}

We remark that for any given $\gamma\in (0,1]$ there is a constant $C\in(0,\infty)$ such that
\begin{align}\label{eq:sincos}
|\sin x|\leq C(|x|^\gamma\wedge 1),\quad |1-\cos x|\leq C(|x|^\gamma\wedge 1),\quad x\in \R.
\end{align}

and it follows that
\begin{align}
&\leq C|u_1-u_2|^{\gamma p}\sum_{\pi}\int_{T_{\e_m}^p\cap \Delta_p(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}e^{-\frac{1}{2}\sum_{j=1}^{2p-1}|R_i|\left|\overline{y}^i\right|^2}\left(\prod_{j=1}^p|y_j|^\gamma |t_j-s_j|^\gamma \widetilde{g}_{\ell}(y_j)\right),\label{eq:lpeta}
\end{align}

where $\widetilde{g}_\ell(y)=1\{|y|\geq a_\ell^{-\frac{1}{7}}\}$.

Then, the statement follows by modifying the argument of (Step 3) and (Step 4) in the proof of Lemma <ref> by $\gamma\mapsto \frac{\gamma}{2}$, $a^{-\frac{1}{6}}\mapsto a_{\ell}^{-\frac{1}{7}}$.

For ${\gamma}\in \left(\frac{1}{3},\frac{1}{2}\right)$, there exists   $\beta>0$ such that the following holds. For any $h\in K$ and $M>0$, there exists $C>0$ such that

\begin{align*}
&\leq C|u_1-u_2|^4a_m^{\frac{4}{7}}+C|u_1-u_2|^{4\gamma}a_m^{\frac{1-2\gamma}{7}}\e_m^{-3}+C_{h,{\gamma}}|u_1-u_2|^{4{\gamma}}\e_m^{\beta}.
\end{align*}

for $u_1,u_2\in [-M,M]$, $n\geq m\geq 1$.

From Proposition <ref>, we have

\begin{align*}
&\leq C|u_1-u_2|^{p}a_m^{\frac{p}{7}}+C|u_1-u_2|^{p \gamma}a_m^{\frac{1-2\gamma}{28}p}\e_m^{-\frac{3p}{4}}+3^{p-1}E\left[|\widehat{J}_{\e_n,a_\ell}(u_1,u_2,h)-\widehat{J}_{\e_m,a_\ell}(u_1,u_2,h)|^{p}\right]
\end{align*}

for $\ell\geq n\geq m\geq 1$, and $p\in 2\N$.
In the same manner as (<ref>), we have
\begin{align}
&=\sum_{\pi}\int_{T_{\e_m,\e_n}^{p}\cap \Delta_{p}(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}
\left(\prod_{j=1}^{p} \left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)f_{a_\ell}(y_j)\right)
\label{eq:lpeta2}
\end{align}

where we set
\begin{align*}
&T_{\e_m,\e_n}:=\{(s,t)\in [0,1]^2: \e_n\leq t-s\leq \e_m\}.
\end{align*}

<cit.> allows us to apply the dominated convergence theorem to (<ref>) in $\ell$. Thus, we find that

\begin{align*}
&\leq C|u_1-u_2|^{p}a_m^{\frac{p}{7}}+C|u_1-u_2|^{p\gamma}a_m^{\frac{1-2\gamma}{28}p}\e_m^{-\frac{3p}{4}}\\
&\quad +3^{p-1}\sum_{\pi} \int_{T_{\e_m,\e_n}^{p}\cap \Delta_{p}(\pi)}\dd {\mathbf{s}}\dd \mathbf{t} \int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^{p}\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
\end{align*}

From this inequality we see that it is sufficient to estimate the integrals in

\begin{align}\label{eq:l4inte}
&\int_{T_{\e_m,\e_n}^{p}\cap \Delta_{p}(\pi)}\dd {\mathbf{s}}\dd \mathbf{t} \int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p \left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
\end{align}

with $p=4$ due to the type of $\pi$.

In the case of <ref>:
We look at the integral in $y_j$ for $j\in \{1,2,3,4\}$ such that $(s_j,t_j)\cap \bigcup_{k\not=j}[s_k,t_k]=\emptyset$.
We only prove the case $j=4$, because the other cases are similar.
Let us observe that we have

\begin{align}
&\int_{\R^{12}}\dd \mathbf{y}\left(\prod_{j=1}^4\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&=\int_{\R^3}\dd {y}_4\left(e^{i\left\lan y_4,u_1h_{\mathbf{s},\mathbf{t}}^{(4)}\right\ran}-e^{i\left\lan y_4,u_2h_{\mathbf{s},\mathbf{t}}^{(4)}\right\ran}\right)e^{-\frac{1}{2}(t_4-s_4)|y_4|^2}\notag\\
&\quad \times \int_{\R^9}\dd y_1\dd y_2\dd y_3\left(\prod_{j=1}^3\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right) \label{eq:Type2}\\
&\quad \times \exp\left({-\frac{1}{2}\sum_{\begin{smallmatrix}j=1,\dots,7\\ j\not=f(4),r(4)\end{smallmatrix}}|R_j|\left|\overline{y}^j\right|^2}\right). \notag
\end{align}

It is easy to see that
\begin{align*}
&\left|\int_{\R^3}\dd {y}_4\left(e^{i\left\lan y_4,u_1h_{\mathbf{s},\mathbf{t}}^{(4)}\right\ran}-e^{i\left\lan y_4,u_2h_{\mathbf{s},\mathbf{t}}^{(4)}\right\ran}\right)e^{-\frac{1}{2}(t_4-s_4)|y_4|^2}\right|\\
&\leq \frac{C}{|t_4-s_4|^\frac{5}{2}} |u_2^2-u_1^2| |h_{t_4}-h_{s_4}|^2
\end{align*}

and that its integral in $s_4,t_4$ is bounded by $CM|u_2-u_1| \e_m^{\frac{1}{2}}$ for $u_1,u_2\in [-M,M]$. Thus, we can reduce it to the case that $p=3$.

In the case of <ref>:
The union $\bigcup_{j=1}^4[s_j,t_j]$ is decomposed into two disjoint sets as $[s_{j_1}, t_{j_1}] \cup [s_{j_2}, t_{j_2}]$ and $[s_{j_3}, t_{j_3}] \cup [s_{j_4}, t_{j_4}]$ where $\{ j_1,j_2,j_3,j_4\} =\{ 1,2,3,4\}$.
Hence, similarly to (<ref>),

\begin{align*}
&\int_{\R^{12}}\dd \mathbf{y}\left(\prod_{j=1}^4\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
e^{-\frac{1}{2}\sum_{j=1}^{7}|R_j|\left|\overline{y}^j\right|^2} \\
&= \int_{\R^{6}}\dd y_{j_1} \dd y_{j_2} \left(\prod_{k=1}^2\left(e^{i\left\lan y_{j_k},u_1 h_{\mathbf{s},\mathbf{t}}^{(j_k)}\right\ran}-e^{i\left\lan y_{j_k},u_2h_{\mathbf{s},\mathbf{t}}^{(j_k)}\right\ran}\right)\right)
e^{-\frac{1}{2}\sum_{k=1}^{3}|R_k|\left|\overline{y}^k\right|^2} \\
&\quad \times \int_{\R^{6}}\dd y_{j_3} \dd y_{j_4} \left(\prod_{k=3}^4\left(e^{i\left\lan y_{j_k},u_1 h_{\mathbf{s},\mathbf{t}}^{(j_k)}\right\ran}-e^{i\left\lan y_{j_k},u_2h_{\mathbf{s},\mathbf{t}}^{(j_k)}\right\ran}\right)\right)
\end{align*}

which are estimated by those in the case that $p=2$.

In the case of <ref>: Here, we consider the estimate of (<ref>) for $2\leq p\leq 4$.
We apply the same argument as in (<ref>) and (<ref>) so that

\begin{align}
&\left|\int_{T_{\e_m,\e_n}^p\cap \Delta_p(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq \sum_{\ell=m}^\infty \left|\int_{T_{\e_\ell,\e_{\ell+1}}^p\cap \Delta_p(\pi)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{3p}}\dd \mathbf{y}\left(\prod_{j=1}^p\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq C |u_1-u_2|^{\gamma p} \sum_{\ell=m}^\infty \e_\ell^{p\gamma }\int_{J(T_{\e_\ell,\e_{\ell+1}}^p(\pi,\mathbf{d}))}\dd z_{\pi(1)}\prod_{i=1}^{2p-1}\dd |R_{i}| \notag\\
&\quad \hspace{5cm} \times \int _{\R^{3p}}\dd \mathbf{y}\exp\left(-\frac{1}{2}\sum_{i=1}^{2p-1}|\overline{y}^i|^2|R_i|\right)\prod_{j=1}^{2p-1}\left(1+|\overline{y}_j|^2\right)^\frac{\gamma}{2}\notag\\
&\leq C |u_1-u_2|^{\gamma p} \sum_{\ell=m}^\infty\e_\ell^{p\gamma } \int_{\R^{3p}}\dd \mathbf{y}\prod_{i=1}^{p}\left(1+|\overline{y}^{f(i)}|^2\right)^{-1+\frac{\gamma}{2}}\left(1+|\overline{y}^{r(i)}|^2\right)^{-1+\frac{\gamma}{2}}\exp\left(-\frac{\e_{\ell+1}|\overline{y}^{d(i)}|^2}{4p^2}\right).\label{eq:typ1bdd1}
\end{align}

We will use Hölder's inequality with $a,b\geq 1$ such that

\begin{align*}
\frac{1}{a}+\frac{1}{b}=1, \quad 2\left(-1+\frac{\gamma}{2}\right)a+2<-1,&\quad 2\left(-1+\frac{\gamma}{2}\right)b+2<-1, \\
p\gamma >\frac{3(p-2)}{2b},&
\end{align*}

where  such $a,b$ exist if $\gamma\in \left(\frac{1}{3},\frac{1}{2}\right)$.
For example, we can take $\gamma=\frac{3}{7}$, $a=b=2$.
Then, (<ref>) is bounded by

\begin{align*}
&C|u_1-u_2|^{\gamma p}\left(\int_{\R^{3p}}\dd \mathbf{y}\prod_{i=1}^{p}\left(1+|\overline{y}^{f(i)}|^2\right)^{(-1+\frac{\gamma}{2})a}\right)^\frac{1}{a}\\
& \times\sum_{\ell=m}^\infty \e_\ell^{\gamma p}\left(\int_{\R^{3p}}\dd \mathbf{y}\left(1+|\overline{y}^{r(i)}|^2\right)^{(-1+\frac{\gamma}{2})b}\exp\left(-b\frac{\e_\ell|\overline{y}^{d(i)}|^2}{8p^2}\right)\right)^\frac{1}{b}\\
&\leq C|u_1-u_2|^{\gamma p}\sum_{\ell=m}^\infty\e_\ell^{\gamma p-\frac{1}{2b}(3p- \mathrm{rank}\left(\{\overline{y}^{r(1)},\dots,\overline{y}^{r(p)}\}\right) }\\
&\leq C|u_1-u_2|^{\gamma p}\e_m^{\gamma p-\frac{3(p-2)}{2b}}
\end{align*}

for $\pi$ of Type<ref> and

\begin{align*}
\mathrm{rank}\left(\{\overline{y}^{r(1)},\dots,\overline{y}^{r(p)}\}\right)\geq 6.
\end{align*}

