11institutetext: Univ. Grenoble Alpes, Inserm, U1216, Grenoble Institut
Neurosciences, 38000, FR 22institutetext: Pixyl, Research and Development
Laboratory, 38000 Grenoble, FR 33institutetext: Univ. Grenoble Alpes, Inria,
CNRS, Grenoble INP, LJK, 38000 Grenoble, FR

# Robust Conformal Volume Estimation in 3D Medical Images

Benjamin Lambert 11 2 2 Florence Forbes 33 Senan Doyle 22 Michel Dojat 11

###### Abstract

Volumetry is one of the principal downstream applications of 3D medical image
segmentation, for example, to detect abnormal tissue growth or for surgery
planning. Conformal Prediction is a promising framework for uncertainty
quantification, providing calibrated predictive intervals associated with
automatic volume measurements. However, this methodology is based on the
hypothesis that calibration and test samples are exchangeable, an assumption
that is in practice often violated in medical image applications. A weighted
formulation of Conformal Prediction can be framed to mitigate this issue, but
its empirical investigation in the medical domain is still lacking. A
potential reason is that it relies on the estimation of the density ratio
between the calibration and test distributions, which is likely to be
intractable in scenarios involving high-dimensional data. To circumvent this,
we propose an efficient approach for density ratio estimation relying on the
compressed latent representations generated by the segmentation model. Our
experiments demonstrate the efficiency of our approach to reduce the coverage
error in the presence of covariate shifts, in both synthetic and real-world
settings. Our implementation is available at
https://github.com/benolmbrt/wcp_miccai.

###### Keywords:

Uncertainty Predictive Interval Covariate Shift

##  1 Introduction

An important downstream application of medical image segmentation is the
extraction of volume measurements for lesions or organs. Lesion volumetry
plays a pivotal role in various medical scenarios, including predicting the
outcome after stroke [12], grading brain tumors [5], or monitoring the
progression of Multiple Sclerosis [20]. Brain volumetry can also be useful to
monitor atrophy [9], and analyzing the volume of organs is useful for aging
studies [27]. However, automated segmentations can be error-prone, which
inevitably leads to imprecise volumetric measurements. A potential solution
would be to associate predictive intervals (PIs) with the estimations to take
into account this uncertainty.

Conformal prediction (CP) [19, 23] is an uncertainty paradigm allowing to
associate PIs with regressed scores (here, volumes). The most popular variant
of CP, Split CP [23], relies on a set-aside calibration dataset (generally a
subset of the training dataset) that is used to calibrate the intervals so
that they match the target coverage level on fresh test data. However, it is
based on the exchangeability hypothesis, following which calibration and test
data are drawn independently from the same distribution. In general, this is
not the case for medical image processing applications, where domain shifts
are extremely common, due to variations in the data acquisition protocol or
the presence of pathologies unseen during training [29]. When calibration and
test data points are not exchangeable, the accuracy of the conformal procedure
collapses drastically [4, 26], which hinders the relevancy of conformalized
PIs in medical applications.

As a potential solution, Weighted Conformal Prediction (WCP) has been proposed
to account for shifts between calibration and test distributions [4, 26]. It
is based on the reweighting of calibration samples according to the estimated
density ratio $dP_{\text{test}}/dP_{\text{train}}$. As a result, calibration
samples close to the test samples are attributed with higher importance in the
conformal procedure. A flourishing literature can be found for density ratio
estimation, with popular approaches including the training of a classifier to
distinguish between training and test distributions [6, 2], moment [14] or
ratio matching [15]. More recently, Deep Learning (DL) approaches are also
investigated to estimate density ratios [10, 22]. However, we note that
applications of WCP to medical image segmentation are still lacking, which may
be due to the difficulty of estimating the density ratio for high-dimensional
imaging data.

In this work, we propose to investigate the use of WCP to tackle covariate
shifts in medical image segmentation tasks, with the ultimate goal of
computing calibrated PIs for lesion volumes. As a contribution, we propose an
efficient way of computing the density ratio in high-dimensional medical
images, by relying on latent representations generated by the segmentation
model.

##  2 Conformal Prediction for volumetry in medical images

###  2.1 Problem definition

We consider a 3D segmentation problem with $N$ classes where our objective is
to estimate the true volumes $Y\in\mathbb{R}^{N-1}$ of each foreground class
based on the predicted segmentation. Within this framework, for an estimation
$X$ of the volume, we define a predictive interval $\Gamma_{\alpha}(X)$ as a
range of values that are constructed to contain the true volume $Y$ with a
user-defined degree of confidence $1-\alpha$ (e.g $90\%$ or $95\%$). More
formally, given a set of estimated volumes $X_{1}\ldots X_{n}$ and their
corresponding ground truth volumes $Y_{1}\ldots Y_{n}$,
$\Gamma_{\alpha}(\cdot)$ should be learned such that it satisfies [1]:

$1-\alpha\leq P(Y_{\text{test}}\in\Gamma_{\alpha}(X_{\text{test}}))\leq
1-\alpha+\frac{1}{n+1}$ (1)

###  2.2 Predictive Interval computation using a multi-head segmentation
architecture

In practice, a PI associated with a volume $X_{i}$ is composed of a lower
bound $l_{i}$, and an upper bound $u_{i}$. For a practical estimation of these
three quantities ($X_{i}$, $l_{i}$ and $u_{i}$), [18] proposed to train a
multi-head segmentation network that predicts 3 output masks: a restrictive
one (low recall, high precision) to estimate the lower bound, a permissive one
(high recall, low precision) to estimate the upper bound, and a balanced one
for the estimation of the mean (Figure 1). The key element is to perform
training using the Tversky loss $T_{\alpha,\beta}$ [24] allowing to control
the penalties applied to false positives (FP) and negatives (FN) contained in
each mask through the loss parameters $\alpha$ and $\beta$, respectively.
Writing $p_{\textit{lower}}$, $p_{\textit{mean}}$ and $p_{\textit{upper}}$ the
outputs of each head and $y$ the ground-truth segmentation, the loss is
defined:

$\mathcal{L}=T_{1-\gamma,\gamma}(p_{\textit{lower},y})+T_{0.5,0.5}(p_{\textit{mean},y})+T_{\gamma,1-\gamma}(p_{\textit{upper},y})$
(2)

where $\gamma$ is a hyperparameter set to $0.2$ controlling the penalties
applied to FP and FN during the training of the lower and upper bound heads.

Figure 1: Illustration of the proposed framework. A multi-head segmentation
model predicts three distinct masks for each label: a restrictive mask
associated with the lower bound volume (red), a permissive mask associated
with the upper bound volume (blue), and a balanced mask for the average volume
(green). For Weighted Conformal Prediction, a compressed latent representation
is extracted from the penultimate convolution filter.

