# NoisyTune: A Little Noise Can Help You Finetune
Pretrained Language Models Better

Chuhan Wu† Fangzhao Wu‡ Tao Qi† Yongfeng Huang† Xing Xie‡
†Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China
‡Microsoft Research Asia, Beijing 100080, China
{wuchuhan15, wufangzhao<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author.

###### Abstract

Effectively finetuning pretrained language models (PLMs) is critical for their
success in downstream tasks. However, PLMs may have risks in overfitting the
pretraining tasks and data, which usually have gap with the target downstream
tasks. Such gap may be difficult for existing PLM finetuning methods to
overcome and lead to suboptimal performance. In this paper, we propose a very
simple yet effective method named NoisyTune to help better finetune PLMs on
downstream tasks by adding some noise to the parameters of PLMs before
finetuning. More specifically, we propose a matrix-wise perturbing method
which adds different uniform noises to different parameter matrices based on
their standard deviations. In this way, the varied characteristics of
different types of parameters in PLMs can be considered. Extensive experiments
on both GLUE English benchmark and XTREME multilingual benchmark show
NoisyTune can consistently empower the finetuning of different PLMs on
different downstream tasks.

##  1 Introduction

In recent years, pretrained language models (PLMs) have achieved huge success
in NLP Qiu et al. (2020). Many PLMs such as BERT Devlin et al. (2019), RoBERTa
Liu et al. (2019) and UniLM Dong et al. (2019) which are pretrained from
large-scale unlabeled corpus in a self-supervised way, have significantly
improve various downstream tasks such as reading comprehension Xu et al.
(2019), machine translation Brown et al. (2020), text classification Bao et
al. (2020), dialog Wu et al. (2020) and recommendation Wu et al. (2021) by
finetuning on these tasks.

How to effectively finetune PLMs to better empower downstream tasks is an
important research problem Zheng et al. (2021). Many existing NLP methods
usually directly finetune PLMs with the labeled data in downstream tasks Sun
et al. (2019). Only a few works explore more effective and robust PLM
finetuning methods Chen et al. (2020); Lee et al. (2020); Aghajanyan et al.
(2021); Zhang et al. (2021); Xu et al. (2021). For example, Chen et al. (2020)
proposed RecAdam that adds a penalty item to minimize the $L_{2}$ distance
between the fine-tuned models and the pretrained models, where the penalty
intensity is time-variant during finetuning. Lee et al. (2020) proposed Mixout
which randomly replaces part of the parameters in the finetuned model with
their original weights in the PLMs. These PLM finetuning methods mainly focus
on preventing PLMs from overfitting the limited labeled data in downstream
tasks. Besides the overfitting of downstream task data, a rarely studied
problem is that the PLMs usually overfit the pretraining tasks and data Qi et
al. (2020), which may have significant gap with the downstream task and data.
It is not easy for existing PLM finetuning methods to overcome such gap
Roberts et al. (2020), which may lead to suboptimal performance especially
when labeled data in downstream tasks is insufficient.

Figure 1: Schematic comparisons between standard PLM finetuning and our
NoisyTune.

In order to handle this problem, in this paper we propose a very simple yet
effective method named NoisyTune, which can help better finetune PLMs for
downstream tasks. Different from the standard finetuning paradigm (Fig. 1 (a))
which directly finetunes PLMs on the downstream task data, the key idea of
NoisyTune is to add a small amount of noise to perturb PLMs parameters before
finetuning (Fig. 1 (b)). It can help prevent PLMs from overfitting the tasks
and data in the pretraining stage, and reduce the gap between pretraining and
downstream tasks. Since PLMs have different types of parameters which usually
own different characteristics, in NoisyTune we use a matrix-wise perturbing
method that adds uniform noise with different intensities to different
parameter matrices according to their standard deviations for better
adaptation. We conduct extensive experiments on two widely used NLP
benchmarks, namely, GLUE Wang et al. (2018) for English language understanding
and XTREME Hu et al. (2020) for multilingual language understanding. The
results show NoisyTune can empower the finetuning of different PLMs on many
different downstream NLP tasks to consistently achieve better performance. In
addition, the results show NoisyTune can be easily combined with many existing
PLM finetuning methods and further improve their performance.

##  2 NoisyTune

The goal of NoisyTune is for more effective finetuning of PLMs on downstream
tasks. The motivation of NoisyTune is that PLMs are well pretrained on some
unlabeled corpus with some self-supervision tasks, and they may overfit these
pretraining data and tasks Qi et al. (2020), which usually have gap with the
downstream task and data. It may be difficult for PLMs to effectively adapt to
downstream tasks especially when labeled data in these tasks are limited,
which is usually the case. Motivated by the dueling bandits mechanism Yue and
Joachims (2009) that adds randomness to the model for exploration, as shown in
Fig. 1, we propose to add some noise to the parameters of PLMs before
finetuning them on downstream tasks to do some “exploration” in parameter
space and reduce the risk of overfitting the pretraining tasks and data.

PLMs usually have different kinds of parameter matrices, such as query, key,
value, and feedforward network matrices Devlin et al. (2019). Different
parameter matrices in the PLMs usually have different characteristics and
scales. For example, some researchers found that the self-attention parameters
and the feed-forward network parameters in Transformers have very different
properties, such as rank and density Wang et al. (2020). Thus, adding unified
noise to all parameter matrices in PLMs may not be optimal for keeping their
good model utility. To handle this challenge, we propose a matrix-wise
perturbing method that adds noise with different intensities to different
parameter matrices according to their variances. Denote the parameter matrices
(or scalars/vectors) in a PLM as
$[\mathbf{W}_{1},\mathbf{W}_{2},...,\mathbf{W}_{N}]$, where $N$ is the number
of parameter matrix types. Denote the perturbed version of the parameter
matrix $\mathbf{W}_{i}$ as $\tilde{\mathbf{W}}_{i}$, which is computed as
follows:

$\tilde{\mathbf{W}}_{i}=\mathbf{W}_{i}+U(-\frac{\lambda}{2},\frac{\lambda}{2})*std(\mathbf{W}_{i}),$
(1)

where $\rm{std}$ stands for standard deviation. The function $U(a,b)$
represents uniform distribution noise ranged from $a$ to $b$, and $\lambda$ is
a hyperparameter that controls the relative noise intensity.111Note that
$U(a,b)$ is a matrix with the same shape with $\mathbf{W}_{i}$ rather than a
scalar. We can see that in NoisyTune parameters in PLMs with higher variance
will be added with stronger noise. In addition, in some PLMs there are some
constant matrices, such as token type embeddings in RoBERTa Liu et al. (2019).
They will not be perturbed because their standard deviation is 0. It can
ensure that these constant matrices will not be accidentally activated by
additional noise.

NoisyTune is a simple and general plug-and-play technique that can be applied
to the finetuning of any PLM on any task, simply by inserting the following
PyTorch-style code before finetuning:

⬇

model.state_dict[name][:] +=

(torch.rand(para.size())-0.5)

*noise_lambda*torch.std(para)

##  3 Experiments

###  3.1 Datasets and Experimental Settings

We conduct extensive experiments on two widely used benchmarks for PLM
evaluation. The first one is GLUE Wang et al. (2018), which is a benchmark for
English language understanding that contains different tasks like natural
language inference, sentiment analysis and sentence similarity evaluation. The
second one is XTREME Hu et al. (2020), which is a benchmark for multilingual
language understanding. It covers 40 languages and contains four groups of
tasks, including sentence classification, structured prediction, sentence
retrieval and question answering. More details of these benchmarks can refer
to their original papers and official websites. Since the test labels of GLUE
are not released, following Bao et al. (2020) we report results on the dev set
of GLUE. The XTREME results are evaluated on the test set. The hyperparameter
$\lambda$ is 0.15 on GLUE and is 0.1 on XTREME. The searching range of
hyperparameters in our work are listed in Table 1.

Hyperparameters | Range
---|---
Learning rate | {7e-6, 1e-5, 2e-5, 3e-5}
Epoch | {3, 5, 7, 10, 15, 20}
Batch size | {8, 16, 32}
Noisy intensity | {0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3}

Table 1: Searching ranges of different hyperparameters in our experiments.

Model | MNLI | QNLI | QQP | RTE | SST | MRPC | CoLA | STS | WNLI |
---|---|---|---|---|---|---|---|---|---|---
Acc | Acc | Acc | Acc | Acc | Acc | MCC | PCC | Acc | Avg.
BERT | 84.4 | 91.5 | 90.9 | 67.7 | 93.0 | 87.1 | 58.1 | 89.4 | 54.4 | 79.6
BERT+NoisyTune | 84.7 | 91.8 | 91.2 | 68.8 | 93.4 | 88.0 | 59.0 | 90.1 | 56.1 | 80.3
XLNET | 86.6 | 91.6 | 91.2 | 72.9 | 94.4 | 88.1 | 59.6 | 89.6 | 57.5 | 81.3
XLNET+NoisyTune | 86.9 | 91.9 | 91.4 | 73.8 | 94.7 | 88.6 | 60.1 | 90.0 | 58.6 | 81.8
RoBERTa | 87.5 | 92.7 | 91.7 | 77.1 | 94.5 | 90.1 | 62.9 | 90.8 | 59.2 | 82.9
RoBERTa+NoisyTune | 87.8 | 93.1 | 91.9 | 78.8 | 94.9 | 90.6 | 63.6 | 91.1 | 60.3 | 83.6
ELECTRA | 88.4 | 92.9 | 91.7 | 75.2 | 94.9 | 88.2 | 64.2 | 90.1 | 62.0 | 83.1
ELECTRA+NoisyTune | 88.7 | 93.2 | 92.1 | 76.4 | 95.2 | 88.7 | 64.9 | 90.5 | 63.4 | 83.7
|  |  |  |  |  |  |  |  |  |

Table 2: Results of different methods on the GLUE dev set.