Therefore, the remainder of $\pi$ with Type<ref> is the one such that
\begin{align}
\mathrm{rank}\left(\{\overline{y}^{r(1)},\dots,\overline{y}^{r(p)}\}\right)=3. \label{eq:rank3}
\end{align}

We remark that (<ref>) holds in the following cases.

\begin{align}
&\tag{$\Delta _2$} \label{eq:pi2} \text{$p=2$ and $\pi$ belongs to \textbf{Type}\ref{item:T1}},\\
&\tag{$\Delta_3(\pi_1)$} \label{eq:pi31} \text{$p=3$ and $s_1<s_2<t_2<s_3<t_3<t_1$},\\
&\tag{$\Delta_3(\pi_2)$} \label{eq:pi32} \text{$p=3$ and $s_2<s_1<t_2<s_3<t_3<t_1$},\\
&\tag{$\Delta_4(\pi_1)$} \label{eq:pi41} \text{$p=4$ and $s_1<s_2<t_2<s_3<t_3<s_4<t_4<t_1$},\\
&\tag{$\Delta_4(\pi_2)$} \label{eq:pi42} \text{$p=4$ and $s_2<s_1<t_2<s_3<t_3<s_4<t_4<t_1$}.
\end{align}

Furthermore, by taking time-reversal (see Definition <ref>) (<ref>) and (<ref>) are reduced to the case where

\begin{align*}
\mathrm{rank}\left(\{\widetilde{\overline{y}}^{\widetilde{r}(1)},\dots,\widetilde{\overline{y}}^{\widetilde{r}(p)}\}\right)\geq 6.
\end{align*}

Hence, the remaining cases are (<ref>), (<ref>) and (<ref>).
The estimates for these cases are proved by Propositions <ref>, <ref>, <ref>, <ref> below, and this finishes then this proof.

\begin{align}
&\Delta_2(\pi_1):=\{0<s_1<s_2<t_2<t_1\leq 1\} , \notag\\
&\Delta_2(\pi_2):=\{0<s_1<s_2<t_1<t_2\leq 1\} . \notag
\end{align}

For any $\gamma\in \left(0,\frac{1}{2}\right)$, there exists $C>0$ such that for any $n\geq 1$

\begin{align}
&\left|\int_{T_{\e_{n-1},\e_n}^2\cap \Delta_2(\pi_1)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{6}}\dd \mathbf{y}\left(\prod_{j=1}^2\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq C|u_1-u_2|^{2\gamma}\e_{n-1}^{\gamma}\label{eq:intpi21}
\end{align}

Take $\gamma\in (0,\frac{1}{2})$.  We set $s=s_2-s_1=|R_1|$, $t=t_2-s_2=|R_2|$, $u=t_1-t_2=|R_3|$. Then, the left-hand side of (<ref>) is dominated by
\begin{align*}
&C|u_1-u_2|^{2\gamma}\e_{n-1}^{2\gamma} \int_{0<s+u<\e_{n-1}}\dd s\dd u\int_{\e_n<t<\e_{n-1} }\dd t \\
&\quad \hspace{4cm} \times \int_{\R^6}\dd x\dd y|x|^\gamma |y|^\gamma \exp\left(-\frac{s+u}{2}|x|^2-\frac{t}{2}|x+y|^2\right)\\
&\leq  C|u_1-u_2|^{2\gamma}\e_{n-1}^{2\gamma} \int_{0<s+u<\e_{n-1}}\dd s\dd u\int_{\e_n<t<\e_{n-1} }\dd t\ \\
&\quad \hspace{4cm} \times \int_{\R^6}\dd x\dd z|x|^\gamma (|x|^\gamma+|z|^\gamma) \exp\left(-\frac{s+u}{2}|x|^2-\frac{t}{2}|z|^2\right)\\
&\leq C|u_1-u_2|^{2\gamma} \e_{n-1}^\gamma.
\end{align*}

Here, note the choice of $\e_n$ (see (<ref>)).

For any $\gamma\in \left(0,\frac{1}{2}\right)$, there exists $C>0$ such that for any $n\geq 1$

\begin{align}
&\left|\int_{T_{\e_{n-1},e_n}^2\cap \Delta_2(\pi_2)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{6}}\dd \mathbf{y}\left(\prod_{j=1}^2\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq C|u_1-u_2|^{2\gamma}\e_{n-1}^{\gamma}.\label{eq:intpi22}
\end{align}

Take $\gamma\in (0,\frac{1}{2})$.  We set $s=s_2-s_1=|R_1|$, $t=t_1-s_2=|R_2|$, $r=t_2-t_1=|R_3|$. Then, the left-hand side of (<ref>) is dominated by

\begin{align*}
&C|u_1-u_2|^{2\gamma} \int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r\\
&\quad \hspace{1cm} \times \int_{\R^6}\dd x\dd y(s+t)^\gamma (t+r)^\gamma |x|^\gamma |y|^\gamma
\exp\left(-\frac{s}{2}\left|x\right|^2-\frac{t}{2}|x+y|^2 - \frac{r}{2}|y|^2\right) \\
&= C|u_1-u_2|^{2\gamma} \int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r\\
&\quad \times \int_{\R^6}\dd x\dd y(s+t)^\gamma (t+r)^\gamma |x|^\gamma |y|^\gamma
\exp\left(-\frac{s+t}{2}\left|x+\frac{t}{s+t}y\right|^2-\frac{st+tr+rs}{2(s+t)}|y|^2\right).
\end{align*}

Changing variables $x+\frac{t}{s+t}y\mapsto z$, $y\mapsto y$, we can get an upper bound of the integral,
\begin{align*}
&\int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r (s+t)^\gamma (t+r)^\gamma (st+tr+rs)^{-\frac{3+\gamma}{2}}\\
&+\int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r (s+t)^\gamma t^\gamma (t+r)^\gamma (st+tr+rs)^{-\frac{3+2\gamma}{2}}\\
&\leq \e_{n-1}^{2\gamma} \int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r  (st+tr+rs)^{-\frac{3+\gamma}{2}} \\
&\quad +\e_{n-1}^{3\gamma} \int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r  (st+tr+rs)^{-\frac{3+2\gamma}{2}}.
\end{align*}

These terms are estimated by $C\e _{n-1}^\gamma$, because for $a\in \left(0,\frac{1}{2}\right)$

\begin{align*}
&\int_{\begin{smallmatrix} \e_n<s+t<\e_{n-1}\\ \e_n<t+r<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r  (st+tr+rs)^{-\frac{3}{2}-a} \\
&\leq C\int_{ \e_n<s+t<\e_{n-1}}\dd s\dd t   (s+t)^{-1}(st)^{-\frac{1}{2}-a}\\
&=C\int_{ \e_n<R<\e_{n-1},0<s<R}\dd R\dd s  \left[ R^{-1}(s(R-s))^{-\frac{1}{2}-a}\right]\\
&=CB\left(\frac{1}{2}-a,\frac{1}{2}-a\right)\int_{\e_{n}<R<\e_{n-1}}\dd RR^{-1-2a}\leq C\e_n^{-2a}.
\end{align*}

where $B(a,b)=\int_0^1 x^{a-1}(1-x)^{b-1}\dd x$ is the Bessel function for $a,b>0$.
Hence, we have the assertion.

We can prove the following propositions by the same argument as in Proposition <ref>.

For $\gamma \in (0,1)$, there exists $C>0$ such that for any $n\geq 1$

\begin{align}
&\left|\int_{T_{\e_{n-1},e_n}^3\cap \Delta_3(\pi_1)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{9}}\dd \mathbf{y}\left(\prod_{j=1}^3\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq C|u_1-u_2|^{3\gamma} \e_n^{\frac{1+3\gamma}{2}} . \label{eq:intpi31}
\end{align}

Take $\gamma\in (0,1)$.  We set $r_1=s_2-s_1$, $s=t_2-s_2$, $r_2=s_3-t_2$, $t=t_3-s_3$, $r_3=t_1-t_1$. Then, the left-hand side of (<ref>) is dominated by
\begin{align*}
&C|u_1-u_2|^{3\gamma} \int_{\begin{smallmatrix} s,t\in [\e_n,\e_{n-1}]\\ 0<r_1+r_2+r_3<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd \mathbf{r}\int_{\R^9}\dd x\dd y\dd z (s+t+r_1+r_2+r_3)^\gamma s^\gamma t^\gamma \\
&\quad \hspace{2cm} \times |x|^\gamma |y|^\gamma |z|^\gamma \exp\left(-\frac{s}{2}|x+z|^2-\frac{t}{2}|y+z|^2-\frac{r_1+r_2+r_3}{2}|z|^2\right).
\end{align*}

Changing variables $x+z\mapsto \widetilde{x}$, $y+z\mapsto \widetilde{y}$, we can get an upper bound of the integral,
\begin{align*}
&\int_{\begin{smallmatrix} s,t\in [\e_n,\e_{n-1}]\\ 0<r_1+r_2+r_3<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd \mathbf{r}\int_{\R^9}\dd x\dd y\dd z (s+t+r_1+r_2+r_3)^\gamma s^\gamma t^\gamma \\
&\quad \hspace{2cm} \times|x|^\gamma |y|^\gamma |z|^\gamma %\notag\\
\exp\left(-\frac{s}{2}|x+z|^2-\frac{t}{2}|y+z|^2-\frac{r_1+r_2+r_3}{2}|z|^2\right)\\
&\leq  \int_{\begin{smallmatrix} s,t\in [\e_n,\e_{n-1}]\\ 0<r_1+r_2+r_3<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd \mathbf{r}\int_{\R^9}\dd \widetilde{x}\dd \widetilde{y}\dd z (s+t+r_1+r_2+r_3)^\gamma s^\gamma t^\gamma \\
&\quad \hspace{2cm} \times(|\widetilde{x}|^\gamma+|z|^\gamma) (|y|^\gamma+|z|^\gamma) |z|^\gamma %\notag\\
\exp\left(-\frac{s}{2}|\widetilde{x}|^2-\frac{t}{2}|\widetilde{y}|^2-\frac{r_1+r_2+r_3}{2}|z|^2\right)\\
&\leq  C\e_{n-1}^{3\gamma}\int_{\begin{smallmatrix} s,t\in [\e_n,\e_{n-1}]\\ 0<r_1+r_2+r_3<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd \mathbf{r}\int_{\R^9}\dd \widetilde{x}\dd \widetilde{y}\dd z\\
&\hspace{3em} (|\widetilde{x}|^\gamma+|z|^\gamma) (|y|^\gamma+|z|^\gamma) |z|^\gamma %\notag\\
\exp\left(-\frac{s}{2}|\widetilde{x}|^2-\frac{t}{2}|\widetilde{y}|^2-\frac{r_1+r_2+r_3}{2}|z|^2\right)\\
&\leq C\e_{n}^{\frac{1+3\gamma}{2}}.
\end{align*}

Here, note the choice of $\e_n$ (see (<ref>)).