To ensure that the computed PIs will achieve the user-defined level of
coverage on test data, the conformal calibration of intervals can be performed
[1]. It operates by first defining a score function
$s_{i}=\max(l_{i}-Y_{i},Y_{i}-u_{i})$. This score is a way to estimate the
accuracy of the interval $[l_{i},u_{i}]$ for the true quantity $Y_{i}$, with
larger scores indicating larger discrepancy. The scores are computed on a set-
aside calibration dataset comprising $n$ pairs of images and associated ground
truths. It allows to compute the $\frac{\lceil(n+1)(1-\alpha)\rceil}{n}$-th
quantile of the empiral scores:
$\hat{q}=\text{Quantile}(s_{1},s_{2},...,s_{n};\frac{\lceil(n+1)(1-\alpha)\rceil}{n})$.
In practice, $\hat{q}$ acts as a corrective factor applied to the PIs so that
they encompass the desired fraction of the true volumes on the calibration
dataset. At test time, the calibrated PI is computed as follows:

$\Gamma_{\alpha}(X_{i})=[l_{i}-\hat{q},u_{i}+\hat{q}]$ (3)

As a result, the intervals expand as $\hat{q}$ increases. Supposing the test
samples are exchangeable with the calibration samples, the marginal coverage
property (Equation 1) is guaranteed.

###  2.3 Weighted Conformal Prediction to tackle covariate shift

WCP has been proposed to take into account the non-exchangeability of
calibration and test data [1, 4, 26]. The core concept of WCP is to reweight
the calibration dataset to more accurately match the test one. This is
achieved by estimating the density ratio
$w=dP_{\text{test}}/dP_{\text{train}}$ for each calibration and test sample.
In practice, writing $X_{1},...,X_{n}$ the $n$ calibration samples and $x$ the
fresh test point, importance weights are computed as:

$\displaystyle p_{i}^{w}(x)$
$\displaystyle=\frac{w(X_{i})}{\sum_{i=1}^{N}w(X_{j})+w(x)}$ (4)

Essentially, the weight is large when the calibration sample $X_{i}$ is likely
under the test distribution. Then, the corrective value $\hat{q}$ can be
reframed as the $1-\alpha$ quantile of the reweighted distribution [1]:

$\hat{q}(x)=\inf\biggl{\\{}s_{j}:\sum_{j=1}^{n}p_{i}^{w}(x)\mathbbm{1}\\{s_{i}\leq
s_{j}\\}\geq 1-\alpha\biggr{\\}}$ (5)

Note that when all weights are equal to $\frac{1}{n+1}$, the standard CP
procedure is recovered. A convenient way to estimate this ratio is to use an
auxiliary classifier that only requires that unlabeled samples from the test
distribution are available during the calibration step [26]. The idea is to
train a probabilistic classification model to classify samples between the
training and test distributions. That is, writing $X_{1},...,X_{n}$ and
$X_{n+1},...,X_{n+m}$ the training and test data points, one can form a
classification dataset composed of the pairs $\\{X_{i},C_{i}\\}$ where
$C_{i}=0$ for $i=1,...,n$ and $C_{i}=1$ for $i=n+1,...,n+m$. Writing
$\widehat{p}(x)=\mathcal{P}(C=1|X=x)$ the probability predicted by a
classifier model trained on the $\\{X_{i},C_{i}\\}$ dataset that the input
sample $x$ belongs to the test distribution, the weight function can be
expressed as [25]:

$\widehat{w}(x)=\frac{\widehat{p}(x)}{1-\widehat{p}(x)}$ (6)

However, this approach has several limitations. First, it requires access to a
sufficient amount of calibration and test samples to allow for a supervised
classification strategy. Second, training the classifier is cumbersome when
dealing with high-dimensional medical images. In this setting, the dedicated
classification approach would be the training of a deep learning Convolution
Neural Network (CNN), requiring numerous examples of both classes (calibration
and test). Moreover, the training of the auxiliary classifier should be
performed during the CP procedure to allow for weight estimation.
Incorporating the training of a CNN in the CP procedure is thus highly
inefficient. As a conclusion, this classification task is computationally too
costly when dealing with 3D medical images. Building on these limitations, we
next investigate a more efficient approach making use of the latent
representations extracted by the deployed segmentation model.

###  2.4 Efficient density ratio estimation using latent representations

As training the auxiliary classifier directly from the input images is too
costly, more efficient approaches have to be investigated. One idea would be
to use a compressed representation of the input image that still preserves
important structural information. A lead in this direction is the use of low-
dimensional latent representations generated by the segmentation model during
the inference process, which has been proven to be a highly efficient summary
allowing the detection of out-of-distribution images [7, 13, 3, 28].
Therefore, using compressed latent representations in place of the high-
dimensional 3D images seems promising as our end goal is to address covariate
shifts. To test this framework, we collect the activations of the penultimate
convolution layer. The feature maps have a shape of $K\times H\times W\times
D$, where $H$, $W$, and $D$ are the spatial dimensions of the 3D image and $K$
the number of kernels in the layer. This feature map is reduced to a
compressed vector $z$ of dimension $K$ by performing an averaging over the
spatial dimensions (see Figure 1). This approach allows training the auxiliary
classifier on compressed representations of the input MRIs, which can be
performed efficiently during the WCP procedure.

##  3 Experiments

###  3.1 Synthetic dataset with controlled covariate shift

To prove the relevancy of the proposed approach, we first rely on a synthetic
setting allowing us to control covariate shift precisely. The task that we
propose here is the segmentation of spheres inside cubic volumes of shape
$32\times 32\times 32$, with the end goal of computing a PI for the volume of
each sphere. The covariate of interest here is the signal-to-noise ratio (SNR)
between the background of the image and the foreground spheres. A total of
$4000$ synthetic images are generated. We then split this dataset into an in-
distribution (ID) split (3000 images) containing images with high SNRs, and a
shifted test dataset (1000 images) containing images with lower SNRs. The ID
dataset is further split into training, calibration, and ID test parts, with
1000 images each. Several examples of synthetic images with varying SNRs are
presented in Figure 2, along with the densities of SNR in the ID and shifted
datasets.

Figure 2: Left: Examples of synthetic images with varying Signal-to-Noise
ratios (SNRs) and associated ground truths. Right: Distribution of SNRs in the
in-distribution and shifted synthetic datasets.