Model | Sentence Pair | Structured Prediction | Sentence Retrieval | Question Answering |
---|---|---|---|---|---
XNLI | PAWS-X | POS | NER | BUCC | Tatoeba | XQuAD | MLQA | TyDiQA |
Metrics | Acc | Acc | F1 | F1 | Acc | Acc | F1/EM | F1/EM | F1/EM | Avg.
Fine-tune multilingual model on English training set (Cross-lingual Transfer)
XLM-Rbase | 74.8 | 84.8 | 75.5 | 61.6 | 77.6 | 73.8 | 71.9/56.6 | 65.2/47.0 | 55.5/38.4 | 70.0
XLM-Rbase+NoisyTune | 75.2 | 85.1 | 76.0 | 62.1 | 78.2 | 74.5 | 72.3/57.1 | 65.5/47.4 | 56.0/39.2 | 70.5
XLM-Rlarge | 79.0 | 86.3 | 72.7 | 62.3 | 79.2 | 76.0 | 76.2/60.4 | 71.4/53.0 | 65.0/45.0 | 72.4
XLM-Rlarge+NoisyTune | 79.3 | 86.5 | 73.5 | 63.2 | 79.9 | 76.8 | 76.7/61.0 | 71.9/53.6 | 65.4/45.6 | 73.0
Fine-tune multilingual model on all training sets (Translate-Train-All)
XLM-Rbase | 78.5 | 88.2 | 76.2 | 62.6 | 79.6 | 79.4 | 75.0/61.5 | 67.8/50.1 | 63.8/47.6 | 73.3
XLM-Rbase+NoisyTune | 78.9 | 88.6 | 76.8 | 63.1 | 80.0 | 79.8 | 75.4/61.8 | 68.0/50.4 | 64.1/48.1 | 73.7
XLM-Rlarge | 82.3 | 90.3 | 77.3 | 67.3 | 82.5 | 82.7 | 80.0/65.6 | 72.9/54.4 | 66.3/47.6 | 76.4
XLM-Rlarge+NoisyTune | 82.5 | 90.5 | 77.8 | 67.9 | 82.9 | 83.0 | 80.4/66.1 | 73.3/54.9 | 66.8/48.2 | 76.8
|  |  |  |  |  |  |  |  |  |

Table 3: Results of different methods on the XTREMRE test set.

Following Zheng et al. (2021), in sentence retrieval tasks we first train the
models on the XNLI dataset, and then use the average of token representations
produced by the hidden layer that yields the best performance. In order not to
harm the alignment of token embeddings across different languages, we do not
add noise to the token embeddings in multilingual PLMs. We repeat experiments
5 times with different random seeds and report the average scores.

###  3.2 Performance Evaluation

On the GLUE benchmark, we compare the performance of directly finetuning the
base version of BERT Devlin et al. (2019), XLNET Yang et al. (2019), RoBERTa
Liu et al. (2019) and ELECTRA Clark et al. (2020) with that of finetuning them
after applying NoisyTune. On the XTREME benchmark, we compare the performance
of directly finetuning both base and large versions of XLM-R Conneau et al.
(2020) with that of their variants obtained by applying NoisyTune. The results
on these two benchmarks are shown in Tables 2 and 3, respectively. On the
XTREME datasets, we report two types of results. The first one is zero-shot
crosslingual transfer from English to other languages, and the second one is
learning models on both English and translated data.

According to these results, NoisyTune can consistently improve the performance
of different PLMs on different tasks in both English and multilingual
settings. In addition, the performance improvement brought by NoisyTune is
usually larger on relatively small datasets (e.g., RTE, CoLA and WNLI). These
results indicate that when labeled data in downstream tasks is insufficient,
it is quite difficult to effectively finetune PLMs starting from the original
parameters which usually overfit the pretraining tasks and data. The
experimental results validate that NoisyTune can properly perturb PLMs with a
little noise to explore different parameter spaces and reduce the overfitting
problem, making PLMs easier to be adapted to downstream tasks.

###  3.3 Which Noise to Use and How?

In this section we study which kind of noise is more suitable for NoisyTune.
In addition, we explore whether our proposed matrix-wise perturbing method is
better than using a unified global noise for all model parameters in PLMs. We
compare five methods, including (1) NoisyTune without any noise; (2) NoisyTune
with a global Gaussian noise; (3) NoisyTune with a global uniform noise; (4)
NoisyTune with matrix-wise Gaussian noise; (5) NoisyTune with matrix-wise
uniform noise. The results on GLUE are shown in Fig. 2, and the results on
XTREME show similar patterns. We find that adding global noise with the same
distribution to all the PLM parameters will harm the model performance. This
is because different parameter matrices in PLMs have very different
distributions and characteristics Wang et al. (2020). Simply adding a unified
global noise to all the parameter matrices is not optimal. The results show
that matrix-wise noise is a much better choice, since the different
characteristics of different parameter matrices can be taken into
consideration. In addition, we find an interesting phenomenon that adding
uniform noise is better than Gaussian noise. This may be because Gaussian
noise has wider ranges and some extreme values may affect the model
performance. Thus, we use matrix-wise uniform noise in NoisyTune.

###  3.4 Combination with Existing PLM Finetuning Methods

From Fig. 1, it is very clear that NoisyTune is independent of the specific
PLM finetuning method, since it is applied at the stage before finetuning PLM
on the task-specific data. Thus, it is very easy to combine NoisyTune with any
kind of existing PLM finetuning method. In this section, we explore whether
NoisyTune has the potential to empower the existing PLM finetuning techniques
to achieve better performance. Here we select two well-known PLM finetuning
for experiments, i.e., RecAdam Chen et al. (2020) and Mixout Lee et al.
(2020). The experimental results are summarized in Fig. 3. We find that
combining NoisyTune with existing PLM finetuning techniques can further
improve their performance. This is because NoisyTune aims to address the
overfitting of pretraining signals while these methods aim to prevent
overfitting in downstream tasks. Thus, NoisyTune and these PLM finetuning
methods are complementary, and they can be empowered by NoisyTune to achieve
better performance.

###  3.5 Empirical Analysis of NoisyTune

Next, we empirically analyze why NoisyTune can help PLM finetuning. We compare
the accuracy of BERT with and without NoisyTune finetuned with different
percentage of samples on the MRPC dataset.222We observe similar patterns on
other datasets. The results are shown in Fig. 4. We find NoisyTune can
consistently improve PLMs under different amounts of data, especially when
less training data is used. This is because the perturbed PLMs may have lower
risks of overfitting the pretraining tasks and have better generalization
abilities, which is especially beneficial for finetuning PLMs on downstream
tasks with limited data.

Figure 2: Different noise types and perturbing methods. Figure 3: NoisyTune
can empower many existing PLM finetuning methods to achieve better
performance. Figure 4: Influence of NoisyTune on finetuning. Figure 5:
Relative changes of the L1-norm of different types of parameters in PLM during
finetuning. Figure 6: Influence of noise intensity $\lambda$.

To further study the impact of NoisyTune on PLM finetuning, we show the
relative changes of the L1-norms of different kinds of parameters in the BERT
model during finetuning on the MRPC dataset in Fig. 5.333The patterns on other
datasets are similar. Since the noise we added to PLMs in NoisyTune is zero-
mean uniform noise, the absolute parameter L1-norm will not change too much.
However, we can see that the relative change of L1-norms becomes smaller when
NoisyTune is applied, which indicates that the PLMs can find the (sub)optimal
parameters for downstream tasks more easily. This result validates directly
finetuning PLMs may need more updates to adapt to downstream tasks, which is
due to the overfitting of pretraining tasks, and NoisyTune can provide a
simple way to alleviate this problem and help finetune PLMs on downstream
tasks more effectively.

###  3.6 Hyperparameter Analysis

We study the influence of the most important hyperparameter in NoisyTune,
i.e., $\lambda$, which controls the relative noise intensity. The average GLUE
scores w.r.t. different $\lambda$ values are shown in Fig. 6. We find that
when $\lambda$ is too small or too large, the performance is not optimal. This
is because when $\lambda$ is too small, it is difficult for PLMs to do
parameter space exploration and overcome the overfitting problem. While when
$\lambda$ is too large, the useful pretrained knowledge in PLMs may be
overwhelmed by random noise. Values between 0.1 and 0.15 are more suitable for
NoisyTune on the GLUE datasets.