There exists $C>0$ such that for any $n\geq 1$

\begin{align}
&\left|\int_{T_{\e_{n-1},e_n}^4\cap \Delta_4(\pi_1)}\dd {\mathbf{s}}\dd \mathbf{t}\int_{\R^{12}}\dd \mathbf{y}\left(\prod_{j=1}^4\left(e^{i\left\lan y_j,u_1h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}-e^{i\left\lan y_j,u_2h_{\mathbf{s},\mathbf{t}}^{(j)}\right\ran}\right)\right)
&\leq C|u_1-u_2|^4 \e_n^3 . \label{eq:intpi41}
\end{align}

We set $v_1=s_2-s_1$, $s=t_2-s_2$, $v_2=s_3-t_2$, $t=t_3-s_3$, $v_3=s_4-t_3$, $r=t_4-s_4$, $v_4=t_1-t_4$. Then, the lefthand side of (<ref>) is dominated by
\begin{align*}
&C|u_1-u_2|^{4} \int_{\begin{smallmatrix} s,t,r\in [\e_n,\e_{n-1}]\\ 0<v_1+v_2+v_3+v_4<\e_{n-1}\end{smallmatrix}}\dd s\dd t \dd r\dd \mathbf{v}\int_{\R^{12}}\dd x\dd y\dd z\dd w\\
&\hspace{2em} (s+t+r+v_1+v_2+v_3+v_4) s t r|x| |y| |z| |w|\notag\\
&\hspace{2em}\times \exp\left(-\frac{s}{2}|x+w|^2-\frac{t}{2}|y+w|^2-\frac{r}{2}|z+w|^2-\frac{v_1+v_2+v_3+v_4}{2}|z|^2\right).
\end{align*}

In a similar fashion to the discussion in Proposition <ref>, we can then prove the statement of Proposition <ref>.

§.§ Remarks on the treatment of Dirac functions

In this section, we will look at the self-intersection local time for pinned Brownian motion. In particular, its continuity will allow us to use the notation

\begin{align*}
E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\prod_{i=1}^n \delta_{x_i}(\omega_{t_i})\right]
\end{align*}

(see Remark <ref>) where $\overline{J}^{\e,\lambda}_{0,1}$ is given after (<ref>), which is useful in our analysis.

Similarly to [46], we define a self-intersection local time of conditioned Brownian motion.
Fix $\e>0$. For each $0<u<v<1$,  $x,y\in \R^3$, $a>0$, we define
\begin{align*}
H_{u,v,x,y}^{\e,a}:=\int_\e^1\dd t\int_0^{t-\e}\dd sp_{a}\left(\Psi_{u,v,x,y}(\omega)(t)-\Psi_{u,v,x,y}(\omega)(s)\right),
\end{align*}

where for $\omega\in C([0,1],\R^3)$
\begin{align*}
\Psi_{u,v,x,y}(\omega)(\tau)=\begin{cases}
\dis x\frac{\tau}{u}+\omega_\tau-\frac{\tau}{u}\omega_u,\quad &0\leq \tau\leq u\\
\dis x\left(1-\frac{\tau-u}{v-u}\right)+y\frac{\tau-u}{v-u}+\left(\omega_\tau-\omega_u-\frac{\tau-u}{v-u}(\omega_v-\omega_u)\right),\quad &u\leq \tau\leq v\\
\dis y+\omega_\tau-\omega_v,\quad &v\leq \tau.
\end{cases}
\end{align*}

By modifying the proof of Proposition <ref> or <cit.>, we can prove the following.

Fix $\e>0$ and $0\leq u<v\leq 1$. Then, there exists a version of $H_{u,v,x,y}^{\e,a}$ which is jointly continuous in $x,y\in \R^3$, and $a\geq 0$. Moreover, the limit as $\searrow 0$ denoted by $H_{u,v,x,y}^{\e}$ exists and also has these properties.

We omit the proof since it follows by the arguments of <cit.>.
In fact we have the estimate that there exists $\gamma\in (0,\frac{1}{4})$ such that for any $p\geq 2$,
\begin{align*}
E\left[\left|H_{u,v,x,y}^{\e,a}-H_{u,v,x',y'}^{\e,a'}\right|^p\right]\leq C\left(|x-x'|^{\gamma p}+|y-y'|^{\gamma p}+|a-a'|^\frac{\gamma p}{2}\right).
\end{align*}

The map $\Psi_{u,v,x,y}$ maps a Brownian motion to conditional Brownian motion which bridges between $(0,0)$, $(u,x)$, $(v,y)$. Lemma <ref> can be generalized for any $n$ intermediate points.  For $0\leq t_1<\dots<t_n\leq 1$, $x_1,\dots,x_n\in \R^3$, we denote by $H^{\e,a}_{(t_i,x_i)}$ a modified version of $H$. We omit the upper index $a$ for $a=0$.

Let $\Phi$ be any bounded measurable function. Then, we have
\begin{align*}
&\lim_{a\to 0}E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi \prod_{i=1}^np_a(\omega_{t_i},x_i)\right]\\
&= E\left[\left.e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi \right|\omega_{t_i}=x_i,i=1,\dots,n\right]p_{t_1}(0,x_1)\prod_{j=2}^{n}p_{t_j-t_{j-1}}(x_{j-1},x_j)
\end{align*}

for $0\leq t_1<\dots<t_n\leq 1$, $x_1,\dots,x_n\in\R^3$.

From construction, we  have that
\begin{align*}
&\int_{\R^{3n}}\dd \mathbf{y}E\left[\left.e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi \prod_{i=1}^np_a(\omega_{t_i},x_i)\right|\omega_{t_i}=y_i,i=1,\dots,n\right]p_{t_1}(0,y_1)\prod_{j=2}^{n}p_{t_j-t_{j-1}}(y_{j-1},y_j)\\
&=\int_{\R^{3n}}\dd \mathbf{y}E\left[e^{-\lambda {H}^{\e}_{(t_i,y_i)}+\lambda \kappa_1(\e)-\lambda^2\kappa_2(\e)}\Phi \right]\prod_{i=1}^np_a(y_i,x_i)p_{t_1}(0,y_1)\prod_{j=2}^{n}p_{t_j-t_{j-1}}(y_{j-1},y_j).
\end{align*}

Letting $a\searrow 0$, we see that the statement holds.

Corollary <ref> allows us to use the formal expression in Remark <ref> below.

Let us use the following notation:

\begin{align*}
&E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\prod_{i=1}^n \delta_{x_i}(\omega_{t_i})\right]\\
&:=E\left[\left.e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi \right|\omega_{t_i}=x_i,i=1,\dots,n\right]p_{t_1}(0,x_1)\prod_{j=2}^{n}p_{t_j-t_{j-1}}(x_{j-1},x_j).
\end{align*}

Also, we set

\begin{align*}
&E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\prod_{i=1}^n \delta_0(\omega_{s_i}-\omega_{t_i})\right] :=\int_{\R^{3n}}E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\prod_{i=1}^n \left(\delta_{x_i}(\omega_{s_i})\delta_{x_i}(\omega_{t_i})\right)\right]\dd x_1\dots \dd x_n
\end{align*}

for $s_1,\dots,s_n,t_1,\dots,t_n\in [0,1]$.

Similarly to Corollary <ref>, we can show that
\begin{align*}
&E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\prod_{i=1}^n \delta_{x_i}(\omega_{r_i})\prod_{j=1}^mJ_{s_j,t_j}^{\e} \Phi\right]=\int_{T_\e^m}\dd \mathbf{s}\dd \mathbf{t} E\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\prod_{i=1}^n \delta_{x_i}(\omega_{r_i})\prod_{j=1}^m\delta_0(\omega_{s_j}-\omega_{t_j}) \Phi\right]
\end{align*}

for $r_1,\dots,r_n\in [0,1]$ and $x_1,\dots,x_n\in \R^3$. We omit the detail of the proof.

§.§ Differentiability

In this subsection, we will look at the differentiability of
\begin{align*}
\rho(\e):=E\left[\exp\left(-\overline{J}_{0,1}^{\e,\lambda}\right)\Phi\right]
\end{align*}

in $\e\in (0,1)$, where $\Phi$ is any bounded random variable.

$\rho$ is differentiable at $\e\in (0,1)$ for any bounded random variable $\Phi$ and satifies

\begin{align*}
\frac{\dd}{\dd \e}\rho(\e)&=\lambda \int_0^{1-\e}\dd sE\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\delta _0 (\omega_s-\omega_{s+\e})\right]+\left(\lambda\frac{\dd}{\dd \e}\kappa_1(\e)-\lambda^2 \frac{\dd}{\dd \e}\kappa_2(\e)\right)\rho(\e) .
\end{align*}

In [46], $\rho(\e)$ with some $\Phi$ is used to construct the polymer measure $\nu_\lambda$.
The author focused on a lower bound for

\begin{align*}
\rho(\e_1)-\rho(\e_2).
\end{align*}

for $0<\e_1<\e_2$. He avoided the discussion of differentiability of $\rho$ and used the lower bound derived from the derivative of the modified function in which ${J}_{0,1}^\e$ is replaced by $J_{0,1}^{\e,a}$.

We may assume that $\Phi$ is a positive random variable.
Fix $\e\in (0,1)$. Let $\widetilde{\e}\in(0,1-\e)$ be small enough.  Then, we have
\begin{align*}
\rho(\e+\widetilde{\e})-\rho(\e)=-E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left(\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}-\overline{J}^{\e,\lambda}_{0,1}\right)-1\right)\Phi\right].
\end{align*}

We can cope with the deterministic part in $\overline{J}^{\cdot,\lambda}_{0,1}$ easily. Thus, it is enough to focus on
\begin{align*}
E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left(\lambda \left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)\right)-1\right)\Phi\right].
\end{align*}

Since $\lambda \left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)$ is negative, we have
\begin{align*}
&E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left(\lambda \left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)\right)-1\right)\Phi\right]\\
&\geq \lambda E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)\Phi\right]\\
&=-\lambda \int_{T_\e\backslash T_{\e+\widetilde{\e}}}\dd u\dd v E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\delta_{0}(\omega_u-\omega_v)\Phi\right].
\end{align*}

It is easy to see that
\begin{align*}
&\lim_{\widetilde{\e}\to 0}\frac{1}{\widetilde{\e}}\int_{T_\e\backslash T_{\e+\widetilde{\e}}}\dd u\dd v E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\delta_{0}(\omega_u-\omega_v)\Phi\right]\\
&= \int_0^{1-\e}\dd sE\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\delta _0 (\omega_s-\omega_{s+\e})\right].
\end{align*}

On the other hand, we can use the following bound obtained by Rosen <cit.>: For any $k\geq 2$ and $p>4 $, there exists a constant $C>0$ such that

\begin{align}
E\left[\left|{J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right|^k\right]\leq C|T_\e\backslash T_{\e+\widetilde{\e}}|^{\frac{k}{p}}\leq C\widetilde{\e}^{\frac{k}{p}},\label{eq:Jdiffmoment}
\end{align}

where $|T_\e\backslash T_{\e+\widetilde{\e}}|$ is the Lebesgue measure of  $T_\e\backslash T_{\e+\widetilde{\e}}$ in $\R^2$.
Therefore, for any $\delta>0$,
\begin{align*}
\varlimsup_{\widetilde{\e}\to 0}\frac{1}{\widetilde{\e}}E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)-1\right)\Phi:{J}^{\e}_{0,1}-{J}^{\e+\widetilde{\e}}_{0,1}>\delta\right]=0.
\end{align*}

Let $r>0$ be a small constant and take $\delta>0$ such that $1-e^{-x}\geq (1-r)x$ for $0\leq x\leq \delta$. Then,
\begin{align*}
&E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left(\lambda \left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)\right)-1\right)\Phi\right]\\
&\leq -(1-r)\lambda E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left({J}^{\e}_{0,1}-{J}^{\e+\widetilde{\e}}_{0,1}\right)\Phi:{J}^{\e}_{0,1}-{J}^{\e+\widetilde{\e}}_{0,1}\leq \delta\right]+o(\widetilde{\e})\\
&\leq  -(1-r)\lambda E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left({J}^{\e}_{0,1}-{J}^{\e+\widetilde{\e}}_{0,1}\right)\Phi\right]+o(\widetilde{\e}),
\end{align*}

where we have used (<ref>) in the last line.
Thus, we have
\begin{align*}
&\varlimsup_{\widetilde{\e}\to 0}\frac{1}{\widetilde{\e}}E\left[\exp\left(-\overline{J}^{\e+\widetilde{\e},\lambda}_{0,1}\right)\left(\exp\left(\lambda \left({J}^{\e+\widetilde{\e}}_{0,1}-{J}^{\e}_{0,1}\right)\right)-1\right)\Phi\right]\\
&\leq (1-r)\lambda \int_0^{1-\e}\dd sE\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\delta _0 (\omega_s-\omega_{s+\e})\right].
\end{align*}

Since $0<r<1$ is arbitrary, the right differentiability follows. The left differentiability follows by the same kind of argument.