###  3.2 Real-world covariate shift in brain tumor segmentation tasks

To test the framework on real-world medical image data, we address multi-class
tumor segmentation in brain MRI. Our dataset consists of glioblastoma and
meningioma subjects gathered from the open-source BraTS 2023 datasets [17,
21]. Each subject has four MRI sequences: T1-weighted, T2-weighted, FLAIR, and
T1 with contrast enhancement. Ground truth masks include necrosis, edematous,
and gadolinium-enhancing tumor classes. For covariate shift analysis, we
divide subjects into an ID dataset (320 glioblastoma subjects, 748 meningioma
subjects, $30\%$-$70\%$ repartition) and a shifted test dataset (873
glioblastoma subjects, 196 meningioma subjects, $82\%-18\%$ repartition). The
covariate shift thus corresponds to the difference in frequencies of each
subtype of tumor in the ID and shifted datasets. The ID dataset is further
divided into training (568), calibration (250), and ID test (250) subsets.

###  3.3 Experimental Protocol and Metrics

We use MONAI’s [8] Dynamic U-Net [11] as segmentation backbone, modified to
have three output heads. The penultimate convolution layer contains $64$
kernels, meaning that the extracted latent representations will also have a
dimension of $64$. The models are trained using Equation 2 and the ADAM
optimizer [16] with a learning rate of $2\times 10^{-4}$. After training, PIs
are calibrated on the calibration dataset, with a target coverage of $95\%$
for the synthetic task, and $90\%$ for brain tumors as the segmentation is
more challenging. Three variants of CP are further compared:

  * •

Standard CP corresponds to the setting where calibration samples are
associated with identical weights, thus not taking into account potential
covariate shifts.

  * •

Weighted CP using Oracle covariates (W-Oracle) uses the ground truth
covariates to train the auxiliary classifier: SNR for the synthetic images,
tumor subtype for brain tumors (0 for glioblastoma, 1 for meningioma).

  * •

Weighted CP using latent representations (W-Latent) leverages the compressed
latent representations to train the auxiliary classifier.

For W-Oracle and W-Latent, we use a Logistic Regression (LR) model as the
auxiliary classifier, trained in a 20-fold cross-validation setting. The
probabilities predicted by LR are clipped in the range $[0.01,0.99]$ to avoid
infinite weights (see Equation 6). To estimate the performance of the CP
procedures, we monitor the empirical coverage on the test datasets (ID and
shifted) as well as the average interval width. We also report the
segmentation performance using Dice scores, and the accuracy of the auxiliary
LR classifier for W-Oracle and W-Latent. The experiments are reproduced for
$R=250$ trials by shuffling the ID calibration and test datasets. The shifted
test dataset is kept identical in each trial.

##  4 Results and Discussion

Tables 1 and 2 present the performance of each CP variant on the synthetic and
brain tumor datasets, respectively. In the absence of covariate shifts (ID
datasets), W-Oracle and W-Latent closely mimic Standard CP, achieving target
coverages with great accuracy ($95\%$ for synthetic data, $90\%$ for brain
tumors). However, in Shift datasets, Standard CP exhibits miscoverage, with
empirical coverages lower than the target level, revealing its inability to
handle non-exchangeable data points. W-Oracle and W-Latent alleviate this
issue, with W-Oracle recovering the exact target coverages on the synthetic
task and the necrosis and edematous brain tumor classes. W-Latent also reduces
the coverage gap, although it doesn’t exactly recover the target coverages. It
can be noticed that this increase robustness is linked with an increase in the
average interval width to achieve the target coverage on shifted test data.

A deeper dive into the functioning of WCP is presented in Figure 3. It
presents the calibration weights’ behavior with and without covariate shifts.
When there are no shifts, all weights are close to the unit, mimicking the
standard CP procedure. When a covariate shift is observed, higher weights are
assigned to calibration samples resembling test samples. For the synthetic
task, higher weights are attributed to calibration weights with low SNRs,
which are similar to the shifted test samples. For tumor segmentation, higher
weights are attributed to glioblastoma subjects, which indeed represent the
majority of the shifted test subjects. W-Oracle and W-Latent provide similar
trends, although W-Latent is more noisy than the Oracle version.

In conclusion, our WCP framework is effective in tackling covariate shifts in
medical image analysis, by addressing covariate shifts either directly or
through latent representations, ensuring the robustness of predictive
intervals. However, one limitation of the presented WCP framework is that it
can only account for moderate covariate shifts. Otherwise, if the covariate
shift is too important between calibration and test samples, the weights will
likely diverge (see Equation 6 when $\hat{p}(x)$ converges to 1) which would
undermine the accuracy of the WCP procedure.

(a) Synthetic dataset

(b) Brain tumor dataset

Figure 3: Weights of calibration samples estimated by W-Oracle and W-Latent, with and without covariate shift, according to the value of the covariate. Table 1: Comparison of standard and weighted Conformal Prediction on the synthetic task, for a target coverage of $95\%$. The mean and standard deviation over 250 trials are presented. Setting | CP version | Accuracy |  Coverage ($\%$) |  Width ($\text{\,}{\mathrm{mm}}^{3}$) | Dice
---|---|---|---|---|---
Calib ID | Standard | - | $95.11\pm 0.93$ | $86.18\pm 3.00$ |
vs. | W-Oracle | $0.50\pm 0.02$ | $95.01\pm 1.01$ | $86.06\pm 2.95$ | $0.92\pm 0.07$
Test ID | W-Latent | $0.50\pm 0.01$ | $95.03\pm 0.93$ | $86.45\pm 3.79$ |
Calib ID | Standard | - | $87.47\pm 1.08$ | $94.76\pm 2.77$ |
vs. | W-Oracle | $0.73\pm 0.01$ | $95.22\pm 1.57$ | $153.28\pm 23.52$ | $0.89\pm 0.11$
Test Shift | W-Latent | $0.72\pm 0.01$ | $93.39\pm 0.90$ | $128.89\pm 11.34$ |
Table 2: Comparison of standard and weighted Conformal Prediction on multi-class tumor volume estimation for a target coverage of $90\%$. The mean and standard deviation over 250 trials are presented. Class | Setting | CP version | Accuracy |  Coverage ($\%$) |  Width ($\text{\,}\mathrm{ml}$) | Dice
---|---|---|---|---|---|---
| Calib ID | Standard | - | $90.40\pm 2.62$ | $4.1\pm 0.7$ |
| vs. | W-Oracle | $0.50\pm 0.04$ | $90.09\pm 2.67$ | $3.9\pm 0.7$ | $0.63\pm 0.33$
Necrosis | Test ID | W-Latent | $0.50\pm 0.03$ | $90.20\pm 2.61$ | $4.0\pm 0.7$ |
| Calib ID | Standard | - | $80.19\pm 2.28$ | $6.2\pm 1.0$ |
| vs. | W-Oracle | $0.70\pm 0.01$ | $89.53\pm 2.45$ | $13.0\pm 2.6$ | $0.71\pm 0.28$
| Test Shift | W-Latent | $0.81\pm 0.00$ | $88.88\pm 2.88$ | $12.4\pm 3.0$ |
| Calib ID | Standard | - | $90.48\pm 2.77$ | $18.9\pm 1.8$ |
| vs. | W-Oracle | $0.50\pm 0.04$ | $90.19\pm 2.85$ | $18.6\pm 1.9$ | $0.81\pm 0.22$
Edematous | Test ID | W-Latent | $0.50\pm 0.03$ | $90.29\pm 2.81$ | $18.7\pm 2.0$ |
| Calib ID | Standard | - | $80.56\pm 2.35$ | $26.2\pm 2.2$ |
| vs. | W-Oracle | $0.70\pm 0.01$ | $89.58\pm 2.44$ | $39.7\pm 5.4$ | $0.80\pm 0.20$
| Test Shift | W-Latent | $0.81\pm 0.00$ | $85.52\pm 3.91$ | $32.9\pm 6.0$ |
| Calib ID | Standard | - | $90.54\pm 2.57$ | $5.0\pm 0.4$ |
| vs. | W-Oracle | $0.50\pm 0.04$ | $90.29\pm 2.64$ | $4.9\pm 0.4$ | $0.88\pm 0.20$
GD-enhancing | Test Shift | W-Latent | $0.50\pm 0.03$ | $90.36\pm 2.55$ | $4.9\pm 0.4$ |
| Calib ID | Standard | - | $81.29\pm 1.76$ | $7.0\pm 0.4$ |
| vs. | W-Oracle | $0.70\pm 0.01$ | $87.25\pm 3.39$ | $9.0\pm 1.5$ | $0.85\pm 0.18$
| Test Shift | W-Latent | $0.81\pm 0.00$ | $86.19\pm 3.90$ | $8.6\pm 1.6$ |