##  4 Conclusion

In this paper, we propose a very simple but effective method named NoisyTune,
which can help better finetune PLMs on downstream tasks by adding a little
noise to them before finetuning. In NoisyTune, we propose a matrix-wise
perturbing method that adds noise with different intensities to different
kinds of parameter matrices in PLMs according to their variances. NoisyTune is
a very general method, and is PLM model agnostic, downstream task agnostic,
and finetuning method agnostic. Extensive experiments on both monolingual GLUE
benchmark and multilingual XTREME benchmark demonstrate NoisyTune can
consistently empower the finetuning of different PLMs on various downstream
tasks to achieve better performance.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grant numbers U1936216, U1936208, and 61862002, and the research
initiation project of Zhejiang Lab (No. 2020LC0PI01).

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# Solving stochastic equations with unbounded nonlinear perturbations

M. Fkirine and S. Hadd  Department of Mathematics, Faculty of Sciences, Ibn
Zohr University, Hay Dakhla, B.P. 8106, 80000 Agadir, Morocco;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is interested in semilinear stochastic equations having unbounded
nonlinear perturbations in the deterministic part and/or in the random part.
Moreover the linear part of these equations is governed by a not necessarily
analytic semigroup. The main difficulty with these equations is how to define
the concept of mild solutions due to the chosen type of unbounded
perturbations. To overcome this problem, we first proved a regularity property
of the stochastic convolution with respect to the domain of “admissible”
unbounded linear operators (not necessarily closed or closable). This is done
using Yosida extensions of such unbounded linear operators. After proving the
well-posedness of these equations, we also establish the Feller property for
the corresponding transition semigroups. Several examples like heat equations
and schrödinger equations with nonlocal perturbations terms are given.
Finally, we give an application to a general class of semilinear neutral
stochastic equations.

###### Key words and phrases:

Semilinear stochastic equations, Unbounded nonlinear perturbation, Hilbert
space, Semigroup, Equations with delays

###### 2010 Mathematics Subject Classification:

60H15, 35F20, 47H14, 35R60, 35B25, 30K40

The first author was supported by National Center for Scientific and Technical
Research (CNRST), Morocco.

##  1\. Introduction

We are concerned with the well-posedness and Feller property of two classes of
abstract stochastic equations under unbounded nonlinear perturbations. We will
work with generators that are not necessarily analytic, so we do not assume
any maximal regularity property for the linear part of the stochastic
equation. More precisely, we consider the following abstract stochastic
equations

$\displaystyle dX(t)=AX(t)dt+\mathscr{M}(\mathscr{B}X(t))dW(t),\quad
X(0)=\xi,\quad t\in[0,T],$ (1.1)

and

$\displaystyle\begin{cases}dX(t)=[AX(t)+F(\mathscr{C}X(t))]dt+\mathscr{M}(X(t))dW(t),&t\in[0,T]\\\
X(0)=\xi\end{cases}$ (1.2)

in a real separable Hilbert space $H$. Here, $A:D(A)\subset H\to H$ is the
generator of a strongly continuous semigroup $\mathbb{T}:=(T(t))_{t\geq 0}$ on
$H$ (not necessarily analytic), $\mathscr{C},\mathscr{B}:\mathscr{Z}\subset
H\to H$ are linear operators (not necessarily closed or closable nor
represented by fractional operators), $D(A)\subset\mathscr{Z}\subset H,$
$F:H\to H$ and $\mathscr{M}:H\to L_{2}^{0}(V,H)$ are nonlinear mappings, where
$L_{2}^{0}(V,H)$ is the space of all Hilbert-Schmidt operators acting between
$Q^{\frac{1}{2}}(V)$ and $H$, see notation below. Moreover, $W(t)$ is a
$Q$-Wiener process in $V$ defined on a filtered probability space
$(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\geq 0},\mathbb{P})$. The initial
condition $\xi$ can be any $\mathcal{F}_{0}$-measurable $H$-valued random
variable.

By selecting

$\displaystyle G=F\circ\mathscr{C}:\mathscr{Z}\to H,$ (1.3)

the equation (1.2) becomes

$\displaystyle\begin{cases}dX(t)=[AX(t)+G(X(t))]dt+\mathscr{M}(X(t))dW(t),&t\in[0,T]\\\
X(0)=\xi.\end{cases}$ (1.4)

In the particular case of $\mathscr{C}\in{\mathcal{L}}(H)$, that is $D(G)=H$,
the equation (1.4) has been extensively studied. We cannot give a complete
description of the literature, however we cite the following references [10],
[7], [23] [35, Chap.4], [34], and [33] in which the existence and uniqueness
of solution as well as the regularity of solutions are established. We also
mention that the case when the nonlinear application has a small domain
$D(G)=\mathscr{Z}\subset H$ and an image in $\mathscr{Z}$ (this is
$G:\mathscr{Z}\to\mathscr{Z}$) is also considered by many authors, see e.g.
[13], [31], [30], [32] and [8]. The common strategy used in the references
mentioned above is to consider the following assumptions: the part of $A$ in
$\mathscr{Z}$ generates a strongly continuous semigroup on $\mathscr{Z},$ $G$
is a locally Lipschitz or a continuous function from $\mathscr{Z}$ to
$\mathscr{Z},$ and the initial condition $\xi$ is a $\mathscr{Z}$-valued
random. As shown in [10, Theorem 7.15], to work with initial conditions
$\xi\in L^{2}(\mathcal{F}_{0},H)$, extra assumptions on
$G:\mathscr{Z}\to\mathscr{Z}$ are required, and in this case we only talk
about a special class of solutions called “generalized” solutions. Also the
equation (1.1) is studied in [10, page 176] and [12] where the generator $A$
satisfies some maximal regularity properties.

In the present paper, we will use the following assumptions for the semilinear
equation (1.4):

  * (S)

The operator $A:D(A)\subset H\to H$ generates a strongly continuous semigroup
$\mathbb{T}:=(T(t))_{t\geq 0}$ on $H,$ such that

$\displaystyle\|T(t)\|\leq Me^{\beta t},\qquad\forall t\geq 0$

for some $\beta>\omega_{0}(A)$ and $M\geq 1$.

  * (G)

The nonlinear map $G:\mathscr{Z}\to H$ is given by (1.3).

  * (L)

$F:H\rightarrow H$, $\mathscr{M}:H\rightarrow L_{2}^{0}(V,H)$ are nonlinear,
Lipschitz continuous operators, i.e,

$\|F(x)-F(y)\|+\|\mathscr{M}(x)-\mathscr{M}(y)\|_{2}\leq
k\|x-y\|,~{}~{}\text{for all }x,y\in H.$

  * (A)

The operator $C:=\mathscr{C}$ with domain $D(C):=D(A)$ is an admissible
operator for $A$. That is, for some (hence all) $\alpha>0$ there exists
$\gamma:=\gamma(\alpha)>0$ such that

$\int_{0}^{\alpha}\|CT(t)x\|^{2}dt\leq\gamma^{2}\|x\|^{2},\qquad(x\in D(A)).$
(1.5)

  * (A’)

$B:=\mathscr{B}$ with domain $D(B):=D(A)$ is a zero-class admissible operator
for $A$. That is, $B$ satisfies the condition (1.5) with a constant
$\tilde{\gamma}(\alpha)\to 0$ as $\alpha\to 0$.

We mention that in the condition (G), we may have the following situation
$\mathscr{Z}\subsetneqq{\rm Im}(G)\subseteq H$. This case goes beyond the
approach used in the references cited above. Also, unlike these references, we
do not assume that the space $\mathscr{Z}$ is stable by the semigroup
$\mathbb{T}$, so we can not use the part of $A$ in $\mathscr{Z}$ to define a
semigroup on $\mathscr{Z}$. This fact may implies that the determinist and
stochastic convolutions may exceed $\mathscr{Z}$. This offers many
difficulties in defining naturally a mild solution for the problem (1.4) (or
(1.2)). To overcome this problem, we will use the fact that $\mathscr{C}$
satisfies the condition (A) and the concept of Yosida extensions (see (1.7))
to give sense to the eventual solution to the semilinear problem (1.2). In
fact, we first show in Theorem 2.4 that under the conditions (A) and (S), we
have

$\displaystyle W_{A}^{\Phi}(t):=\int_{0}^{t}T(t-s)\Phi(s)dW(s)\in
D(C_{\Lambda}),\qquad a.e.\;t\geq 0,\quad\mathbb{P}-a.s,$

where $C_{\Lambda}$ is the Yosida extension of $C$ for $A$. Furthermore, we
have the regularity estimate (2.2). We note that in the case of analytic
semigroup $\mathbb{T}$ and $\mathscr{C}=(-A)^{\alpha}$ for some
$\alpha\in(0,\frac{1}{2})$, the above result is proved if [10, Theorem 6.14],
where the proof is heavily based on properties of analytic semigroups. This
case can also considered as a corollary of our result Theorem 2.4, because the
analyticity of $\mathbb{T}$ implies that $(-A)^{\alpha}$ satisfies the
condition (A).

In Theorem 2.7, we assumed that the operator $B$ satisfies the condition (A’)
and proved that the stochastic equation (1.1) has a unique mild solution. The
proof is mainly based on Theorem 2.4 and a Banach’s fixed point theorem. In
fact, compared with [3], [10, Theorem 6.21] and [12], we dot not assume any
regularity of the semigroup $\mathbb{T}$. We note that the condition (A’) is
somehow similar to a condition introduced in [12].