§ SOME ESTIMATES WITH RESPECT TO $\NU_{\E,\LAMBDA}$

§.§ Recursive estimates

For $x\in\R^3$, we set $p_t(x):=(2\pi t)^{-\frac{3}{2}}\exp\left(-\frac{|x|^2}{2t}\right)$, and

\begin{align*}
g_t^{\e,\lambda}(x):=E\left[\exp\left(-\overline{J}_{0,t}^{\e,\lambda}\right)\delta_{x}(\omega_t )\right] .
\end{align*}

In [46] and [27] it was shown that for any given $\lambda\in [0,\infty)$ there is a constant $C\in (0,\infty)$ such that

\begin{align}
\left|g_{t}^{\e,\lambda}(x)-p_t(x)\right|\leq Ct^{\frac{1}{2}}p_{2t}(x)\label{eq:partpest}
\end{align}

for all $t\in [0,1]$ and $x\in \R^3$.
Here, we remark that the constant $C$ in (<ref>) can be chosen independently of $\lambda$, at least for small $\lambda$.
For $n\geq 1$, we define an $n$-points “transition density" by

\begin{align*}
g_{\mathbbm{t}}^{T,\e,\lambda}(\mathbf{x}):=E\left[\exp\left(-\overline{J}_{0,T}^{\e,\lambda}\right)\delta_{x_1}(\omega_{t_1}) \cdots \delta_{x_n}(\omega_{t_n}) \right],
\end{align*}

for $\mathbbm{t}=(t_1,\cdots,t_n)$ with $0<t_1<\cdots<t_n\leq T\leq 1$ and $\mathbf{x}=(x_1,\dots,x_n)\in (\R^{3})^{n}$.
The following lemma is provided in <cit.>.

For any $n\geq 1$, there exists a constant $C\in (0,\infty)$ such that
\begin{align*}
g_{\mathbbm{t}}^{T,\e,\lambda}(\mathbf{x})\leq Cp_{2t_1}(x_1)p_{2(t_2-t_1)}(x_2-x_1)\cdots p_{2(t_n-t_{n-1})}(x_n-x_{n-1}).
\end{align*}

for $\mathbf{x}=(x_1,\dots,x_n)\in (\R^3)^n$, $\lambda\in [0,\infty)$ and $\e\in (0,1)$.

Let $\psi:[0,1]\times \R^3\to \R$ be bounded and infinitely often differentiable with bounded derivatives of all orders, and for $0\leq s<t\leq 1$, we define $\Psi_{s,t}$ by
\begin{align}
\Psi_{s,t}:=\exp\left(\int_s^t \psi(u,\omega_u)\dd u\right).\label{eq:psidef}
\end{align}

As was shown in <cit.>, (<ref>) implies that for any such $\psi$ and $\lambda\in [0,\infty)$ there is a constant $C\in (0,\infty)$ such that
\begin{align}
\left|E\left[\exp\left(-\overline{J}_{0,1}^{\e_1,\lambda}\right)\Psi_{0,1}\right]-E\left[\exp\left(-\overline{J}_{0,1}^{\e_2,\lambda}\right)\Psi_{0,1}\right]\right|\leq C(\e_1^{\delta}+\e_2^{\delta})
\label{eq:psiest}
\end{align}

for all $\e_1,\e_2\in (0,1)$, where $\delta\in (0,1)$. This implies that the limit

\begin{align}
N(\lambda)=\lim_{\e\to 0}E\left[\exp\left(-\overline{J}_{0,1}^{\e,\lambda}\right)\right] \in (0,\infty ) \label{eq:Nlam}
\end{align}

exists for all $\lambda\in[0,\infty)$. Strict positivity was proved in [27].
In this section, we will give a similar estimate for more general functions $\Phi$.

Let $f:\mathbb{R}^3\to \R$ be a bounded measurable function. Let $\Delta_{u,v;s,t}:=\{(\sigma,\tau):u\leq \sigma\leq v,s\leq \tau\leq t\}$, and for any fixed $a\in (0,1)$ set

\begin{align}
\Phi_{u,v;s,t}:=\int_{T_{a}\cap \Delta_{u,v;s,t}}f(\omega_\sigma-\omega_\tau)\dd \sigma \dd \tau \label{eq:PhiF}
\end{align}

where $T_a := \{ (s,t)\in [0,1]^2: t-s \geq a\}$ as above.
For short, we set $\Phi:=\Phi_{0,1;0,1}$ and $\Phi_{s,t}:=\Phi_{s,t;s,t}$. We set
\begin{align}
\rho(\e):=E\left[\exp\left(-\overline{J}_{0,1}^{\e,\lambda}\right)\Phi\right].\label{eq:phidef}
\end{align}

The main aim of this section is to prove the following lemma, that will be used in the proof of Theorem <ref> below.

There are constants $C\in (0,\infty)$ and $\widetilde{\delta} \in (0,1)$ such that

\begin{align}
|\rho(\e_1)-\rho({\e_2})|\leq C\sup_{x\in\mathbb{R}^3}|f(x)|(\e_1^{\widetilde{\delta}}+\e_2^{\widetilde{\delta}})\label{eq:phiest}
\end{align}

for $\e_1,\e_2\in (0,1)$.

The proof of Lemma <ref> will be provided in Section <ref> below.
The estimate (<ref>), with $|\Phi|$ replaced by $\Psi_{0,1}$ in (<ref>), was already proved by Bolthausen in <cit.>. By modifying his argument one can see that the discussion in <cit.> is also suitable for the present case. Since $|\Phi_{0,{t+s}}|$ is not equal to $|\Phi_{0,s}||\Phi_{s,s+t}|$, some arguments given in the proof of Lemma <ref> below are different from those given in <cit.>.

* Lemma <ref> is true for $|\Phi|$ where $\Phi=\Phi_1-\Phi_2$ for $\Phi_1,\Phi_2$ of the form of the $\Phi$ entering in (<ref>).
A typical example of $\Phi$ to which Lemma <ref> applies is:
\begin{align*}
\Phi:=\widetilde{J}_{\e_n,a_n}(u,h)-\widetilde{J}_{\e_{n-1},a_{n-1}}(u,h)\quad h\in K
\end{align*}

where $\widetilde{J}$ was defined just after Corollary <ref>.

* We are able to generalize (<ref>) as follows:
Let $h$ be a given function in $K_0$ (with $K_0$ is as in Theorem <ref> and $K$ as in Section <ref>) and $U_{s,t}(h) :=h'_t\omega_t-h'_s\omega_s-\int_s^t \omega_\tau h''_\tau\dd \tau$ with $U_1(h) :=U_{0,1}(h)$.
Let $F$ be an element of $C_b^1(\R^{n+2})$, the set of continuously differentiable functions $F$ with $\dis \sup_{x\in \R^{n+2}}|F(x)|+\sum_{|\alpha|=1}\sup_{x\in \R^{n+2}}|\partial _{\alpha}F(x)|<\infty$.
Then, we have (with $\overline{J}^{\e,\lambda}_{0,1}$ is given after (<ref>))

\begin{align*}
&\left| E\left[\exp\left(-\overline{J}_{0,1}^{\e_1,\lambda}\right)F(\Phi,U_1,\omega_{t_1},\cdots,\omega_{t_n})\right]-E\left[\exp\left(-\overline{J}_{0,1}^{\e_2,\lambda}\right)F(\Phi,U_1,\omega_{t_1},\cdots,\omega_{t_n})\right]\right|\\
&\leq C\left(\sup_{x\in \R^{n+2}} |F(x)|+\max|f|\sup_{x\in \R^{n+2}}|\partial _{x_1}F(x)|+\sum_{i=2}^{n+2}\sup_{x\in \R^{n+2}}|\partial _{x_i}F(x)|\right)(\e_1^{\delta_0}+\e_2^{\delta_0}),
\end{align*}

for any $0\leq t_1<\cdots<t_n\leq 1$.

* We also have another generalization:
For $i=1,\cdots,m$, let $f_i:\R^3\to \R$ be any bounded measurable function. For $a_i\in (0,1)$ and $i=1,\cdots,m$, set

\begin{align*}
\Psi^{(i)}_{u,v;s,t}:=\int_{T_{a_i}\cap \Delta_{u,v;s,t}}f_i(\omega_\tau-\omega_{\sigma})\dd \sigma\dd \tau .
\end{align*}

Then, we have that for any given $\ell\geq 1$, there are constants $C\in (0,\infty)$ and $\widetilde{\delta} \in (0,1)$ such that (with $\overline{J}^{\e,\lambda}_{0,1}$ is given after (<ref>))

\begin{align*}
&\left|E\left[\exp\left(-\overline{J}_{0,1}^{\e_1,\lambda}\right) \prod_{i=1}^m\Psi^{(i)}_{0,1;0,1}\right]-E\left[\exp\left(-\overline{J}_{0,1}^{\e_2,\lambda}\right)\prod_{i=1}^m\Psi^{(i)}_{0,1;0,1}\right]\right|\\
&\leq C\prod_{i=1}^m \max_{x\in R^3}|f_i(x)|\left(\e_1^{\widetilde{\delta} }+\e_2^{\widetilde{\delta} }\right)
\end{align*}

for $\e_1,\e_2\in (0,1)$.
For the proof, we remark that the same argument as the proof of Lemma <ref> can be applied in view of the fact that

\begin{align*}
\dis \left|\prod_{i=1}^n (x_i+v_i)-\prod_{i=1}^n x_i\right|\leq \sum_{i=1}^n |v_i|\prod_{j\not= i} |x_j+v_j|\vee |x_j|
\end{align*}

for any $x_i, v_i \in {\mathbb R}$ ($i=1,2,\dots ,n$).

Lemma <ref> is the same as the one in [23] and the idea of the proof is based on the one in [23] (where the detailed estimation of some integrals was omitted, but we confirm that it can be provided).