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######  Theorem 4.13 (Convergence in probability in supremum norm).

For any $T>0,\ \eta>0,\ N>0$, and $M>0$,

$\displaystyle\lim_{\epsilon\to 0}\sup_{\|u^{0}\|_{\mathrm{C}([0,1])}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\|u_{\epsilon,\phi}-u_{0,\phi}\|_{\mathrm{C}([0,T]\times[0,1])}>\eta\bigg{)}=0.$
(4.21)

###### Proof.

In Lemma 4.12, we have shown that for any $\|u^{0}\|_{\mathrm{C}([0,1])}\leq
N,$ the family

$\mathopen{}\mathclose{{}\left\\{u_{\epsilon,\phi}-u_{0,\phi}:\phi\in\mathscr{P}_{2}^{M},\epsilon\in(0,1]}\right\\},$

is tight in $\mathrm{C}([0,T]\times[0,1])$. Therefore, by applying Prokhorov’s
Theorem, we get that for any sequences $\epsilon_{m}\rightarrow
0,\phi_{m}\in\mathscr{P}_{2}^{M}$ and $\|u^{0}_{m}\|_{\mathrm{L}^{p}}\leq N$,
there exists a subsequence (relabeled $(\epsilon_{m},\phi_{m},u^{0}_{m})$)
such that $u_{\epsilon_{m},\phi_{m}}-u_{0,\phi}$ converges in distribution.
Let us denote the limit function by $u^{*}$. Since we have the embedding
$\mathrm{C}([0,T]\times[0,1])\hookrightarrow\mathrm{C}([0,T],\mathrm{L}^{p}([0,1])),$
the above convergence implies $u_{\epsilon_{m},\phi_{m}}-u_{0,\phi}\Rightarrow
u^{*}$ in the $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ topology also.
Therefore, Theorem 4.11 yields

$\sup_{t\in[0,T]}\|u^{*}(t)\|_{\mathrm{L}^{p}}^{p}=0$

with probability $1$ and it implies that
$\|u^{*}\|_{\mathrm{C}([0,T]\times[0,1])}=0$ with probability one. Since
$\|u_{\epsilon,\phi}-u_{0,\phi}\|_{\mathrm{C}([0,T]\times[0,1])}$ is
converging to zero which is a constant (non-random), therefore

$\lim_{m\rightarrow\infty}\mathbb{P}\mathopen{}\mathclose{{}\left(\|u_{\epsilon_{m},\phi_{m}}-u_{0,\phi}\|_{\mathrm{C}([0,T]\times[0,1])}>\eta}\right)=0.$

Due to the arbitrariness of $\epsilon_{m}\downarrow
0,\|u^{0}_{m}\|_{\mathrm{C}([0,1])}\leq R$ and
$\phi_{m}\in\mathscr{P}_{2}^{M}$, we obtain the required result (4.21). ∎

###### Proof of Theorem 4.8.

: Combining Theorems 4.6 and 4.13 yield the required result. ∎

##  5\. Bounded Non-linearity with Space-time White Noise

In this section, we define the noise as the derivatives of Brownian sheet
$\frac{\partial^{2}\mathrm{W}(\cdot,\cdot)}{\partial t\partial
x}:=\mathcal{W}(\cdot,\cdot)$ on the probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$, that is,
space-time white noise (see Subsection 1.2). Note that the growth condition of
the coefficient $g$ was linear in the previous sections, but we assume that
the coefficient $g$ is of bounded nonlinearity in this section. Moreover, we
prove our main results of this section for the case $p\geq 2\delta+1$. Let us
start by introducing the assumptions on the noise coefficient $g$:

######  Hypothesis 5.1.

The function $g:[0,T]\times[0,1]\times\mathbb{R}\to\mathbb{R}$ is a measurable
function satisfying

$\displaystyle|g(t,x,r)|\leq K,\ \ \text{ and }\ \ |g(t,x,r)-g(t,x,s)|\leq
L|r-s|,$ (5.22)

for all $t\in[0,T],\ x\in[0,1]$ and $r,s,\in\mathbb{R}$, where $K,L$ are some
positive constants.

For the basic definitions of EULP and ULDP, we refer Section 4. Let us comment
on the solvability under the new Hypothesis 5.1. In this part we are not using
the global solvability result obtained in Section 2, since the noise is space-
time white noise. For this part, we recall the following solvability result
from [20] under Hypothesis 5.1.