By using the above result on the stochastic convolution, the assumptions (A),
(L), (G), and (S), and an approximation method, we prove (see Theorem 3.4) the
existence and uniqueness of a process $X\in\mathcal{C}_{\mathcal{F}}([0,T],H)$
such that $X(t)\in D(C_{\Lambda})$ a.e. $t>0,$ $\mathbb{P}$-a.s., and
satisfying

$\displaystyle
X(t)=T(t)\xi+\int^{t}_{0}T(t-s)F(C_{\Lambda}X(s))ds+\int^{t}_{0}T(t-s)\mathscr{M}(X(s))dW(s)$

for any $t\geq 0$ and $\xi\in L^{2}(\mathcal{F}_{0},H)$. The particular case
when $F\in{\mathcal{L}}(H)$ is recently proved in [25, Theorem 3.6]. On the
other hand, in the deterministic case, the authors of [36] considered the
equation (1.2) with $\mathscr{M}\equiv 0$ and assumed that $\mathscr{C}$
satisfies (A). They proved the existence and uniqueness of a continuous mild
solution, but the above variation constant formula is not established. We also
mention that the proof of Theorem 3.4 can also be used for the deterministic
case which we consider as an alternative proof to the corresponding proof in
[36]. Another result in the deterministic case can be found in [5, Section 11]
and [29, Chap.7], in which a very particular case is considered.

In Section 4, we consider the transition semigroup associated with the
equation (1.2),

$\displaystyle P_{t}\phi=\mathbb{E}\left(\phi(X(t,x))\right)$

for any $t\geq 0,\,x\in H$ and any $\phi\in{\mathcal{C}}_{b}(H),$ the space of
all bounded and Borel measurable functions $\phi:H\to\mathbb{R}$. In
Proposition 4.1, a continuous dependence of the solutions on initial
conditions is established, which implies the Markov and Feller properties, in
particular $P_{t}{\mathcal{C}}_{b}(H)\subset{\mathcal{C}}_{b}(H)$.

We end this paper with an application to semilinear stochastic equations of
neutral type. We will use product spaces to introduce a semigroup approach to
the well-posedness of such equations.

In the literature, one can find the typical semilinear stochastic equation

$\displaystyle\begin{cases}dX(t,x)=\left[\partial_{x}^{2}X(t,x)+\frac{1}{2}\left(\partial_{x}X(t,x)\right)^{2}\right]dt+dW(t,x),&t\in[0,T],\quad
x\in\mathbb{R}\\\ X(0,x)=\xi(x)\in L^{2}(\mathbb{R}),\end{cases}$ (1.6)

where $W(t)$ is a $Q$-Wiener process on a Hilbert space $L^{2}(\mathbb{R})$.
In fact, equation (1.6) is called the Kardar-Parisi-Zhang equation (KPZ
equation) and was introduced in [24] as a model of the interface growing in
the phase transitions theory. See also [2], [14], [20], [21] for more details
about such equations. The singular term $(\partial_{x}X(t,x))^{2}$ makes the
study of the well-posed of the equation (1.6) very difficult. In fact, this
question was an open problem for over 15 years until Hairer’s key paper [20]
appears. It is known that the equation (1.6) can be reformulated as a Burgers
equation, which is investigated in [6], see also [10, Chap.13] and [9,
Chap.14].

If we take $A=\partial_{x}^{2},$ $f(x)=x^{2}$, $\mathscr{C}=\partial_{x}$, and
$F(\phi)(x)=f(\phi(x)),$ then the KPZ equation (1.6) has the same form as our
abstract equation (1.2). The only difference is that in the equation (1.2) the
function $F$ is locally Lipschitz function. In our case, we only treat the
case when $F$ is global Lipschitz function (see the condition (L)).

The organization of this paper is as follows: In Section 2 we first prove a
technical result on stochastic convolutions and then use this result to prove
the well-posedness of the stochastic equation (1.1), while in Section 3 we
prove the well-posedness of the equation (1.2). In section 4, we establish the
Feller property of the transition semigroup associated with the solution of
the equation (1.2). The last section is devoted to the well-posedness of a
large class of neutral stochastic equations.

Notation. Throughout this paper, $H$ denotes a real separable Hilbert space
with norm $\|\cdot\|$ and $(A,D(A))$ is the generator of a $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ on $H$ with growth bound $\omega_{0}$. For
$\lambda\in\rho(A)$ (the resolvent set of $A$) we set
$R(\lambda,A):=(\lambda-A)^{-1}$. The Yosida extension of a linear operator
$C:D(A)\to H$ for $A$ is the following linear operator

$\displaystyle\begin{split}D(C_{\Lambda})&:=\left\\{x\in
H:~{}~{}\lim_{\lambda\to+\infty}C\lambda R(\lambda,A)x\quad\text{exists
in}\quad H\right\\},\\\
C_{\Lambda}x&:=\lim_{\lambda\in\mathbb{R},\,\lambda\to+\infty}C\lambda
R(\lambda,A)x.\end{split}$ (1.7)

Clearly, we have $D(A)\subset D(C_{\Lambda})\subset H$ and $C_{\Lambda}x=Cx$
for any $x\in D(A)$.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space equipped
with a normal filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$. Let
$(\beta_{k}(t))_{k}$ be a sequence of real-valued one-dimensional standard
Brownian motions mutually independent over $(\Omega,\mathcal{F},\mathbb{P})$.
Set

$W(t)=\sum_{k=0}^{\infty}\sqrt{\lambda_{k}}\beta_{k}(t)e_{k},\qquad t\geq 0,$

where $(\lambda_{k})_{k>0}$ are nonnegative real numbers and $(e_{k})_{k>0}$
is a complete orthonormal basis in $V$. Let $Q\in\mathcal{L}(V)$ be an
operator defined by $Qe_{k}=\lambda_{k}e_{k}$ with finite trace
$\operatorname{tr}Q=\sum_{k=1}^{\infty}\lambda_{k}<\infty$. Then the above
$V$-valued stochastic process $W(t)$ is called a $Q$-Wiener process. Denote by
$L_{2}^{0}(V,H)$ the Hilbert space of all Hilbert-Schmidt operators from $V$
to $H$ equipped with the norm:

$\|D\|_{2}:=\|D\|_{L_{2}^{0}(V,H)}=\left(\sum_{k=1}^{+\infty}\|\sqrt{\lambda_{k}}De_{k}\|^{2}\right)^{\frac{1}{2}},\qquad
D\in L_{2}^{0}(V,H).$

Let $L^{2}(\mathcal{F}_{T},H):=L^{2}(\Omega,\mathcal{F}_{T},H)$ the Hilbert
space of all $\mathcal{F}_{T}$-measurable square integrable.
$L^{2}_{\mathcal{F}}([0,T];U)$ is the space of the $\mathcal{F}_{t}$-adapted,
$U$-valued measurable process $u(t,\omega)$ on $[0,T]$ such that
$\mathbb{E}\int_{0}^{T}\|u(t,\omega)\|^{2}dt<+\infty$.
$\mathcal{C}_{\mathcal{F}}([0,T];U)$ is the space of the
$\mathcal{F}_{t}$-adapted, $U$-valued measurable process $u(t,\omega)$ on
$[0,T]$ such that $\left(\mathbb{E}\|u(t,\omega)\|^{2}\right)^{\frac{1}{2}}$
is continuous. For brevity, we suppress the dependence of all mappings on
$\omega$ throughout the manuscript.

##  2\. Regularity of the stochastic convolutions for admissible operators

We start this section by giving some known consequences of the condition (A)
introduced in the first section. The constant $\gamma>0$ in (1.5) is a non-
decreasing function on $\alpha$ and $\lim_{\alpha\to
0}\alpha^{r}\gamma(\alpha)=0$ for any $r>0$. If $C\in{\mathcal{L}}(D(A),H)$
satisfies the condition (A) then by Hölder’s inequality one can see that there
exists $\alpha_{0}>0$ and a constant $\tilde{\gamma}\in(0,1)$ such that
$\|CT(\cdot)x\|_{L^{1}([0,\alpha_{0}],H)}\leq\tilde{\gamma}\|x\|$ for any
$x\in D(A)$. Thus $C$ is a Miyadera-Voigt perturbation for $A$. Hence the
operator $A^{C}:=A+C$ with domain $D(A^{C}):=D(A)$ generates a strongly
continuous semigroup on $H,$ see [11, p.199].

As shown in [40], the condition (A) implies that

$\displaystyle\begin{split}&T(t)x\in D(C_{\Lambda})\quad
a.e.\;t>0,\cr&\int_{0}^{\alpha}\|C_{\Lambda}T(t)x\|^{2}dt\leq\gamma(\alpha)^{2}\|x\|^{2},\quad\forall
x\in H,\;\forall\alpha>0,\end{split}$ (2.1)

where $C_{\Lambda}$ is the Yosida extension of $C$ for $A$, see notation in
the first section. The following result taken from [15, prop.3.3] shows a
regularity property for the deterministic convolution.