§.§ Proof of Lemma <ref>

For short, we will denote $\sup_{x\in \R^3}|f(x)|$ by $\max|f|$. Also, we remark that $\rho(\e)\leq C\max|f|$ for $0<\e<\frac{1}{2}$.
From Lemma <ref>,  we have the following relation:

\begin{align}
\frac{\dd}{\dd \e}\rho(\e)&=\lambda \int_0^{1-\e}\dd sE\left[e^{-\overline{J}^{\e,\lambda}_{0,1}}\Phi\delta _0 (\omega_s-\omega_{s+\e})\right]+\left(\lambda\frac{\dd}{\dd \e}\kappa_1(\e)-\lambda^2 \frac{\dd}{\dd \e}\kappa_2(\e)\right)\rho(\e) .\label{eq:derofrho}
\end{align}

We remark that in [46] the differentiability of $\rho(\e)$ is not provided.

We decompose $\overline{J}$ as follows (see Figure <ref>):

\begin{align}
\overline{J}_{0,1}^{\e,\lambda}&=\lambda J_{0,1}^\e-\lambda \kappa_1(\e)+\lambda^2\kappa_2(\e)\notag\\
&=\lambda J_{0,s}^\e-\lambda s \kappa_1(\e)+\lambda^2 s\kappa_2(\e)\notag\\
&\quad +\lambda J_{s+\e,1}^\e-\lambda (1-s-\e) \kappa_1(\e)+\lambda^2 (1-s-\e)\kappa_2(\e)\notag\\
&\quad +\lambda J_{0,s;s+\e,1}^\e+\lambda J_{0,s;s,s+\e}^\e+\lambda J_{s,s+\e;s+\e,1}^\e-\e (\lambda \kappa_1(\e)-\lambda^2\kappa_2(\e)) \notag\\
&=\overline{J}_{0,s}^{\e,\lambda}+\overline{J}_{s+\e,1}^{\e,\lambda}+\lambda J_{0,s;s+\e,1}^{\e}+\lambda J_{0,s;s,s+\e}^\e+\lambda J_{s,s+\e;s+\e,1}^\e-\e (\lambda \kappa_1(\e)-\lambda^2\kappa_2(\e))\label{eq:jtilde}
\end{align}

and  set
\begin{align*}
\widetilde{J}_{0,1}^{s,\e,\lambda}:=\overline{J}_{0,s}^{\e,\lambda}+\overline{J}_{s+\e,1}^{\e,\lambda}+\lambda J_{0,s;s+\e,1}^{\e}.
\end{align*}

The readers have to be careful not to confuse it with the similar notation $\widetilde{J}_{\e,a}(u,h)$ given just after Corollary <ref>.

A decomposition of $J_{0,1}^\e$.

It is obvious from the definition of $\kappa_1$ in (<ref>) and $\kappa _2$ entering (<ref>) that

\begin{align}
K_{\e,\lambda}:=\e (\lambda \kappa_1(\e)-\lambda^2\kappa_2(\e)) =O(\e^{\frac{1}{2}}).\label{eq:Kelam}
\end{align}

Using the fact that $1-x\leq e^{-x}\leq 1-x+\frac{1}{2}x^2$ for $x>0$, one shows that
\begin{align}
E\left[e^{-\overline{J}_{0,1}^{\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) \Phi\right]
&=E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) \Phi\right]e^{K_{\e,\lambda}}\notag\\
&\quad -\lambda E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) (J_{0,s;s,s+\e}^{\e}+J_{s,s+\e;s+\e,1}^{\e})\Phi\right]e^{K_{\e,\lambda}}\notag\\
&\quad +R_{\lambda,s,\e},\label{eq:Jdecom}
\end{align}

where the remainder term $R_{\lambda,s,\e}$ is dominated by

\begin{align*}
&\frac{\lambda^2}{2}E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) (J_{0,s;s,s+\e}^{\e}+J_{s,s+\e;s+\e,1}^{\e})^2|\Phi|\right]e^{K_{\e,\lambda}}.
\end{align*}

Substituting this into (<ref>), we have
\begin{align}
\frac{\dd }{\dd \e}\rho(\e)&=\lambda\int_0^{1-\e}\dd s E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) \Phi\right]e^{K_{\e,\lambda}}\notag\\
&-\lambda^2 \int_0^{1-\e  }\dd s E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) (J_{0,s;s,s+\e}^{\e}+J_{s,s+\e;s+\e,1}^{\e})\Phi\right]e^{K_{\e,\lambda}}\notag\\
&\quad +\lambda \int_0^{1-\e}\dd s R_{\lambda,s,\e}+\left(\lambda\frac{\dd}{\dd \e}\kappa_1(\e)-\lambda^2 \frac{\dd}{\dd \e}\kappa_2(\e)\right)\rho(\e). \label{eq:rhoexpand}
\end{align}

We set

\begin{align*}
&I_{\e,s}^{(1)}:=E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0(\omega_s-\omega_{s+\e})\Phi\right] ,\\
&I_{\e,s}^{(2)}:=E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0(\omega_s-\omega_{s+\e})(J_{0,s;s,s+\e}^{\e}+J_{s,s+\e;s+\e,1}^{\e})\Phi\right] ,\\
\end{align*}

and will prove that there exists $\widetilde{\delta} \in (0,1)$ such that
\begin{align}
\int_0^{1-\e}I_{\e,s}^{(1)}\dd s&=-(1-\e)\rho(\e)\frac{\dd}{\dd \e}\kappa_1(\e)+\lambda(1-\e)\e\kappa_1(\e)\frac{\dd}{\dd \e}\kappa_1(\e)\rho(\e) \notag\\
&\quad +O(1)\max|f|\e^{\widetilde{\delta}-1}\label{eq:1st}\\
&=-(1-\e)(1-\e\lambda\kappa_1(\e))\rho(\e)\frac{\dd }{\dd\e}\kappa_1(\e)+O(1)\max|f|\e^{\widetilde{\delta}-1} \notag\\
\int_0^{1-\e}I_{\e,s}^{(2)}\dd s&=-\frac{\dd }{\dd \e}\kappa_2(\e)\rho(\e) +O(1)\max|f|\e^{\widetilde{\delta}-1} \label{eq:2nd}\\
\int_0^{1-\e}I_{\e,s}^{(3)}\dd s&\leq  O(1)\max|f|\e^{-\frac{1}{2}} . \label{eq:3rd}
\end{align}

(<ref>)-(<ref>) will be proven in Sections <ref>–<ref> below.
Once we obtain (<ref>)-(<ref>), we deduce from (<ref>) that

\begin{align*}
\frac{\dd }{\dd \e}\rho(\e)=
&-\lambda\left((1-\e)(1-\e\lambda\kappa_1(\e))e^{K_{\e,\lambda}}-1\right)\rho(\e)\frac{\dd }{\dd\e}\kappa_1(\e)\\
&+ \lambda^2\left(e^{K_{\e,\lambda}}-1\right)\frac{\dd }{\dd \e}\kappa_2(\e)\rho(\e)\\
&+O (1)\max|f|\e^{\widetilde{\delta}-1}e^{K_{\e,\lambda}}+		O(1)\max|f|\e^{-\frac{1}{2}}e^{K_{\e,\lambda}}.
\end{align*}

On the other hand, we find from (<ref>), (<ref>), and (<ref>) that

\begin{align*}
(1-\e)(1-\e\lambda\kappa_1(\e))e^{K_{\e,\lambda}}-1&=(1-\e)(1-\e\lambda \kappa _1(\e))\left(1+\e \left(\lambda\kappa _1(\e)-\lambda^2\kappa_2(\e)\right)+O(\e)\right)-1\\
&=O\left(\e\log \frac{1}{\e}\right)\\
\end{align*}

and hence we have

\begin{align*}
\frac{\dd }{\dd \e}\rho(\e)&= O \left(\e\log \frac{1}{\e}\right) \rho(\e)\frac{\dd}{\dd \e}\kappa_1(\e)+\lambda^2 O (1) \e^\frac{1}{2} \rho(\e)\frac{\dd }{\dd \e}\kappa_2(\e)+O(1)\max|f|\e^{-\max\{\frac{1}{2},1-\widetilde{\delta}\}}\\
\end{align*}

for some $\widehat{\delta}\in \left(0,\min\left\{\frac{1}{2},\widetilde{\delta}\right\}\right)$, where we have used (<ref>).
From this, we complete the proof of Lemma <ref>.

§.§.§ Proof of (<ref>)

The Cauchy-Schwarz inequality implies that

\begin{align*}
\left|I_{\e,s}^{(3)}\right| &\leq E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0(\omega_s-\omega_{s+\e})(J_{0,s;s,s+\e}^{\e}+J_{s,s+\e;s+\e,1}^{\e})^2|\Phi|\right]\\
&\leq 2E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) (|J_{0,s;s,s+\e}^{\e}|^2+|J_{s,s+\e;s+\e,1}^{\e}|^2)|\Phi|\right].
\end{align*}

By definition, we know that
\begin{align*}
|J_{0,s;s,s+\e}^{\e}|^2=\int_{([0,s]\times [s,s+\e])^2}\prod_{i=1}^2 1\{\tau_i-\sigma_i\geq \e\}\delta _0 (\omega_{\sigma_i}-\omega_{\tau_i})\dd \tau_i\dd \sigma_i
\end{align*}

and by using  Lemma <ref>, we obtain that

\begin{align}
&E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta _0(\omega_s-\omega_{s+\e})\delta _0 (\omega_{\sigma_1}-\omega_{\tau_1})\delta _0 (\omega_{\sigma_2}-\omega_{\tau_2})\right]\notag\\
\begin{cases}
\displaystyle C\int_{\R^3}\dd x\int_{\R^3}\dd y\int_{\R^3}\dd z p_{2\sigma_1}(x)p_{2(\sigma_2-\sigma_1)}(x,y)p_{2(s-\sigma_{2})}(y,z) \\
\hspace{7em} \times p_{2(\tau_1-s)}(z,x)p_{2(\tau_2-\tau_1)}(x,y)p_{2(s+\e-\tau_2)}(y,z)\\
\hspace{10em}\text{if }0<\sigma_1<\sigma_2<s<\tau_1<\tau_2<s+\e ; \\[2mm]
\displaystyle C\int_{\R^3}\dd x\int_{\R^3}\dd y\int_{\R^3}\dd z p_{2\sigma_1}(x)p_{2(\sigma_2-\sigma_1)}(x,y)p_{2(s-\sigma_{2})}(y,z)\\
\hspace{7em} \times p_{2(\tau_2-s)}(z,y)p_{2(\tau_1-\tau_2)}(y,x)p_{2(s+\e-\tau_1)}(x,z)\\
\hspace{10em}\text{if }0<\sigma_1<\sigma_2<s<\tau_2<\tau_1<s+\e .
\end{cases}\label{eq:elbound}
\end{align}

By the relations
\begin{align*}
&p_s(x)p_t(x)=\frac{1}{(2\pi (s+t))^{3/2}}p_{\frac{st}{s+t}}(x),\\
&\int_{\R^3}p_s(x,y)p_t(y,z)\dd y=p_{s+t}(x,z),
\end{align*}

the right-hand side in (<ref>) is given by
\begin{align*}
\intertext{for $0<\sigma_1<\sigma_2<s<\tau_1<\tau_2<s+\e$ and }
\end{align*}

for $0<\sigma_1<\sigma_2<s<\tau_2<\tau_1<s+\e$.

The transitions for the case that $0<\sigma_1<\sigma_2<s<\tau_1<\tau_2<s+\e$. When we regard it as an electrical circuit, the computation technique of the resistance helps us to compute the integrals with respect to  the space variables $x,y,z$.