######  Theorem 5.2 (Theorem 3.5, [20]).

Let us assume that the initial data $u^{0}\in\mathrm{L}^{p}([0,1])$, for
$p\geq 2\delta+1$. Then, there exists a unique $\mathrm{L}^{p}([0,1])$-valued
$\mathscr{F}_{t}$-adapted continuous process $u_{\epsilon}$, for
$\epsilon\in(0,1]$ satisfying (1.2) such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C(T).$ (5.23)

Our aim is to prove ULDP under the new Hypothesis 5.1 on the noise
coefficient.

Let us define
$\displaystyle\zeta_{\epsilon,\phi}(t,x):=\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi}(s,y))\mathrm{W}(\mathrm{d}s,\mathrm{d}y)$,
and set

$\displaystyle z_{\epsilon,\phi}:=u_{\epsilon,\phi}-\zeta_{\epsilon,\phi}.$
(5.24)

In order to prove that $u_{\epsilon,\phi}(\cdot)$ is bounded in probability
uniformly in $\epsilon,u^{0}$ and $\phi$, it is enough to show that both
$z_{\epsilon,\phi}(\cdot)$ and $\zeta_{\epsilon,\phi}(\cdot)$ are bounded in
probability uniformly in $\epsilon,u^{0}$ and $\phi$. We start with a result
which gives the required bound for $\zeta_{\epsilon,\phi}(\cdot)$.

Using Hypothesis 5.1, Corollary 4.3, [15] for every $p\geq 1,\ T>0$, there
exists a constant $C$ such that the following holds:

$\displaystyle\mathbb{E}\big{[}|\zeta(t,x)-\zeta(s,y)|^{2p}\big{]}\leq
C\big{(}|t-s|^{\frac{1}{4}}+|x-y|^{\frac{1}{2}}\big{)}^{2p},\ \ \text{ for all
}\ \ s,t\in[0,T],\ \ x,y\in[0,1].$

Hence by Kolmogorov’s continuity theorem (see [14, 38]), we have

$\displaystyle\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\epsilon\in(0,1]}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{E}\bigg{[}\sup_{(t,x)\in[0,T]\times[0,1]}|\zeta_{\epsilon,\phi}(t,x)|^{p}\bigg{]}<\infty.$
(5.25)

Define

$\displaystyle\zeta^{*}_{\epsilon,\phi}:=\sup_{(t,x)\in[0,T]\times[0,1]}|\zeta_{\epsilon,\phi}(t,x)|,$
(5.26)

then, applying Markov’s inequality yields $\zeta^{*}_{\epsilon,\phi}$ is
bounded in probability uniformly in $\epsilon,u^{0}$ and $\phi$. Let us now
prove that $z_{\epsilon,\phi}(\cdot)$ is bounded in probability uniformly in
$\epsilon,u^{0}$ and $\phi$.

######  Lemma 5.3.

The random field
$\sup\limits_{t\in[0,T]}\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}$ is bounded
in probability, that is, for any given $T>0$, $M>0$, and $N>0$,

$\displaystyle\lim_{R\to\infty}\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\mathbb{P}\bigg{[}\sup_{t\in[0,T]}\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}>R\bigg{]}=0,$
(5.27)

for all $p\geq 2\delta+1$.

Before embarking to the proof, let us provide a useful result which will be
used in the proof.

######  Remark 5.4.

Let us take $u\in\mathrm{C}_{0}^{\infty}([0,1])$. Then
$|u|^{p/2}\in\mathrm{H}_{0}^{1}([0,1])$, for all $p\in[2,\infty]$, and from
the Sobolev embedding, we have
$\mathrm{H}_{0}^{1}([0,1])\hookrightarrow\mathrm{L}^{p}([0,1])$, for all
$p\in[2,\infty]$, we obtain

$\displaystyle\|u\|^{q}_{\mathrm{L}^{pq}}=\||u|^{\frac{q}{2}}\|_{\mathrm{L}^{2p}}^{2}\leq
C\int_{0}^{1}|\partial_{x}|u(x)|^{\frac{q}{2}}|^{2}\mathrm{d}x.$

But it can be easily seen that $|\partial_{x}|u|^{\frac{q}{2}}|^{2}\leq
C_{q}|u|^{q-2}|\partial_{x}u|^{2}$. Therefore we have

$\displaystyle\|u\|^{q}_{\mathrm{L}^{pq}}\leq
C_{q}\int_{0}^{1}|\partial_{x}u(x)|^{2}|u(x)|^{q-2}\mathrm{d}x,$

for any $p\in[2,\infty]$. In particular, we have

$\displaystyle\int_{0}^{t}\|u(s)\|_{\mathrm{L}^{\infty}}^{q}\mathrm{d}s\leq
C_{q}\int_{0}^{t}\||u(s)|^{\frac{q-2}{2}}\partial_{x}u(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s,$
(5.28)

for all $t\in[0,T]$.

###### Proof of Lemma 5.3.

Note that the process $z_{\epsilon,\phi}(\cdot)$ satisfies the following
system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
z_{\epsilon,\phi}}{\partial
t}(t,x)&=\nu\frac{\partial^{2}z_{\epsilon,\phi}}{\partial
x^{2}}(t,x)+\beta\mathfrak{c}((z_{\epsilon,\phi}+\zeta_{\epsilon,\phi})(t,x))-\frac{\alpha}{\delta+1}\frac{\partial}{\partial
x}\mathfrak{p}((z_{\epsilon,\phi}+\zeta_{\epsilon,\phi})(t,x))\\\
&\quad+g(t,x,(z_{\epsilon,\phi}+\zeta_{\epsilon,\phi})(t,x))\phi(t,x),\\\
z_{\epsilon,\phi}(t,0)&=z_{\epsilon,\phi}(t,1)=0,\ \ t\in(0,T],\\\
z_{\epsilon,\phi}(0,x)&=u^{0}(x),\ \ x\in[0,1].\end{aligned}}\right.$ (5.29)

One can follow similar arguments in the proof of Proposition 3.4, [20] to
obtain the $\mathrm{L}^{p}$-energy estimate. In fact, the term

$\displaystyle
I(t):=\int_{0}^{t}\big{(}g(s,(z_{\epsilon,\phi}+\zeta_{\epsilon,\phi})(s))\phi(s),|z_{\epsilon,\phi}(s)|^{q-2}z(s)\big{)}\mathrm{d}s,$
(5.30)

while taking the inner product with
$|z_{\epsilon,\phi}|^{q-2}z_{\epsilon,\phi}$ in (5.29), can be handled in the
following way:

$\displaystyle|I(t)|$ $\displaystyle\leq
K\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2(q-1)}}^{q-1}\mathrm{d}s$
$\displaystyle\leq
K\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}^{\frac{q(2\delta+2-q)}{4\delta}}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q+2\delta}}^{\frac{(q-2)(2\delta+q)}{4\delta}}\mathrm{d}s$
$\displaystyle\leq\frac{p\beta}{16}\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q+2\delta}}^{q+2\delta}\mathrm{d}s+C\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{\frac{4\delta}{4\delta-q+2}}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}^{\frac{q(2\delta+2-q)}{4\delta-q+2}}\mathrm{d}s$
$\displaystyle\leq\frac{p\beta}{16}\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q+2\delta}}^{q+2\delta}\mathrm{d}s+C\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s+C\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}^{q}\mathrm{d}s,$
(5.31)

provided $2\leq q\leq 2\delta+2$, where we have used the uniform bound for
$g$, interpolation, Hölder’s and Young’s inequalities. Now taking the inner
product with $|z_{\epsilon,\phi}|^{q-2}z_{\epsilon,\phi}$ and performing
calculations similar to the proof of Proposition 3.4, [20] and (5), we deduce
for all $t\in[0,T]$ and $q\in[2,2\delta+2]$

$\displaystyle\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{q}}^{q}+\frac{\nu
q(q-1)}{2}\int_{0}^{t}\||z_{\epsilon,\phi}(s)|^{\frac{q-2}{2}}\partial_{x}z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s+\beta
p\gamma\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}^{q}\mathrm{d}s$
$\displaystyle\quad+\frac{\beta
q}{8}\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q+2\delta}}^{q+2\delta}\mathrm{d}s$
$\displaystyle\leq
e^{CT}\bigg{\\{}\|u^{0}\|_{\mathrm{L}^{q}}^{q}+CT+CT\sup_{t\in[0,T]}\|\zeta_{\epsilon,\phi}(t)\|_{\mathrm{L}^{q(\delta+1)}}^{q(\delta+1)}+CT\sup_{t\in[0,T]}\|\zeta_{\epsilon,\phi}(t)\|_{\mathrm{L}^{q+2\delta}}^{q+2\delta}$
$\displaystyle\qquad+C\int_{0}^{T}\|\phi(t)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}t\bigg{\\}}=:\mathcal{K}(u^{0},\phi,\zeta^{*}_{\epsilon,\phi},T),$
(5.32)

where $C$ is a positive constant.

From (5.28), one can easily deduce that

$\displaystyle\sup_{s\in[0,T]}\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{q}}^{q}+\int_{0}^{T}\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{\infty}}^{q}\mathrm{d}t\leq\mathcal{K}(u^{0},\phi,\zeta^{*}_{\epsilon,\phi},T),$
(5.33)

for all $q\in[2,2\delta+2].$

Let us now consider the case $q=2\delta+3$. We just need to estimate the term
$I(t)$ given in (5.30) in a proper way. Using Hölder’s and Young’s
inequalities and (5.33), we estimate

$\displaystyle|I(t)|$ $\displaystyle\leq
K\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}\||z_{\epsilon,\phi}(s)|^{2\delta+1}z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2}}\mathrm{d}s$
$\displaystyle\leq
K\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{\infty}}^{\delta+1}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2(\delta+1)}}^{\delta+1}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{\infty}}^{2(\delta+1)}\mathrm{d}s+C\sup_{s\in[0,t]}\|z_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2(\delta+1)}}^{2(\delta+1)}\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s,$
(5.34)

for all $t\in[0,T]$. Therefore, (5) holds true for all $q\in[2,2\delta+3]$,
and a similar calculation yields for $q=2\delta+4$. Continuing like this, one
can show that (5) holds true for all $q\in[2,\infty)$.

Since, $\zeta^{*}_{\epsilon,\phi}(\cdot)$ is bounded in probability, an
application of Markov’s inequality results that
$\sup\limits_{t\in[0,T]}\|z_{\epsilon,\phi}(t)\|_{\mathrm{L}^{q}}$, for
$q\in[2,\infty)$ is also uniformly bounded in probability. ∎

Combining (5.25), (5.26) and Lemma 5.3, one can establish Corollary 4.10, that
is, the random field
$\sup\limits_{t\in[0,T]}\|u_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}$ is bounded
in probability for $p\geq 2\delta+1$.

In order to prove our main Theorem 4.7 under Hypothesis 5.1, it is sufficient
to verify Condition 4.4. Now, for the verification of Condition 4.4, we need
to prove Theorem 4.11 under Hypothesis 5.1, that is, the uniform convergence
in probability.

###### Proof of Theorem 4.11.

For any $\phi\in\mathscr{P}_{2}^{M},\ \|u^{0}\|_{\mathrm{L}^{p}}\leq N$, for
$p\geq 2\delta+1$ and $\epsilon\in(0,1]$, we have

$\displaystyle u_{\epsilon,\phi}(t,x)-u_{0,\phi}(t,x)$
$\displaystyle=\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\big{(}\mathfrak{p}(u_{\epsilon,\phi}(s,y))-\mathfrak{p}(u_{0,\phi}(s,y))\big{)}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big{(}\mathfrak{c}(u_{\epsilon,\phi}(s,y))-\mathfrak{c}(u_{0,\phi}(s,y))\big{)}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big{(}g(s,y,u_{\epsilon,\phi}(s,y))-g(s,y,u_{0,\phi}(s,y))\big{)}\phi(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi}(s,y))\mathrm{W}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle=\frac{\alpha}{\delta+1}J_{\epsilon}^{1}(t,x)+\beta
J_{\epsilon}^{2}(t,x)+J_{\epsilon}^{3}(t,x)+\sqrt{\epsilon}J_{\epsilon}^{4}(t,x).$
(5.35)

Now, we consider $J_{\epsilon}^{1}(\cdot,\cdot)$. From (4.8)-(4.10), we have

$\displaystyle\|J_{\epsilon}^{1}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}\leq
C\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{\delta}{2p}}\|u_{\epsilon,\phi}(s)-u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s,$
(5.36)

where $\chi_{A}$ is the characteristic function of the set $A$. Using (4.9),
(4.2) and (4.2), we obtain

$\displaystyle\|J_{\epsilon}^{2}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\leq
C\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}\bigg{\\{}(t-s)^{-\frac{\delta}{2p}}(1+\delta)(1+\gamma)+\gamma+(t-s)^{-\frac{\delta}{p}}(1+2\delta)\bigg{\\}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s,$
(5.37)

where we have used Hölder’s inequality. Using Lemma 3.4, [16], Hypothesis 5.1
(bound of $g$) and (4.9), we find

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|J_{\epsilon}^{3}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\bigg{\|}\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)g(s,y,u_{\epsilon,\phi}(s,y))\mathrm{W}(\mathrm{d}s,\mathrm{d}y)\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq C(p,T,K),$ (5.38)

for $p>2$. Combining (5)-(5) and applying Fubini’s theorem followed by
Gronwall’s inequality, we arrive at