######  Proposition 2.1.

Assume that $A$ and $C\in{\mathcal{L}}(D(A),H)$ satisfy the conditions (S) and
(A), respectively. Then

$\displaystyle(\mathbb{T}*f)(t):=\int_{0}^{t}T(t-s)f(s)ds\in
D(C_{\Lambda}),\quad a.e.\;t\geq 0,$
$\displaystyle\left\|C_{\Lambda}(\mathbb{T}*f)\right\|_{L^{2}([0,\alpha],H)}\leq
c(\alpha)\|f\|_{L^{2}([0,\alpha],H)},$

for all $f\in L^{2}_{loc}(\mathbb{R}^{+},H)$ and $\alpha>0$ with
$c(\alpha):=\alpha^{\frac{1}{2}}\gamma(\alpha)>0$ is independent of $f$.
Moreover, $\lim_{\alpha\to 0}c(\alpha)=0$.

In the following we will prove a somehow an analogue regularity for the
stochastic convolution. Before doing this, let gather some facts about
fractional powers of operators.

######  Remark 2.2.

Assume that $A$ generates an analytic semigroup $\mathbb{T}:=(T(t))_{t\geq 0}$
on a Hilbert space $H$ and let $H_{\beta}:=D((-A)^{\beta})$ for some
$\beta\in(0,\frac{1}{2})$. This space is endowed with the norm
$\|x\|_{\beta}=\|(-A)^{\beta}x\|$. We recall that
$\|(-A)^{\beta}T(t)\|\leq\frac{M}{t^{\beta}}$ for $t>0,$ due to the
analyticity of $\mathbb{T}$, see e.g. [11]. Now clearly
$K:=(-A)^{\beta}|_{D(A)}$ satisfies the condition (A), and its Yosida
extension for $A$ is exactly $(-A)^{\beta},$ see [1, p.187]. We have the
following facts:

  * (i)

Assume that $H_{\beta}\subseteq\mathscr{Z}\subset H$ and
$\mathscr{C}:\mathscr{Z}\to H$ is a linear operator and define
$C:=\mathscr{C}|_{D(A)}$. Then $C$ satisfies the condition (A). In fact, let
$\alpha>0$ and $x\in D(A)$. Then

$\displaystyle\int^{\alpha}_{0}\|CT(t)x\|^{2}dt$
$\displaystyle=\int^{\alpha}_{0}\|\mathscr{C}T(t)x\|^{2}dt$
$\displaystyle\leq\|\mathscr{C}\|_{{\mathcal{L}}(H_{\beta},H)}\int^{\alpha}_{0}\|(-A)^{\beta}T(t)x\|^{2}dt.$

Furthermore, we have $H_{\beta}\subset D(C_{\Lambda})$ and
$C_{\Lambda}=\mathscr{C}$ on $H_{\beta}$.

  * (iii)

Let $\theta\in(\frac{1}{2},1)$ another exponent. Take
$\mathscr{Z}:=H_{\theta}$, so that $\mathscr{Z}\subset H_{\beta}\subset H$,
due to $\theta>\beta$. Note that the operator $(-A)^{\theta}|_{D(A)}$ is not
admissible for $A$, [4]. Thus a linear operator $\mathscr{C}:\mathscr{Z}\to H$
is not necessarily satisfies the condition (A).

In the rest of this section we are concerned with proving a regularity
property of the following stochastic convolution for general semigroups

$W_{A}^{\Phi}(t):=\int_{0}^{t}T(t-s)\Phi(s)dW(s).$

In the case of an analytic semigroup $\mathbb{T}$, we have the following
classical result in stochastic analysis proved in Theorem 5.15 and Theorem
6.14 of [10].

######  Theorem 2.3.

Let $\Phi(t)$, $t\geq 0$ an $L_{2}^{0}(V,H)$-valued predictable process such
that

$\mathbb{E}\int_{0}^{t}\|\Phi(s)\|^{2}_{2}ds<+\infty,\qquad t\geq 0.$

  * i)

If $A$ generates an analytic semigroup on $H$, then $W_{A}(t)\in
D((-A)^{\theta})$ for any $\theta\in(0,\frac{1}{2})$, $\mathbb{P}$-almost
surely and

$\mathbb{E}\int_{0}^{\alpha}\left\|(-A)^{\theta}W_{A}^{\Phi}(t)\right\|^{2}dt\leq\gamma(\alpha)^{2}\mathbb{E}\int_{0}^{\alpha}\|\Phi(t)\|_{2}^{2}dt,$

for all $\alpha>0$ and some constant $\gamma(\alpha)>0$.

  * ii)

If $A$ generates a contractive and analytic semigroup on $H$, then
$W_{A}^{\Phi}(t)\in D((-A)^{\frac{1}{2}})$, $\mathbb{P}$-almost surely and

$\mathbb{E}\int_{0}^{\alpha}\left\|(-A)^{\frac{1}{2}}W_{A}^{\Phi}(t)\right\|^{2}dt\leq\gamma(\alpha)^{2}\mathbb{E}\int_{0}^{\alpha}\|\Phi(t)\|_{2}^{2}dt,$

for all $\alpha>0$ and some constant $\gamma(\alpha)>0$.

The following result shows a regularity property for the stochastic
convolution without assuming any regularity on the semigroup $(T(t))_{t\geq
0}$ and for general unbounded linear operators $C$ which are not necessarily
closed or closable.

######  Theorem 2.4.

Let $A$ and $C\in{\mathcal{L}}(D(A),H)$ satisfy (S) and (A), and let
$(\Phi(t))_{t\geq 0}$ be an $L_{2}^{0}(V,H)$-valued predictable process such
that

$\mathbb{E}\int_{0}^{t}\|\Phi(s)\|^{2}_{2}ds<+\infty,\qquad t\geq 0.$

For all $\alpha>0,$

$\displaystyle\begin{split}&W_{A}^{\Phi}(t)\in D(C_{\Lambda}),\quad
a.e.\,t\geq 0,\;\mathbb{P}-a.s,\quad
and\cr&\mathbb{E}\int_{0}^{\alpha}\left\|C_{\Lambda}W_{A}^{\Phi}(t)\right\|^{2}dt\leq\gamma(\alpha)^{2}\mathbb{E}\int_{0}^{\alpha}\|\Phi(t)\|_{2}^{2}dt.\end{split}$
(2.2)

###### Proof.

Let $\mathcal{N}^{2}(0,\alpha;H)$ be the space of all predictable process
$\Phi:[0,T]\times\Omega\to L_{2}^{0}(V,H)$ such that

$\displaystyle\|\Phi\|_{\alpha}:=\mathbb{E}\int_{0}^{\alpha}\|\Phi(s)\|^{2}_{2}ds<+\infty.$

Let $\Phi\in\mathcal{N}^{2}(0,\alpha;H)$ and $\beta>\omega_{0}(A)$. By the
same arguments as in [25], for any $\lambda\in(\beta,\infty)$,

$\displaystyle\mathbb{E}\int_{0}^{\alpha}\|C\lambda
R(\lambda,A)W_{A}^{\Phi}(t)\|^{2}dt\leq\gamma(\alpha)^{2}\||\lambda
R(\lambda,A)\Phi(s)|\|_{\alpha}.$ (2.3)

This implies that for $\lambda,\mu\in(\beta,\infty),$

$\displaystyle\begin{split}\mathbb{E}\int_{0}^{\alpha}\|C\lambda&R(\lambda,A)W_{A}^{\Phi}(t)-C\mu
R(\mu,A)W_{A}^{\Phi}(t)\|^{2}dt\cr&\leq\gamma(\alpha)^{2}\|\lambda
R(\lambda,A)\Phi(s)-\mu R(\mu,A)\Phi(s)\|_{\alpha}.\end{split}$ (2.4)

Moreover,

$\displaystyle\|\lambda R(\lambda,A)$ $\displaystyle\Phi(s)-\mu
R(\mu,A)\Phi(s)\|^{2}_{2}$
$\displaystyle=\sum_{k=1}^{+\infty}\lambda_{k}\|\lambda
R(\lambda,A)\Phi(s)e_{k}-\mu R(\mu,A)\Phi(s)e_{k}\|^{2}.$

Since $\operatorname{tr}Q=\sum_{k=1}^{+\infty}\lambda_{k}<+\infty$, then
$(\lambda R(\lambda,A)\Phi(s))_{\lambda>\beta}$ is a Cauchy sequence in
$L_{2}^{0}(V,H)$. On the other hand, we have
$\Phi\in\mathcal{N}^{2}(0,\alpha;H)$, and hence $(\lambda
R(\lambda,A)\Phi(s))_{\lambda>\beta}$ is a Cauchy sequence in
$\mathcal{N}^{2}(0,\alpha;H)$ as well. Now, the inequality (2.4) implies that
$(C\lambda R(\lambda,A)W_{A}^{\Phi}(\cdot))_{\lambda>\beta}$ is a Cauchy
sequence in $L^{2}_{\mathcal{F}}([0,\alpha];H)$. Thus there exists
$(\lambda_{k})_{k\in\mathbb{N}}\subset\rho(A)\cap\mathbb{R}$ such that the
sequence $\left(C\lambda_{k}R(\lambda_{k},A)W_{A}^{\Phi}(t)\right)_{k}$
converges for a.e $t\geq 0$, $\mathbb{P}$-almost surely. Consequently,
$W_{A}^{\Phi}(t)\in D(C_{\Lambda})$ for a.e $t\geq 0$, $\mathbb{P}$-almost
surely. ∎

######  Remark 2.5.

We can use Theorem 2.4, to give an alternative proof to Theorem 2.3. In fact,
if we assume that $A$ generates an analytic semigroup $\mathbb{T}$ on $H$ and
$\theta\in(0,\frac{1}{2}),$ then from Remark (2.2), we know that
$C:=(-A)^{\theta}$ with domain $D(C)=D(A)$ satisfies the condition (A) and
$C_{\Lambda}=(-A)^{\theta}$. Thus $W_{A}^{\Phi}(t)\in
D\left((-A)^{\theta}\right)$. If in addition $\mathbb{T}$ is a contraction
semigroup, then $C:=(-A)^{\frac{1}{2}}$ with domain $D(C)=D(A)$ satisfies the
condition (A), due to Le Merdy [28]. Thus, Theorem 2.4 shows also that
$W_{A}^{\Phi}(t)\in D((-A)^{\frac{1}{2}})$.