Hence, we have

\begin{align*}
&E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) |J_{0,s;s,s+\e}^{\e}|^2 |\Phi|\right] \\
&\leq C \max |f| \int _{0<\sigma _1< \sigma _2 <s <\tau _1 <\tau _2 <s+\e} \dd \sigma _1 \dd \tau _1 \dd \sigma _2 \dd \tau _2\\
&\quad \hspace{4cm}[ {(\sigma_2-\sigma_1)(\tau_2-\tau_1)}({s+\e-\tau_2+s-\sigma_2})\\
&\quad \hspace{4cm}+{({\sigma_2-\sigma_1+\tau_2-\tau_1})(s+\e-\tau_2)(s-\sigma_2)}\\
&\quad \hspace{4cm}+({\sigma_2-\sigma_1+\tau_2-\tau_1})({s+\e-\tau_2+s-\sigma_2})(\tau_1-s)]^{-\frac{3}{2}}\\
&\quad + C \max |f| \int _{0<\sigma_1<\sigma_2<s<\tau_2<\tau_1<s+\e}\dd \sigma _1 \dd \tau _1 \dd \sigma _2 \dd \tau _2\\
&\quad \hspace{4cm} [ (\tau_2-\sigma_2)(\sigma_2-\sigma_1)(\tau_1-\tau_2)\\
&\quad \hspace{4cm}+(\sigma_2-\sigma_1+\tau_1-\tau_2)(s-\sigma_2)(\tau_2-s) \\
&\quad \hspace{4cm}+(\tau_2-\sigma_2)(\sigma_2-\sigma_1+\tau_1-\tau_2)(s+\e-\tau_1)]^{-\frac{3}{2}}.
\end{align*}

Changing variables $\tau_i-s=\e t_i$, $s-\sigma_i=\e s_i$ ($i=1,2$), we have

\begin{align}
&E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta _0 (\omega_s-\omega_{s+\e})|J_{0,s;s,s+\e}|^2|\Phi|\right]\notag\\
&\leq C\max |f| {\e^{-\frac{1}{2}}} \left( \iint _{0<s_2<s_1 <\infty, 0<t_1 <t_2 < 1}\dd s_1 \dd s_2 \dd t_1 \dd t_2 \right. \notag\\
&\quad \hspace{4cm} [ (s_1-s_2)(t_2-t_1)(1-t_2+s_2) +(s_1-s_2+t_2-t_1)(1-t_2)s_2 \notag\\
&\quad \hspace{8cm}+(s_1-s_2+t_2-t_1)(1-t_2+s_2)t_1]^{-\frac{3}{2}}\notag\\
&\quad + \iint _{0<s_2<s_1<\infty, 0<t_2<t_1<1} \dd s_1 \dd s_2 \dd t_1 \dd t_2 \notag\\
&\quad \hspace{4cm} [ (t_2+s_2)(s_1-s_2)(t_1-t_2)+(s_1-s_2+t_1-t_2) s_2 t_2 \notag\\
&\quad \hspace{7cm} \left. \phantom{\int}+(t_2+s_2)(s_1-s_2+t_1-t_2)(1-t_1)]^{-\frac{3}{2}} \right) \notag\\
& = C{\e^{-\frac{1}{2}}}.\label{eq:I3estimate}
\end{align}

For details see Appendix <ref>.

Similarly, we have that

\begin{align*}
E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta _0 (\omega_s-\omega_{s+\e})  |J_{s,s+\e;s+\e,1}^{\e}|^2 |\Phi|\right]\leq C{\e^{-\frac{1}{2}}}.
\end{align*}

In this way we then obtain (<ref>).

§.§.§ Proof of (<ref>)

Before starting the proof of (<ref>), we give two propositions.

We take $(s,t)$ and $u,v$ such as
0≤s≤1-,        1-≤t≤1

0≤u≤s≤v≤t,         v-u≤.

Then we have
\begin{align}
&E\left[\delta _0 (\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\Phi_{0,t} \right]\notag\\
&=p_{v-u}(0)E\left[e^{-\overline{J}_{0,t-(v-u)}^{{\e,\lambda}}-\lambda Y_{u,v}}\Phi_{0,t-(v-u)}\right]+O(1)\max |f|(v-u)^{-\frac{1}{2}},
\end{align}

\begin{align*}
&Y_{u,v}:=Y_{u,v}^{t,\e}=\iint_{\left\{\begin{smallmatrix}(u-\e)\vee 0<\sigma<u\\
u<\tau<(u+\e)\wedge (t-(v-u))\\
\end{smallmatrix}\right\}} \dd \sigma\dd\tau \delta _0 (\omega_\tau - \omega_\sigma) .
\end{align*}

The domains in the integrals  $\overline{J}_{0,u}^{\e,\lambda}$, $\overline{J}_{v,t}^{\e,\lambda}$, and $J_{0,u;v,t}^{\e}$.

The domains in the integral is shifted by the Markov property of Brownian motion.

By definition we know that
\begin{align*}
|\Phi_{0,u;u,v}|+|\Phi_{u,v;v,t}|+\left|\Phi_{u,v;u,v}\right|\leq C\max|f|(v-u).
\end{align*}

This and Lemma <ref> imply that

\begin{align*}
&E\left[\delta _0 (\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\left(|\Phi_{0,u;u,v}|+|\Phi_{u,v;v,t}|+\left|\Phi_{u,v;u,v}\right|\right)\right]\\
&\leq  C\max|f|(v-u) E\left[\delta _0 (\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}} \right] \\
&\leq  C\max|f|(v-u) \int _{\R ^3} \int _{\R ^3} \dd x \dd y g_u(x) p_{v-u}(0) g_{t-v}(y-x) \\
&\leq  C\max |f|(v-u)^{-\frac{1}{2}}.
\end{align*}

Similarly we have

\begin{align*}
&E\left[\delta _0 (\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\left|\int_{T_a\cap \left\{\begin{smallmatrix}(u-a)\vee 0<\sigma<u\\ v<\tau<(v+a)\wedge t\\
\tau-\sigma\leq v-u+a\end{smallmatrix}\right\}}f(\omega_\tau-\omega_\sigma)\dd \tau\dd \sigma\right|\right]\\
&\leq C\max |f| (v-u)^{-\frac{1}{2}}.
\end{align*}

We denote $\Theta_{a,u,v}:=\int_{T_a\cap \left\{\begin{smallmatrix}(u-a)\vee 0<\sigma<u\\ v<\tau<(v+a)\wedge t\\
\tau-\sigma\leq v-u+a\end{smallmatrix}\right\}}f(\omega_\tau-\omega_\sigma)\dd \tau\dd \sigma$. (See Figure <ref> below.)

Decomposition of the integral of $\Phi$.

This shows that

\begin{align*}
&E\left[\delta _0 (\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\Phi_{0,t} \right]\\
&= O(1)\max |f|(v-u)^{-\frac{1}{2}}\\
&\quad +E\left[e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\delta _0 (\omega_v-\omega_{u})\left(\Phi_{0,t}-\Phi_{0,u;u,v}-\Phi_{u,v;v,t}-\Phi_{u,v;u,v}-\Theta_{a,u,v}\right)\right]\\
&= O(1)\max |f|(v-u)^{-\frac{1}{2}}\\
&\quad +E\left[e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\delta _0 (\omega_v-\omega_{u})\left(\Phi_{0,u}+\Phi_{v,t}+\Phi_{0,u;v,t}-\Theta_{a,u,v}\right)\right].
\end{align*}

By the Markov property of Brownian motion and the shift invariance of $J$ we have that conditioned on $\omega_v=\omega_u$, the following equality in distribution holds.

\begin{align*}
&(\overline{J}_{0,u}^{\e,\lambda}+\overline{J}_{v,t}^{\e,\lambda}+\lambda J_{0,u;v,t}^{\e}, \Phi_{0,u},\Phi_{v,t},\Phi_{0,u;v,t}-\Theta_{a,u,v})\\
&\stackrel{d}{=}(\overline{J}_{0,t-(v-u)}^{\e,\lambda}+\lambda Y_{u,v}, \Phi_{0,u},\Phi_{u,t-(v-u)},\Phi_{0,u;u,t-(v-u)}).
\end{align*}

From this we have

\begin{align*}
&E\left[e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,t}^{\e,\lambda}-\lambda J_{0,u;v,t}^{\e}}\delta _0 (\omega_v-\omega_{u})\left(\Phi_{0,u}+\Phi_{v,t}+\Phi_{0,u;v,t}-\Theta_{a,u,v}\right)\right]\\
&=p_{v-u}(0)E\left[e^{-\overline{J}_{0,t-(v-u)}^{\e,\lambda}-\lambda Y_{u,v}} ( \Phi_{0,u}+\Phi_{u,t-(v-u)}+\Phi_{0,u;u,t-(v-u)} ) \right]\\
&=p_{v-u}(0)E\left[e^{-\overline{J}_{0,t-(v-u)}^{\e,\lambda}-\lambda Y_{u,v}} \Phi_{0,t-(v-u)} \right].
\end{align*}

From this equality and the inequalities above we obtain the desired estimate of this proposition.

There exists $\widetilde{\delta} >0$ such that for sufficiently small $\e >0$ and $r\leq 2\e$,

\begin{align*}
\left|E\left[e^{-\bar{J}_{0,1-r}^{\e ,\lambda}}\Phi\right]-E\left[e^{-\bar{J}_{0,1}^{\e ,\lambda}}\Phi\right]\right| \leq C\max|f|\e^{\widetilde{\delta} -\frac{1}{2}}.
\end{align*}

This estimate is proved similarly to <cit.>.
We omit the detailed proof.

Applying Proposition <ref> for $u=s$, $v=s+\e$, and $t=1$, we have

\begin{align*}
&=O(1)\max|f|\e^{-\frac{1}{2}}+p_\e(0)E\left[e^{-\overline{J}_{0,1-\e}^{\e,\lambda}-\lambda Y_{s,s+\e}} \Phi_{0,1-\e}
\right]\\
&=O(1)\max|f|\e^{-\frac{1}{2}} +p_\e(0)\left(E\left[e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1-\e}\right]-\lambda E\left[Y_{s,s+\e}e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1-\e}\right] \right. \\
&\hspace{8cm} \left. +O(1)\lambda^2 E\left[Y_{s,s+\e}^2e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1-\e}\right]\right)\\
&=:O(1)\max|f|\e^{-\frac{1}{2}}+p_\e(0)\left(A_{\e}^{(1)}-\lambda A_{\e,s}^{(2)}+O(1)\lambda^2 A_{\e,s}^{(3)}\right).
\end{align*}

Here, we note that $p_\e(0)=-\frac{\dd}{\dd \e}\kappa_1(\e)$.
\begin{align*}
A_{\e}^{(1)}=E\left[e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1}\right]+O(1)\max|f| \e,
\end{align*}

Lemma <ref> gives that

\begin{align*}
A_{\e}^{(1)}=\rho(\e)+O(1)\max |f|\e^{\frac{1}{2}}.
\end{align*}

By Lemma <ref>, we have

\begin{align}
A_{\e,s}^{(3)}=E\left[Y_{s,s+\e}^2e^{-\overline{J}_{0,1-\e}^{\e,\lambda}} \Phi_{0,1-\e}\right]\leq C\max|f|\e \label{eq:A3}
\end{align}

(see Appendix <ref>) and hence,

\begin{align*}
p_\e(0)\lambda^2 A_{\e,s}^{(3)}\leq C\max|f|\e^{-\frac{1}{2}}
\end{align*}

for $0\leq s\leq 1-\e$.