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\exp\bigg{\\{}C\int_{0}^{T}\mathopen{}\mathclose{{}\left[|t-s|^{-\frac{1}{2}-\frac{\delta}{2p}}+|t-s|^{-\frac{\delta}{2p}}+\gamma+|t-s|^{-\frac{\delta}{p}}}\right]\mathrm{d}s\bigg{\\}}$
$\displaystyle\leq
C(\alpha,\beta,\delta,\gamma,p,T,M,K,P)\epsilon^{\frac{p}{2}}.$ (5.39)

For any $\varpi>0$. Using similar calculations as in (4.2) and Theorem 5.2,
there exists $P>0$ large enough such that

$\displaystyle\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\big{[}1-\mathbb{P}(\Upsilon_{P}^{\epsilon})\big{]}\leq\varpi.$
(5.40)

Using (5), (5.40), and Markov’s inequality, for any
$\epsilon\in(0,1],u^{0}\in\mathrm{L}^{p}([0,1])$ and
$\phi\in\mathscr{P}_{2}^{M}$,

$\displaystyle\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}>\eta\bigg{)}$
$\displaystyle\leq\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\big{[}1-\mathbb{P}(\Upsilon_{P}^{\epsilon})\big{]}+\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\eta\bigg{)}$
$\displaystyle\leq\varpi+\frac{C(\alpha,\beta,\delta,\gamma,p,T,M,P)\epsilon^{\frac{p}{2}}}{\eta^{p}}.$
(5.41)

Due to arbitrary choice of $\varpi$, we obtain the required result (4.6). ∎

###### Proof of Theorem 4.7.

We established Theorem 4.11 under Hypothesis 5.1, which validates Condition
4.4. Therefore, EULP holds and by Theorem 4.3, we conclude that ULDP remains
valid. ∎

###### Proof of Theorem 4.8.

: One can obtain the proof of Lemma 4.12 and Theorem 4.13 by repeating the
same calculations, since the Lipschitz condition of the noise coefficient $g$
is same in both Hypothesis 1.1 and 5.1, and hence the theorem is over. ∎

##  Appendix A Some Useful Results

Let us recall some useful results from [15, 18].

Green’s function estimates: The following estimates have been frequently used
in this work (cf. [15, 16, 24, 20])

$\displaystyle|G(t,x,y)|$ $\displaystyle\leq
Ct^{-\frac{1}{2}}e^{-\frac{|x-y|^{2}}{a_{1}t}},$ (A.1)
$\displaystyle\bigg{|}\frac{\partial G}{\partial y}(t,x,y)\bigg{|}$
$\displaystyle\leq Ct^{-1}e^{-\frac{|x-y|^{2}}{a_{2}t}},$ (A.2)
$\displaystyle\bigg{|}\frac{\partial G}{\partial t}(t,x,y)\bigg{|}$
$\displaystyle\leq Ct^{-\frac{3}{2}}e^{-\frac{|x-y|^{2}}{a_{3}t}},$ (A.3)
$\displaystyle\bigg{|}\frac{\partial^{2}G}{\partial y\partial
t}(t,x,y)\bigg{|}$ $\displaystyle\leq Ct^{-2}e^{-\frac{|x-y|^{2}}{a_{4}t}},$
(A.4) $\displaystyle|G(t,x,z)-G(t,y,z)|$ $\displaystyle\leq
C|x-y|^{\vartheta}t^{-\frac{\vartheta}{2}-\frac{1}{2}}\max\bigg{\\{}e^{-\frac{|x-z|^{2}}{a_{5}t}},e^{-\frac{|y-z|^{2}}{a_{5}t}}\bigg{\\}},$
(A.5) $\displaystyle\bigg{|}\frac{\partial G}{\partial
z}(t,x,z)-\frac{\partial G}{\partial z}(t,y,z)\bigg{|}$ $\displaystyle\leq
C|x-y|^{\vartheta}t^{-1-\frac{\vartheta}{2}}\max\bigg{\\{}e^{-\frac{|x-y|^{2}}{a_{6}t}},e^{-\frac{|x-z|^{2}}{a_{6}t}}\bigg{\\}},$
(A.6)

for all $t\in(0,T],\ x,y,z\in[0,1]$, $C,\;a_{i}$ for $i=1,\ldots,6$, are some
positive constants and $\vartheta\in[0,1]$. The following estimate (see [15])
has been used frequently in the sequel

$\displaystyle\big{\|}e^{-\frac{|\cdot|^{2}}{a(t-s)}}\big{\|}_{\mathrm{L}^{p}}\leq
C(t-s)^{\frac{1}{2p}},$ (A.7)

for any positive constant $a$ and $p\geq 1$.

Now, we define the operators $J_{1}(\cdot)$ and $J_{2}(\cdot)$ by

$\displaystyle J_{1}u(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big{(}(1+\gamma)u^{\delta+1}-\gamma
u-u^{2\delta+1}\big{)}(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle=:\big{(}(1+\gamma)J_{11}-\gamma J_{12}-J_{13}\big{)}u(t,x),$
$\displaystyle J_{2}u(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{0}^{1}\frac{\partial G}{\partial
y}(t-s,x,y)u^{\delta+1}(s,y)\mathrm{d}y\mathrm{d}s,$

for all $t\in[0,T],\ x,y\in[0,1]$, where
$u\in\mathrm{L}^{\infty}(0,T;\mathrm{L}^{p}([0,1]))$, for some $p\geq 1$. A
similar result can be found in [15] (cf. [16, 18, 20]). We are providing for
$J_{13}(\cdot)$ (considering the highest order non-linearity) only, other term
$J_{11}(\cdot)$ can be handled in similar manner. Note that for
$J_{12}(\cdot)$ one can obtain the estimates from [15, 18].

######  Lemma A.1.

Under $\eqref{A1}-\eqref{A7}$, we have

  1. (1)

$J_{13}(\cdot)$ is a bounded operator from
$\mathrm{L}^{\sigma_{3}}(0,T;\mathrm{L}^{q}([0,1]))$ into
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ for $p\in[1,\infty]$,
$q\in[2\delta+1,p]$, $\kappa_{3}=1+\frac{1}{p}-\frac{2\delta+1}{q}$ and
$\sigma_{3}>\frac{2(2\delta+1)}{\kappa_{3}+1}$.