In the following we will give an application to Theorem 2.4. Let
$\mathscr{B}:\mathscr{Z}\to H$ and $\mathscr{M}:H\to L^{0}_{2}(V,H)$ as in the
introductory section.

######  Definition 2.6.

Let $A$ satisfies the condition (S) and denote $B:=\mathscr{B}$ with domain
$D(B):=D(A)$. A process $X\in{\mathcal{C}}_{\mathcal{F}}([0,T],H)$ is called a
mild solution of the stochastic equation (1.1) if the operator $B$ admits an
extension $\tilde{B}:D(\tilde{B})\subset H\to H$ such that $X(t)\in
D(\tilde{B})$ for a.e. $t>0$, $\mathbb{P}$-a.s, $X(t)$ is
${\mathcal{F}}_{t}$-adapted for any $t\in[0,T],$ the maps $t\mapsto X(t)$ and
$t\mapsto\tilde{B}X(t)$ are measurable, $2$-integrable on $\Omega\times[0,T]$,
and satisfies the integral equation

$\displaystyle X(t)=T(t)\xi+\int^{t}_{0}T(t-s)\mathscr{M}(\tilde{B}X(s))dW(s)$

for any $\xi\in L^{2}(\mathcal{F}_{0},H),$ and $t\geq 0$, $\mathbb{P}$-a.s.

The following result introduce conditions for the existence and uniqueness of
the mild solution of the semilinear stochastic equation (1.1).

######  Theorem 2.7.

Assume that $A$ satisfies (S), $\mathscr{M}$ satisfies the condition (L), and
the unbounded linear operator $B$ satisfies the admissibility condition (A’).
Then there exists a unique $X\in{\mathcal{C}}_{\mathcal{F}}([0,T],H)$ of the
stochastic equation (1.1) such that $X(t)\in D(B_{\Lambda})$ for a.e. $t>0$
and $\mathbb{P}$-a.s., and

$\displaystyle
X(t)=T(t)\xi+\int^{t}_{0}T(t-s)\mathscr{M}(B_{\Lambda}X(s))dW(s)$

for any $t\geq 0$ and $\xi\in L^{2}(\mathcal{F}_{0},H),$ where $B_{\Lambda}$
is the Yosida extension of $B$ for $A$.

###### Proof.

Let $u\in L^{2}_{\mathcal{F}}([0,\alpha],H)$ for $\alpha>0$ and consider the
process

$\displaystyle X(t;u)=T(t)\xi+\int^{t}_{0}T(t-s)\mathscr{M}(u(s))dW(s),\quad
t\in[0,\alpha],\quad\xi\in L^{2}(\mathcal{F}_{0},H).$ (2.5)

According to (2.1) and Theorem 2.4, $X(t)\in D(B_{\Lambda})$ for $t>0$,
$\mathbb{P}$-a.s., and there exist constants $c>0$ and $\tilde{\gamma}>0$ such
that

$\displaystyle\mathbb{E}\int^{\alpha}_{0}\|B_{\Lambda}X(t)\|^{2}dt\leq
c\tilde{\gamma}(\alpha)\left(\mathbb{E}\|\xi\|^{2}+\mathbb{E}\|u\|_{L^{2}([0,\alpha],H)}\right)$

for any $\xi\in L^{2}(\mathcal{F}_{0},H)$ and $u\in
L^{2}_{\mathcal{F}}([0,\alpha],H)$, where $\tilde{\gamma}(\alpha)\to 0$ as
$\alpha\to 0$. We now select the following map

$\displaystyle\Gamma:L^{2}_{\mathcal{F}}([0,\alpha],H)\to
L^{2}_{\mathcal{F}}([0,\alpha],H),\quad(\Gamma u)(t)=B_{\Lambda}X(t;u).$

For any $u_{1}$ and $u_{2}$ in $L^{2}_{\mathcal{F}}([0,\alpha],H),$ we have

$\displaystyle\|\Gamma u_{1}-\Gamma
u_{2}\|_{L^{2}_{\mathcal{F}}([0,\alpha],H)}\leq\tilde{\gamma}(\alpha)\|u_{1}-u_{2}\|_{L^{2}_{\mathcal{F}}([0,\alpha],H)}.$

We can choose $\alpha_{0}>0$ such that $0<\tilde{\gamma}(\alpha_{0})<1$, so
that $\Gamma$ is a contraction on $L^{2}_{\mathcal{F}}([0,\alpha],H)$. By the
Banach’s fixed pint theorem, there is a unique $u\in
L^{2}_{\mathcal{F}}([0,\alpha],H)$ such $u=\Gamma u=B_{\Lambda}X(\cdot,u)$.
Now replacing $u$ in (2.5), we have

$\displaystyle
X(t)=T(t)\xi+\int^{t}_{0}T(t-s)\mathscr{M}(B_{\Lambda}X(s))dW(s),\quad
t\in[0,\alpha_{0}],\;\xi\in L^{2}(\mathcal{F}_{0},H).$

By standard arguments the restriction on $\alpha_{0}$ can be removed. ∎

######  Remark 2.8.

  * (i)

The linear case of the equation (1.1) (i.e. $\mathscr{M}=I$) is considered in
[10, Section 6.5, page 176], where additional conditions on the semigroup
$\mathbb{T}$ and the perturbation $B$ are assumed. In fact, in this reference
the authors are mainly based on the concept of maximal regularity and
interpolation spaces to introduce an appropriate condition on $B$. These facts
facilities the use of the classical Banach’s fixed theorem to prove the well-
posedness of the equation as well as the representation of the solution in
terms of a variation of constants formula. Compared with our result, we have
less conditions on $B$ and on the generator $A$, due to the regularity of the
stochastic convolution obtained in Theorem 2.4.

  * (ii)

In the case of $\mathscr{M}\equiv I$ , the equation (1.1) was also treated by
Bonaccorsi [3] Flandoli [12, Theorem 1.2] where the semigroup $(T(t))_{t\geq
0}$ satisfies a regularity condition and the operator $B$ is (somehow) zero-
class admissible for $A$. Moreover, in [3], the author used Malliavin calculus
to represent the solution via an appropriate variation of constants formula.
In Theorem 2.7 we do not assume any regularity on the semigroup $(T(t))_{t\geq
0}$.

  * (iii)

In [22], the author considered the KPZ equation driven by space-time white
noise replaced with its fractional derivatives of order $\gamma>0$ in spatial
variable. This equation has the same regularity as the solution of the
equation (1.1) with $A=\partial_{x}^{2}$, $\mathscr{M}=I$ and
$B=\partial_{x}^{\sigma}$. According to remark 2.2, $\partial_{x}^{\sigma}$ is
a zero-class admissible operator for $\partial_{x}^{2}$ for any $0<\sigma<1$.

######  Example 2.9.

Let $\mathscr{O}\subset\mathbb{R}^{n}$ be an open bounded set with a $C^{2}$
boundary $\partial\mathscr{O}$ and outer unit normal $\nu$ and put
$H=L^{2}(\mathscr{O})$ and $\mathscr{Z}=H^{2}(\mathscr{O})$. We consider the
following nonlinear initial value problem

$\displaystyle\begin{split}&\frac{\partial}{\partial t}X(t,x)=\Delta
X(t,x)+f\left(\displaystyle\int_{\partial\mathscr{O}}\Upsilon(x,y)c_{1}(y)X(t,y)dy\right)\frac{\partial}{\partial
t}W(t,x)\cr&\hskip 227.62204ptx\in\mathscr{O},t\in[0,T],\\\
&X(0,x)=g(x),\hskip 142.26378ptx\in\mathscr{O},\xi\in L^{2}(\mathscr{O}),\\\
&\nabla X(t,x)|\nu(x)=c_{2}(x)X(t,x),\hskip
85.35826ptx\in\partial\mathscr{O},t\in[0,T],\end{split}$ (2.6)

where $\Upsilon\in L^{\infty}(\mathscr{O}\times\partial\mathscr{O})$,
$c_{1},c_{2}\in C_{b}(\partial\mathscr{O})$, $f$ is a real valued function,
globally Lipschitz and $W(t)$ is $Q$-Wiener process on $L^{2}(\mathscr{O})$.
Define the following operators