In the rest, we will prove that
\begin{align*}
O(1)\max |f|\e^\frac{1}{2}, &1-2\e\leq s\leq 1-\e\\
\e\left(\kappa_1(\e)+\frac{2}{(2\pi)^{\frac{3}{2}}}\right)\rho(\e)+O(1)\max|f|\e,\quad &\e\leq s\leq 1-2\e\\
\frac{2}{(\sqrt{2\pi})^3}\left(2\sqrt{s}-\frac{s}{\sqrt{\e}}\right)\rho(\e)+O(1)\max|f|\e,\quad &0\leq s\leq \e
\end{cases}
\end{align*}

which implies that

\begin{align*}
\int_0^{1-\e}A_{\e,s}^{(2)}\dd s=(1-\e)\e\kappa_1(\e)\rho(\e)+O(1)\max|f|\e,
\end{align*}

which then, completes the proof of (<ref>).

\begin{align*}
E\left[Y_{s,s+\e}e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}|\Phi_{0,1-\e}-\Phi_{0,1}|\right]\leq C\max |f|\e^{\frac{3}{2}},
\end{align*}

we have
\begin{align*}
&=E\left[\int\int_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}\dd u\dd v\delta_0(\omega_v-\omega_u)e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1}\right]+O(1)\max|f|\e^{\frac{3}{2}}\\
\end{align*}

As in (<ref>), we have

\begin{align*}
&=E\left[\int\int_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}\dd u\dd v\delta_0(\omega_v-\omega_u)e^{-\overline{J}_{0,1-\e}^{\e,\lambda}}\Phi_{0,1}\right]\\
&=E\left[\int\int_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}\dd u\dd v\delta_0(\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,1-\e}^{\e,\lambda}-\lambda J_{0,u;v,1-\e}^{\e}}\Phi_{0,1} e^{K_{\e,v-u,\lambda}} \right]\\
&\quad -O(1)\lambda E\left[\int\int_{ 0\leq u\leq s\leq v\leq 1-\e, v-u\leq \e}\dd u\dd v\delta_0(\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,1-\e}^{\e,\lambda}-\lambda J_{0,u;v,1-\e}^{\e}}\Phi_{0,1}\right. \\
&\hspace{25em} \times  (J_{0,u;u,v}^\e+J_{u,v;v,1-\e}^\e)e^{K_{\e,v-u,\lambda}} \bigg]\\
&=:A_{\e,s}^{(2,1,1)}-O(1)\lambda A_{\e,s}^{(2,1,2)},
\end{align*}

\begin{align*}
K_{\e,\theta,\lambda}:=\theta (\lambda \kappa_1(\e)-\lambda^2\kappa_2(\e)).
\end{align*}

Using the same argument as in the proof of (<ref>), one proves that

\begin{align}\label{eq:A212}
= O(1)\max |f|\e
\end{align}

for $0\leq s\leq 1-\e$ (see Appendix <ref>).

Applying Proposition <ref>, we get that

\begin{align*}
&=E\left[\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}\delta_0(\omega_{v}-\omega_{u})e^{-\overline{J}_{0,u}^{\e,\lambda}-\overline{J}_{v,1-\e}^{\e,\lambda}-\lambda J_{0,u;v,1-\e}^{\e}}\Phi_{0,1}  e^{K_{\e,v-u,\lambda}}\right]\\
&=\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}p_{v-u}(0)E\left[e^{-\overline{J}_{0,1-\e-(v-u)}^{{\e,\lambda}}-\lambda Y_{u,v}}\Phi_{0,1-\e-(v-u)}\right]e^{K_{\e,v-u,\lambda}}\dd u\dd v\\
&=\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}p_{v-u}(0)E\left[e^{-\overline{J}_{0,1-\e-(v-u)}^{{\e,\lambda}}}\Phi_{0,1-\e-(v-u)}\right]e^{K_{\e,v-u,\lambda}}\dd u\dd v\\
&-O(1)\lambda\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}p_{v-u}(0)E\left[e^{-\overline{J}_{0,1-\e-(v-u)}^{{\e,\lambda}}}\Phi_{0,1-\e-(v-u)}Y_{u,v}\right]e^{K_{\e,v-u,\lambda}}\dd u\dd v\\
&=:A_{\e,s}^{(2,1,1,1)}-O(1)\lambda A_{\e,s}^{(2,1,1,2)}
\end{align*}

By Lemma <ref>, we know that
\begin{align*}
A_{\e,s}^{(2,1,1,2)}&\leq C\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}p_{v-u}(0)E\left[e^{-\overline{J}_{0,1-\e-(v-u)}^{{\e,\lambda}}}\Phi_{0,1-\e-(v-u)}Y_{u,v}\right]\dd u\dd v\\
\end{align*}

for $0\leq s\leq 1-\e$.
\begin{align*}
\iint_{0\leq u\leq s\leq v\leq 1-\e,v-u\leq \e}p_{v-u}(0)\dd u\dd v=\begin{cases}
4\sqrt{1-\e-s}-2\frac{1-\e-s}{\sqrt{\e}},\quad &1-2\e\leq s\leq 1-\e\\
\e\kappa_1(\e)+\frac{2}{(2\pi)^{\frac{3}{2}}}\e,\quad &\e\leq s\leq 1-2\e\\
\frac{2}{(\sqrt{2\pi})^3}\left(2\sqrt{s}-\frac{s}{\sqrt{\e}}\right),\quad &0\leq s\leq \e
\end{cases}
\end{align*}

\begin{align*}
E\left[e^{-\overline{J}_{0,1-\e-(v-u)}^{{\e,\lambda}}}\left|\Phi_{0,1-\e-(v-u)}-\Phi_{0,1}\right|\right]=O(1)\max |f|\e,\quad \text{for }0<v-u<\e,
\end{align*}

Lemma <ref> yields that
\begin{align*}
O(1)\max|f|\e^\frac{1}{2}&1-2\e\leq s\leq 1-\e\\
\e\left(\kappa_1(\e)+\frac{2}{(2\pi)^{\frac{3}{2}}}\right)\rho(\e)+O(1)\max|f|\e,\quad &\e\leq s\leq 1-2\e\\
\frac{2}{(\sqrt{2\pi})^3}\left(2\sqrt{s}-\frac{s}{\sqrt{\e}}\right)\rho(\e)+O(1)\max|f|\e\quad &0\leq s\leq \e
\end{cases}
\end{align*}

\begin{align*}
O(1)\max|f|\e^\frac{1}{2},&1-2\e\leq s\leq 1-\e\\
\e\left(\kappa_1(\e)+\frac{2}{(2\pi)^{\frac{3}{2}}}\right)\rho(\e)+O(1)\max|f|\e,\quad &\e\leq s\leq 1-2\e\\
\frac{2}{(\sqrt{2\pi})^3}\left(2\sqrt{s}-\frac{s}{\sqrt{\e}}\right)\rho(\e)+O(1)\max|f|\e\quad &0\leq s\leq \e.
\end{cases}
\end{align*}

Putting things together, we obtain that
\begin{align*}
O(1)\max|f|\e^\frac{1}{2},&1-2\e\leq s\leq 1-\e\\
\e\left(\kappa_1(\e)+\frac{2}{(2\pi)^{\frac{3}{2}}}\right)\rho(\e)+O(1)\max|f|\e,\quad &\e\leq s\leq 1-2\e\\
\frac{2}{(\sqrt{2\pi})^3}\left(2\sqrt{s}-\frac{s}{\sqrt{\e}}\right)\rho(\e)+O(1)\max|f|\e \quad &0\leq s\leq \e.
\end{cases}
\end{align*}

§.§.§ Proof of (<ref>)

\begin{align*}
&I_{\e,s}^{(2,1)}:=E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) J_{0,s;s,s+\e}^{\e} \Phi\right] ,\\
&I_{\e,s}^{(2,2)}:=E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0 (\omega_s-\omega_{s+\e}) J_{0,s+\e;s+\e,1}^{\e} \Phi\right] .
\end{align*}

As in <cit.>, we choose $1>\beta>\gamma>\frac{1}{2}>\alpha>0$.

\begin{align*}
T^1(\e^\alpha)&:=\{(s,u,v):s<\e^\alpha,0<u<s<v<s+\e,v-u\geq \e\} \\
&\qquad \cup \{(s,u,v):s>1-\e^\alpha,0<u<s<v<s+\e,v-u\geq \e\} , \\
T^2(\e^\alpha,\e^\beta) &:=\{(s,u,v):\e^\alpha<s<1-\e^\alpha,0<u<s-\e^\beta<s<v<s+\e,v-u\geq \e\} ,
\end{align*}

we find that

\begin{align*}
&\int_{T^1(\e^\alpha)}\dd s\dd u\dd v E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}}\delta_0(\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) |\Phi|\right]\\
&\leq C\max|f|\int_{T^1(\e^\alpha)}\dd s\dd u\dd v \int \dd x\dd yp_{2u}(x)p_{2(s-u)}(x,y)p_{2(v-s)}(y,x)p_{2(s+\e-v)}(x,y)\\
&\leq C\max|f| \e^{\alpha-1}
\end{align*}

\begin{align*}
&\int_{T^2(\e^\alpha,\e^\beta)}\dd s\dd u \dd v E\left[ e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_v-\omega_u) |\Phi | \right] \\
&\leq C\max|f|\int_{T^2(\e^\alpha,\e^\beta)}\dd s\dd u \dd v \int \dd x\dd yp_{2u}(x)p_{2(s-u)}(x,y)p_{2(v-s)}(y,x)p_{2(s+\e-v)}(x,y)\\
&\leq C\max|f|\e^{-\frac{\beta+1}{2}}.
\end{align*}

Therefore, we have

\begin{align*}
\int_0^{1-\e}\dd sI_{\e,s}^{(2,1)} &=\int_{0}^{1-\e}\dd s\int_{u<s<v<s+\e,v-u>\e}\dd u\dd vE\left[ e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \Phi\right]\\
&=\int_{\e^\alpha}^{1-\e^\alpha}\dd s\int_{s-\e^\beta}^s\dd u\int_{s\vee (u+\e)}^{s+\e}\dd v E\left[ e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \Phi\right] \\
&\quad +\max|f|O\left(\e^{(\alpha-1)\wedge \left(-\frac{\beta+1}{2}\right)}\right).
\end{align*}

We write

\begin{align*}
T^3 (\e^\alpha,\e^\beta) &:=\{(s,u,v):\e^\alpha<s<1-\e^\alpha,s-\e^\beta<u<s<v<s+\e,v-u\geq \e\} \\
&=\{(s,u,v):\e^\alpha<s<1-\e^\alpha,s-\e^\beta<u<s\vee (u+\e)<v<s+\e\}.
\end{align*}

For $\e^\gamma\leq s\leq 1-\e^\gamma$, we set

\begin{align*}
\check{J}_{0,1}^{s,\e,\lambda} := \overline{J}_{0,s-\e^\gamma}^{\e,\lambda} +\overline{J}_{s+\e^\gamma,1}^{\e,\lambda} +\lambda J_{0,s-\e^\gamma;s+\e^\gamma,1}^{\e}.
\end{align*}