Moreover, for the values of $\kappa_{3}$ given above, the following estimate
hold:

  1. (1)

For every $0\leq t\leq T$, there is a constant $C$ such that

$\displaystyle\|J_{13}v(t,\cdot)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}+\frac{\kappa_{3}}{2}}\|v(s)\|_{\mathrm{L}^{q}}^{2\delta+1}\mathrm{d}s$
$\displaystyle\leq
Ct^{\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\frac{(\kappa_{3}-1)\sigma_{3}}{2(\sigma_{3}-2\delta-1)}+1}\right)\frac{\sigma_{3}-2\delta-1}{\sigma_{3}}}\right\\}}\bigg{(}\int_{0}^{t}\|u(s)\|_{\mathrm{L}^{q}}^{\sigma_{3}}\mathrm{d}s\bigg{)}^{\frac{2\delta+1}{\sigma_{3}}},$

where $\sigma_{3}$ are given above.

  2. (2)

For $0\leq s\leq t\leq T$, $\theta_{3}\in(0,\frac{\kappa_{3}+1}{2})$,
$\sigma_{6}>\frac{2(2\delta+1)}{\kappa_{3}+1-2\theta_{3}}$, there is a
constant $C$ such that

$\displaystyle\|J_{13}v(t,\cdot)-J_{13}v(s,\cdot)\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(t-s)^{\vartheta_{3}}\bigg{(}\int_{0}^{t}\|v(s)\|_{\mathrm{L}^{q}}^{\sigma_{6}}\mathrm{d}s\bigg{)}^{\frac{2\delta+1}{\sigma_{6}}}.$

  3. (3)

For every $0\leq t\leq T$, $\rho_{6}\in(0,\kappa_{3}+1)$,
$\sigma_{9}>\frac{2(2\delta+1)}{\kappa_{3}+1-\rho_{6}}$, there is a constant
$C$ such that

$\displaystyle\|J_{13}v(t,\cdot)-J_{13}v(t,\cdot+z)\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C|z|^{\rho_{6}}\bigg{(}\int_{0}^{t}\|v(s)\|_{\mathrm{L}^{q}}^{\sigma_{9}}\mathrm{d}s\bigg{)}^{\frac{2\delta+1}{\sigma_{9}}},$

for all $z\in\mathbb{R}$. We also set $J_{1i}u(t,y):=0,\text{ for
}i=\\{1,2,3\\}$, whenever $y\in\mathbb{R}\backslash[0,1]$.

######  Lemma A.2.

Under $\eqref{A2}-\eqref{A7}$, for $p\in[1,\infty],\ q\in[\delta+1,p],\
\kappa_{1}=1+\frac{1}{p}-\frac{\delta+1}{q}$ and
$\sigma_{10}>\frac{2(\delta+1)}{\kappa_{1}}$, the operator $J_{2}(\cdot)$ is
bounded from $\mathrm{L}^{\sigma_{10}}(0,T;\mathrm{L}^{q}([0,1]))$ into
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$. Moreover the following estimates
hold:

  1. (1)

For every $0\leq t\leq T$, there is a constant $C$ such that

$\displaystyle\|J_{2}v(t,\cdot)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-1+\frac{\kappa_{1}}{2}}\|v(s)\|_{\mathrm{L}^{q}}^{\delta+1}\mathrm{d}s$
$\displaystyle\leq
Ct^{\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\frac{(\kappa_{1}-2)\sigma_{10}}{2(\sigma_{10}-\delta-1)}+1}\right)\frac{\sigma_{10}-\delta-1}{\sigma_{10}}}\right\\}}\bigg{(}\int_{0}^{t}\|v(s)\|_{\mathrm{L}^{q}}^{\sigma_{10}}\mathrm{d}s\bigg{)}^{\frac{\delta+1}{\sigma_{10}}}$

  2. (2)

For $0\leq s\leq t\leq T$, $\theta_{4}\in(0,\frac{\kappa_{1}}{2})$,
$\sigma_{11}>\frac{2(\delta+1)}{\kappa_{1}-2\theta_{5}}$, there is a constant
$C$ such that

$\displaystyle\|J_{2}v(t,\cdot)-J_{2}v(s,\cdot)\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(t-s)^{\theta_{4}}\bigg{(}\int_{0}^{t}\|v(s)\|_{\mathrm{L}^{q}}^{\sigma_{11}}\mathrm{d}s\bigg{)}^{\frac{\delta+1}{\sigma_{11}}},.$

  3. (3)

For every $0\leq t\leq T,\;\rho_{7}\in(0,\kappa_{1})$ and
$\sigma_{12}>\frac{2(\delta+1)}{\kappa_{1}-\rho_{7}}$, there is a constant $C$
such that

$\displaystyle\|J_{2}v(t,\cdot)-J_{2}v(t,\cdot+z)\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C|z|^{\rho_{7}}\bigg{(}\int_{0}^{t}\|v(s)\|_{\mathrm{L}^{q}}^{\sigma_{12}}\mathrm{d}s\bigg{)}^{\frac{\delta+1}{\sigma_{12}}},$

for all $z\in\mathbb{R}$. We also set $J_{2}v(t,y):=0$, whenever
$y\in\mathbb{R}\backslash[0,1]$.

Let us recall a result from [18], which helps us to obtain the uniform
tightness for the operators $J_{i}$:

######  Lemma A.3 (Corollary 3.2, [18]).

Let $\zeta_{n}(t,y)$ be a sequence of random fields on $[0,T]\times[0,1]$ such
that $\sup\limits_{0\leq t\leq
T}\|\zeta_{n}(t,\cdot)\|_{\mathrm{L}^{q}}\leq\xi_{n}$, for $q\in[1,p],$ where
$\xi_{n}$ is a finite random variable for every $n$. Assume that the sequence
$\\{\xi_{n}\\}$ is bounded in probability, that is,

$\displaystyle\lim_{C\to\infty}\sup_{n}\mathbb{P}(\xi_{n}\geq C)=0.$

Under $\eqref{A1}-\eqref{A4}$, the sequences $J_{i}(\zeta_{n})$, for
$i=\\{1,2\\}$, are uniformly tight in
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1])),$ for $p\geq 1$. Moreover in the case
$p=\infty$, the sequence $J_{i}(\zeta_{n})$ is tight in the space
$\mathrm{C}([0,T]\times[0,1])$.

Acknowledgments: The first author would like to thank Ministry of Education,
Government of India - MHRD for financial assistance. The second author would
like to thank Prof. B. Rajeev (ISI Bangalore) for the financial assistance
through his CRG grant (SERB sanction File No.: CRG/2019/002595), and the
Department of Atomic Energy (DAE), Government of India for assisting through
NBHM post-doctoral fellowship (File No.: 0204/6/2022/R&D-II/5635). M. T. Mohan
would like to thank the Department of Science and Technology (DST), India for
Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award
(IFA17-MA110).

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