$\displaystyle A=\Delta,\quad D(A)=\left\\{\varphi\in
H^{2}(\mathscr{O}):\nabla\varphi(x)|\nu(x)=c_{2}(x)\varphi(x),\quad
x\in\partial\mathscr{O}\right\\},$
$\displaystyle(\mathscr{M}(\phi))(x)=f(\phi(x)),\qquad x\in\mathscr{O},$
$\displaystyle(\mathscr{R}\varphi)(x)=\int_{\partial\mathscr{O}}\Upsilon(x,y)\varphi(y)dy,\qquad
x\in\mathscr{O},$ $\displaystyle(\Theta\varphi)(y)=c_{1}(y)\varphi(y),\quad
y\in\partial\mathscr{O},\quad\varphi\in L^{2}(\partial\mathscr{O})$
$\displaystyle\mathscr{B}=\mathscr{R}\Theta:H^{2}(\mathscr{O})\to
L^{2}(\mathscr{O}),$ $\displaystyle\mathscr{K}:H^{2}(\mathscr{O})\to
L^{2}(\partial\mathscr{O}),\quad(\mathscr{K}\varphi)(x)=c_{2}(x)\varphi(x),\quad
x\in\partial\mathscr{O}.$

It suffices to prove that $A$ is a generator of a strongly continuous
semigroup $\mathbb{T}$ on $H,$ and $B:=\mathscr{B}$ with domain $D(B)=D(A)$ is
a zero-class admissible operator for $A$. To this end, we will use a
perturbation result in [18]. In fact, the following operator

$A_{0}:=\Delta,\quad D(A_{0})=\left\\{\varphi\in
H^{2}(\mathscr{O}):\nabla\varphi(x)|\nu(x)=0,\quad
x\in\partial\mathscr{O}\right\\}$

generates an analytic semigroup $\mathbb{T}_{0}:=(\mathbb{T}_{0}(t))_{t\geq
0}$ on $H$. Now denote by $\varphi=\mathscr{N}\psi\in
H^{\frac{3}{2}}(\mathscr{O})$ the solution of the elliptic boundary value
problem $\varphi=0$ on $\mathscr{O}$ and $\nabla\varphi|\nu=\psi$ on
$\partial\mathscr{O}$ for $\psi\in L^{2}(\partial\mathscr{O})$. Then
$\mathscr{N}$ is continuous from $L^{2}(\partial\mathscr{O})$ to
$D((-A_{0})^{\beta})$ for any $\beta\in(0,\frac{3}{4})$. We select the
operators

$\displaystyle\Xi:=-A_{0}\mathscr{N}:L^{2}(\partial\mathscr{O})\to
D(A_{0}^{\ast})^{\prime},$ $\displaystyle K:=\mathscr{K},\qquad
D(K):=D(A_{0}).$

We will verify that the triple of operators $(A_{0},\Xi,K)$ satisfies the
condition of [18, Theorem 4.1]. In fact, for for any $t>0,$ $\theta\geq 0$ and
$0<\beta<3/4$

$\begin{split}(-A_{0})^{\theta}\mathbb{T}_{0}(t)\Xi\in{\mathcal{L}}(L^{2}(\partial\mathscr{O}),H),\quad\|(-A_{0})^{\theta}\mathbb{T}_{0}(t)\Xi\|\leq\kappa_{\beta}t^{\beta-\theta-1},\end{split}$
(2.7)

where $\kappa_{\beta}>0,$ is a constant, due to [26]. By choosing
$\beta\in(\frac{1}{2},\frac{3}{4})$ and $\theta=0$ (hence $2(1-\beta)<1$), we
obtain

$\displaystyle\left\|\int^{t}_{0}\mathbb{T}_{0}(t-s)\Xi
v(s)ds\right\|_{L^{2}(\Omega)}\leq\delta_{\beta}\|\|v\|_{L^{2}([0,t],L^{2}(\mathscr{O}))}$

for all $t>0$ and $v\in L^{2}([0,t],L^{2}(\mathscr{O})),$ where
$\delta_{\beta}>0$ is a constant, due to (2.7). In the terminology of control
theory, this means that $\Xi$ is an admissible control operator for $A_{0}$.
On the other hand, the operator $\mathscr{K}:D((-A_{0})^{\theta})\to
L^{2}(\mathscr{O})$ is uniformly bounded for any $\theta>\frac{1}{4}$, so that
$\mathscr{K}(-A_{0})^{-\theta}\in{\mathcal{L}}(L^{2}(\mathscr{O}))$ and
$\|\mathscr{K}(-A_{0})^{-\theta}\|\leq\eta$ for a constant $\eta>0$. Let us
now choose $\theta\in(\frac{1}{4},\frac{1}{2})$. We have

$\displaystyle\|Ke^{tA_{0}}\|\leq\eta\frac{M_{0}}{t^{\theta}},\quad(t>0).$
(2.8)

This shows that $K$ satisfies (1.5) with respect to $\mathbb{T}_{0}$ and a
constant $\gamma_{0}(\alpha)\to 0$ as $\alpha\to 0$. On the other hand, by
using Remark 2.2, the Yosida extension of $K$ for $A_{0}$ is exactly
$K_{\Lambda}=\mathscr{K}$. On the other hand, by using (2.7) and the fact that
then $\frac{1}{2}<1-(\beta-\theta)<1$, we obtain

$\displaystyle\left(\int^{\alpha}_{0}\left\|\mathscr{K}\int^{t}_{0}\mathbb{T}_{0}(t-s)\Xi
v(s)ds\right\|^{2}dt\right)^{\frac{1}{2}}\leq\eta\kappa_{\beta}\alpha^{\beta-\theta}\|v\|_{L^{2}([0,\alpha],L^{2}(\partial\mathscr{O}))}$
(2.9)

for any $v\in L^{2}(\partial\mathscr{O})$. This implies that the operator $A$
coincides with the following one

$\displaystyle A_{0}^{cl}:=A_{0}+{\mathbb{B}}\mathscr{K},\quad
D(A_{0}^{cl})=\\{\varphi\in
H^{2}(\mathscr{O}):(A_{0}+\Xi\mathscr{K})\varphi\in H\\},$

which generates a strongly continuous semigroup
$\mathbb{T}=(\mathbb{T}(t))_{t\geq 0}$ on $H$ such that

$\displaystyle\mathbb{T}(t)g=\mathbb{T}_{0}(t)g+\int^{t}_{0}\mathbb{T}_{0}(t-s)\Xi\mathscr{K}\mathbb{T}(s)gds,\qquad(t\geq
0,\;g\in H),$ (2.10)

see the proof of Theorem 4.1 of [18]. According to [39], if we define
$K^{A}:=\mathscr{K}$ with domain $D(K^{A})=D(A)$ then its Yosida extension for
$A$ is $K^{A}_{\Lambda}=K_{\Lambda}=\mathscr{K}$, and

$\displaystyle\int^{\alpha}_{0}\|\mathscr{K}\mathbb{T}(t)g\|^{2}dt\leq\tilde{\gamma}^{2}\|g\|^{2}_{H}$
(2.11)

for any $g\in D(A)$ and $\alpha>0$, and a constant
$\tilde{\gamma}:=\tilde{\gamma}(\alpha)>0$. Let us now prove that
$B:=\mathscr{B}$ with domain $D(B)=D(A)$ is a zero-class admissible operator
for $A$. To this end, we select $B_{0}:=\mathscr{B}$ with domain
$D(B_{0})=D(A_{0})$. As the operator $\mathscr{B}$ and $\mathscr{K}$ have the
same form, then they have the same properties. So that $B_{0}$ satisfies the
condition (2.8) if we replace $K$ by $B_{0}$. Also $\mathscr{B}$ satisfies the
inequality (2.9) if we replace $\mathscr{K}$ by $\mathscr{B}$. Then for any
$g\in D(A)$ and $t\geq 0,$

$\displaystyle B\mathbb{T}(t)g=\mathscr{B}\mathbb{T}(t)g$
$\displaystyle=\mathscr{B}\left(\mathbb{T}_{0}(t)g+\int^{t}_{0}\mathbb{T}_{0}(t-s)\Xi\mathscr{K}\mathbb{T}(s)gds\right)$
$\displaystyle=B_{0}\mathbb{T}_{0}(t)g+\mathscr{B}\int^{t}_{0}\mathbb{T}_{0}(t-s)\Xi\mathscr{K}\mathbb{T}(s)gds$

Remak that (2.9) holds also if we replace $\mathscr{K}$ by $\mathscr{B}$. Now
by using this fact and the estimate (2.11), for any $\alpha>0$, there exists a
constant $\kappa>0$ independent of $\alpha>0$ such that

$\displaystyle\int^{\alpha}_{0}\|B\mathbb{T}(t)g\|^{2}dt\leq\kappa\left(\alpha^{1-2\theta}+\alpha^{2(\beta-\theta)}\right)\|g\|^{2}_{H}\qquad(g\in
D(A)).$

This $B$ is a zero-class admissible operator for $A$, so it satisfies the
condition (A’). Let $B_{\Lambda}$ be the Yosida extension of $B$ for $A$, and
let us prove that $B_{\Lambda}=\mathscr{B}$. In fact, for a large $\lambda>0$
and by taking Laplace transform on the both sides of (2.10), we obtain