Let us recall that $1>\beta>\gamma>\frac{1}{2}>\alpha$.
We note then that the deterministic part in
$\widetilde{J}_{0,1}^{s,\e,\lambda} - \check{J}_{0,1}^{s,\e,\lambda} $ derived from $\kappa_1$ and $\kappa_2$ is of order $\e^{\gamma-\frac{1}{2}} $ and the random part comes from the intersection local time is positive and dominated by
\begin{align}
\lambda \left(J_{0,1}^\e -J_{0,s-\e^\gamma}^{\e}- J_{s+\e^\gamma,1}^\e-J^\e_{0,s-\e^\gamma;s+\e^\gamma,1}\right)&=\lambda \left(J^\e_{0,s-\e^\gamma;s-\e^\gamma,s+\e^\gamma}+J^\e_{s-\e^\gamma,s+\e^\gamma;s+\e^\gamma,1}+J_{s-\e^\gamma,s+\e^\gamma}^\e\right)\notag\\
&= \lambda \left(J^\e_{0,s+\e^\gamma;s-\e^\gamma,s+\e^\gamma}+J^\e_{s-\e^\gamma,s+\e^\gamma;s+\e^\gamma,1}\right)\label{eq:JJhatdiff}
\end{align}

Thus, we have
\begin{align*}
&\int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\widetilde{J}_{0,1}^{s,\e,\lambda}} \delta_0(\omega_s-\omega_{s+\e})\delta_{0}(\omega_u-\omega_v)\Phi\right]
\\
&=\int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(\omega_s-\omega_{s+\e})\delta_0(\omega_u-\omega_v)\Phi\right]\\
&\quad+O\left(\e^{\gamma-\frac{1}{2}}\right)\int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(\omega_s-\omega_{s+\e})\delta_0(\omega_u-\omega_v)|\Phi|\right]\\
&\quad+O(1) \int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} J_{0,s+\e^\gamma;s-\e^\gamma,s+\e^\gamma}^\e\delta_0({\omega_s-\omega_{s+\e}})\delta_0({\omega_u-\omega_v})|\Phi|\right]\\
&\quad +O(1) \int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} J_{s-\e^\gamma,s+\e^\gamma;s+\e^\gamma,1}^\e\delta_0({\omega_s-\omega_{s+\e}})\delta_0({\omega_u-\omega_v})|\Phi|\right].
\end{align*}

Moreover, we can see from Appendix <ref> that there exists $\delta\in (0,1)$ such that

\begin{align}
\int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} J^\e_{0,s+\e^\gamma;s-\e^\gamma,s+\e^\gamma}\delta_0({\omega_s-\omega_{s+\e}})\delta_0({\omega_u-\omega_v})|\Phi|\right]=O(\e^{\delta-1})\label{eq:Remainder1}\\
\int_{T^3_{s,u,v}(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[e^{-\check{J}_{0,1}^{s,\e,\lambda}} J^\e_{s-\e^\gamma,s+\e^\gamma;s-\e^\gamma,1}\delta_0({\omega_s-\omega_{s+\e}})\delta_0({\omega_u-\omega_v})|\Phi|\right]=O(\e^{\delta-1}).\label{eq:Remainder2}
\end{align}

Thus, we have

\begin{align}
\int_0^{1-\e}\dd sI_{\e,s}^{(2,1)} &=\left(1+O\left(\e^{\gamma-\frac{1}{2}}\right)\right) \int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \Phi\right] \label{eq:I3-1}\\
&\quad +\max|f|O(\e^{\widetilde{\delta}-1}). \notag
\end{align}

for some $\widetilde{\delta} >0$.

Also, we can see that

\begin{align}
&\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \Phi\right] \notag\\
&=\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right)\right] \notag\\
&\quad +\max|f|O(\e^{\gamma-1})\notag\\
&=\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v\iint \dd x\dd y E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0 (\omega_s-\omega_{s+\e}) \delta_0 (\omega_u-\omega_v) \right.\notag\\
&\hspace{7em}\left. \phantom{e^{-\check{J}_{0,1}^{s,\e,\lambda}}} \times \delta_0(x-\omega_{s-\e^\gamma})\delta_0({y}-\omega_{s+\e^\gamma}) \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right)\right] \notag\\
&\quad +\max|f|O(\e^{\gamma-1})\notag\\
&=\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \iint \dd x\dd yE\left[\left.  e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(x-\omega_{s-\e^\gamma}) \right. \right. \notag\\
&\quad \hspace{7em}\left. \left. \phantom{e^{-\check{J}_{0,1}^{s,\e,\lambda}}} \times \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right|\omega_{s+\e^\gamma}=y\right]q_{u,s,v}(x,y)\notag\\
&\quad +\max|f|O(\e^{\gamma-1}),\label{eq:I3-2}
\end{align}

\begin{align*}
q_{u,s,v}(x,y) &:=\int \dd z\int \dd w p_{u-(s-\e^\gamma)}(x,z)p_{s-u}(z,w)p_{v-s}(w,z)p_{s+\e-v}(z,w)p_{(s+\e^\gamma)-(s+\e)}(w,y).
\end{align*}

Explicit calculation implies

\begin{align*}
q_{u,s,v}(x,y)=\tau p_{2\e^\gamma-\e-(s-u)+\sigma}(x,y),
\end{align*}

\begin{align*}
\tau &:=\frac{1}{(2\pi)^3}((s+\e-v)(v-s)+(s+\e-v)(s-u)+(v-s)(s-u))^{-\frac{3}{2}},\\
\sigma&:=\frac{(s+\e-v)(v-s)(s-u)}{(s+\e-v)(v-s)+(s+\e-e)(s-u)+(v-s)(s-u)}\leq s-u.
\end{align*}

In particular, we have $ 2\e^\gamma-\e^\beta-\e\leq 2\e^\gamma-\e-(s-u)+\sigma\leq 2\e^\gamma-\e$.

\begin{align*}
r_{u,s,,v}(x,y):=q_{u,s,v}(x,y)-\tau p_{2\e^\gamma}(x,y).
\end{align*}

Then, we derive in a similar way to (<ref>)-(<ref>) that

\begin{align}
&\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \iint \dd x\dd yE\left[\left.  e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(x-\omega_{s-\e^\gamma}) \right. \right. \notag\\
&\quad \hspace{5em}\left. \left. \phantom{e^{-\check{J}_{0,1}^{s,\e,\lambda}}} \times \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right|\omega_{s+\e^\gamma}=y\right]\tau p_{2\e^\gamma}(x,y)\notag\\
&=\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \tau E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}}
\left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right] \notag\\
&=\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \tau E\left[e^{-\overline{J}_{0,1}^{\e,\lambda}}\Phi_{0,1}\right]+O\left(\e^{\gamma -\frac{3}{2}}\right)\max|f|\label{eq:l3-2}\\
&=-\frac{1}{2}\frac{\dd \kappa_2(\e)}{\dd \e}\rho(\e)+O\left(\e^{\gamma-\frac{3}{2}}\right)\max|f|,\label{eq:I3-3}
\end{align}

where we have used (<ref>) in the last line.
For the proof of (<ref>), we note that from (<ref>) we have

\begin{align*}
&E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}}    \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right] \\
&=E\left[ e^{-\overline{J}_{0,1}^{\e,\lambda}}    \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right]\\
&+O\left(E\left[ e^{-\check{J}_{0,1}^{s,\e,\lambda}} \left|\overline{J}_{0,1}^{\e,\lambda}-\check{J}_{0,1}^{s,\e,\lambda}\right|  \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right]\right)
\end{align*}

where we have used the fact that
$\overline{J}_{0,1}^{\e,\lambda}-\check{J}_{0,1}^{s,\e,\lambda}\leq 2\lambda \e^\gamma \kappa_1(\e)-\lambda^2\e^\gamma \kappa_2(\e)\leq 1$ for $\e>0$ small enough
and $\left|e^x-e^y\right|\leq e^{x+1}|x-y|$ for $x+1>y$. Then, it is easy to see that the last is $O\left(\e^{\gamma-\frac{1}{2}}\right)\max|f|$ and the first term is $E\left[ e^{-\overline{J}_{0,1}^{\e,\lambda}}    \Phi \right]+O(\e^\gamma)\max|f|$.

Also, we show that

\begin{align}
&\int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \iint \dd x\dd yE\left[\left.  e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(x-\omega_{s-\e^\gamma}) \right. \right. \notag\\
&\quad \hspace{5em}\left. \left. \phantom{e^{-\check{J}_{0,1}^{s,\e,\lambda}}} \times \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right|\omega_{s+\e^\gamma}=y\right] r_{u,s,v}(x,y)\notag\\
\end{align}

Indeed, it is easy to see that

\begin{align*}
\left|q_{u,s,v}(x,y)-\tau p_{2\e^\gamma}(x,y)\right|&\leq  C\tau \frac{s+\e-u-\sigma}{2\e^\gamma-\e-(s-u)+\sigma}p_{2\e^\gamma-\e-(s-u)+\sigma}(x,y)\\
&\quad +C\tau \frac{(s+\e-u-\sigma)|x-y|^2}{(2\e^\gamma-\e-(s-u)+\sigma)(2\e^\gamma)}p_{2\e^\gamma}(x,y) \\
&\leq C\e^{\beta-\gamma}\tau \left(1+\frac{|x-y|^2}{\e^\gamma}\right)p_{2\e^\gamma}(x,y)\\
\end{align*}

and $\check{J}_{0,1}^{s,\e,\lambda}\geq \overline{J}_{0,s-\e^\gamma}^{\e,\lambda}+\overline{J}_{s+\e^\gamma,1}^{\e,\lambda}$.
Therefore, in view of Lemma <ref> we have

\begin{align*}
&\left| \int_{T^3(\e^\alpha,\e^\beta)}\dd s\dd u\dd v \iint \dd x\dd yE\left[\left.  e^{-\check{J}_{0,1}^{s,\e,\lambda}} \delta_0(x-\omega_{s-\e^\gamma}) \right. \right. \right. \\
&\quad \hspace{4em} \left. \phantom{\int} \left. \left. \phantom{e^{-\check{J}_{0,1}^{s,\e,\lambda}}} \times \left(\Phi_{0,s-\e^\gamma}+\Phi_{s+\e^\gamma,1}+\Phi_{0,s-\e^\gamma;s+\e^\gamma,1}\right) \right|\omega_{s+\e^\gamma}=y\right] r_{u,s,v}(x,y) \right| \\
&\leq \max|f|\int_{T^3 (\e^\alpha,\e^\beta)} \dd s\dd u\dd v \\
&\quad \times \iint \dd x\dd y E\left[ \left.e^{-\overline{J}_{0,s-\e^\gamma}^{\e,\lambda} -\overline{J}_{s+\e^\gamma,1}^{\e,\lambda}} \delta_0(x-\omega_{s-\e^\gamma}) \right|\omega_{s+\e^\gamma}=y\right]\widetilde{R}_{u,s,v}(x,y)\\
&\leq C\max|f|\e^{\beta-\gamma}\int_{T^3 (\e^\alpha,\e^\beta)} \dd s\dd u\dd v  \tau  \\
&\leq O(\e^{\beta-\gamma-1})\max |f|.
\end{align*}

Thus, we obtain from (<ref>)-(<ref>) that

\begin{align*}
\int_{0}^{1-\e}\dd s I_{\e,s}^{(2,1)}=-\frac{1}{2}\frac{\dd}{\dd \e}\kappa_2(\e)\rho(\e)+\max|f|O(\e^{\widetilde{\delta}-1})
\end{align*}

for some $\widetilde{\delta} \in (0,1)$.

Similarly, we can prove that

\begin{align*}