$\displaystyle
R(\lambda,A)=R(\lambda,A_{0})+R(\lambda,A_{0})\Xi\mathscr{K}R(\lambda,A)$

As for $K,$ the Yosida extension of $B_{0}$ for $A_{0}$ is
$(B_{0})_{\Lambda}=\mathscr{B}$. By using the fact
${\mathcal{B}}R(\lambda,A_{0})\Xi\to 0$ as $\lambda\to 0$ (see [39]), for any
$g\in H^{2}(\mathscr{O}),$

$\displaystyle\|\lambda\mathscr{B}R(\lambda,A_{0})\Xi\mathscr{K}R(\lambda,A)g\|\leq\|{\mathcal{B}}R(\lambda,A_{0})\Xi\|\left(\|\mathscr{K}\lambda
R(\lambda,A)g-\mathscr{K}g\|+\|\mathscr{K}g\|\right).$

This implies that for any $g\in H^{2}(\mathscr{O}),$

$\displaystyle\lim_{\lambda\to+\infty}B\lambda
R(\lambda,A)g=\lim_{\lambda\to+\infty}\mathscr{B}\lambda
R(\lambda,A_{0})g=\mathscr{B}g.$

Thus by Theorem 2.7, the solution of the equation (2.6) satisfies $X(t)\in
H^{2}(\mathscr{O})$ for a.e. $t>0$, $\mathbb{P}$-a.s. and

$\displaystyle
X(t)=\mathbb{T}(t)g+\int^{t}_{0}\mathbb{T}(t-s)\mathscr{M}(\mathscr{B}X(s))dW(s)$

for any $t\geq 0,$ and $g\in H$.

##  3\. The well-posedness

In this section we study the well-posedness of the semilinear stochastic
equation (1.2) (hence of (1.4)). Due to the form of the nonlinear term $G$
given in (1.3), it is not clear how to define the mild solution $X(\cdot)$ of
the stochastic equation (1.2). This is because the operator $\mathscr{C}$ is
only defined on a small domain of $H$. However, we will adopt a kind of
solution to the equation (1.2) that will be justified in the main result of
this section.

######  Definition 3.1.

Let the condition (S) be satisfied. A continuous process
$X(\cdot):[0,T]\mapsto H$ is a mild solution of (1.2) if there exists an
operator $\tilde{\mathscr{C}}:D(\tilde{\mathscr{C}})\subset H\to H$, extension
of $\mathscr{C}$ such that

  1. (1)

$X(t)$ is $\mathcal{F}_{t}$-adapted, for each $0\leq t\leq T$, $X(t)\in
D(\tilde{\mathscr{C}})$ a.e $t\geq 0,$ $\mathbb{P}$-almost surely,

  2. (2)

$X(t)$ and $\tilde{\mathscr{C}}X(t)$ are measurable, and

$\int_{0}^{T}\|X(t)\|^{2}dt<\infty,\qquad\int_{0}^{T}\|\tilde{\mathscr{C}}X(t)\|^{2}dt<\infty,\quad\mathbb{P}\text{-a.s},$

  3. (3)

$X(\cdot)$ satisfies the stochastic integral equation

$\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(\tilde{\mathscr{C}}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(X(s))dW(s),$

for all $t\in[0,T]$, $\mathbb{P}$-almost surely.

The following technical results are needed to prove the main results of this
sections.

######  Lemma 3.2.

Let assumptions (S), (A) and (L) be satisfied and let $\xi\in
L^{2}(\mathcal{F}_{0},H)$. For all $u\in\mathcal{C}_{\mathcal{F}}([0,T];H)$
there exists a sequence
$(X_{n})_{n\in\mathbb{N}}\subset\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that
for all $n\in\mathbb{N}$

$X_{n}(t)\in D(C_{\Lambda})\quad\text{a.e.}~{}t\geq 0,\mathbb{P}-\text{almost
surely},~{}\text{and}$
$\displaystyle\begin{split}&X_{0}(t)=T(t)\xi+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s),\cr&X_{n+1}(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X_{n}(s))ds\cr&\hskip
142.26378pt+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s).\end{split}$ (3.1)

###### Proof.

Let $\xi\in L^{2}(\mathcal{F}_{0},H)$ and
$u\in\mathcal{C}_{\mathcal{F}}([0,T];H)$. Using the condition ${\bf(A)}$ and
(2.1), we have $X_{0}(t)\in D(C_{\Lambda})$ for a.e $t\geq 0$ and
$\mathbb{P}$-a.s, and

$\displaystyle\mathbb{E}\int^{\alpha}_{0}\|C_{\Lambda}X_{0}(s)\|^{2}ds<\infty,$
(3.2)

for any $\alpha>0$. This allows us to define the following process

$\displaystyle
X_{1}(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X_{0}(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s),$

for all $u\in{\mathcal{C}}_{\mathcal{F}}([0,T];H)$. It follows from (3.2),
Proposition 2.1 and Proposition 2.4 that $X_{1}(t)\in D(C_{\Lambda})$ a.e
$t\geq 0$, $\mathbb{P}$-a.s., and

$\displaystyle\mathbb{E}\int^{\alpha}_{0}\|C_{\Lambda}X_{1}(s)\|^{2}ds<\infty,$

for any $\alpha>0$. Therefore, by induction, $X_{n}(t)\in D(C_{\Lambda})$ a.e
$t\geq 0$, $\mathbb{P}$-a.s., and

$\displaystyle\mathbb{E}\int^{\alpha}_{0}\|C_{\Lambda}X_{n}(s)\|^{2}ds<\infty,$

and the following sequence

$X_{n+1}(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X_{n}(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s),$

is well defined for any $n\in\mathbb{N}$. ∎

######  Lemma 3.3.

Let assumptions (S), (A) and (L) be satisfied. For any process
$u\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ there exists a unique
$X(\cdot)\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that

$\displaystyle X(t)\in D(C_{\Lambda}),\quad\text{a.e}~{}t\geq
0,~{}\mathbb{P}\text{-almost surely},$ $\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s).$
(3.3)

###### Proof.

Let $X_{n}$ be as in (3.1). Using (2.1) and Proposition 2.1, we have

$\displaystyle\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X_{n}(s)-C_{\Lambda}X_{n-1}(s)\|^{2}ds$
$\displaystyle\qquad\qquad\leq
c(t)^{2}k^{2}\mathbb{E}\int_{0}^{t}\left\|C_{\Lambda}X_{n-1}(\sigma)-C_{\Lambda}X_{n-2}(\sigma)\right\|^{2}d\sigma.$

for all $t\in[0,T]$. By induction, we get

$\displaystyle\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X_{n}(s)-C_{\Lambda}X_{n-1}(s)\|^{2}ds\leq\left(c(t)k\right)^{2n}\gamma(t)^{2}\mathbb{E}\|\xi\|^{2}.$

Since $\gamma(t)$ is nondecreasing in $t$, it follows that
$\gamma(t)\leq\gamma(T)$ and $c(t)\leq c(T)$ for any $t\in[0,T]$. Thus, for
any $t\in[0,T]$,

$\mathbb{E}\int_{0}^{T}\|C_{\Lambda}X_{n}(s)-C_{\Lambda}X_{n-1}(s)\|^{2}ds\leq\left(c(T)k\right)^{2n}\gamma(T)^{2}\mathbb{E}\|\xi\|^{2}.$
(3.4)

Moreover, $\lim_{t\to 0}c(t)=0$, then we can choose $T$ small enough such that
$c(T)k<1$ and consequently, $(C_{\Lambda}X)_{n}$ is a Cauchy sequence in
$L^{2}_{\mathcal{F}}([0,T];H)$. The case of general $T>0$ can be treated by
considering the equation in intervals $[0,T_{0}]$, $[T_{0},2T_{0}]$,$\dots$
with $T_{0}$ such that $c(T_{0})k<1$. On the other hands, simple calculations
lead to

$\sup_{t\in[0,T]}\mathbb{E}\|X_{n+1}-X_{n}(t)\|^{2}\leq
TM^{2}e^{2|\beta|T}k^{2}\int_{0}^{T}\|C_{\Lambda}X_{n}(s)-C_{\Lambda}X_{n-1}(s)\|^{2}ds.$

From (3.4), it follows that $(X_{n})_{n}$ is a Cauchy sequence in
${\mathcal{C}}_{\mathcal{F}}([0,T];H)$.
Now, let

$\displaystyle
X:=\lim_{n\to\infty}X_{n}~{}~{}\text{in}~{}~{}{\mathcal{C}}_{\mathcal{F}}([0,T];H),$
$\displaystyle
Y:=\lim_{n\to\infty}C_{\Lambda}X_{n}~{}~{}\text{in}~{}~{}L^{2}_{\mathcal{F}}([0,T];H).$

It is clear that $\int_{0}^{t}T(t-s)F(C_{\Lambda}X_{n}(s))ds$ converges to
$\int_{0}^{t}T(t-s)F(Y(s))ds$ as $n\to\infty$ for all $t\in[0,T]$. Thus,

$X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(Y(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))ds,$

and $X(t)\in D(C_{\Lambda})$ for $t\geq 0$ a.e, $\mathbb{P}$ almost surely.
Moreover, we have

$\mathbb{E}\int_{0}^{T}\|C_{\Lambda}X_{n}(s)-C_{\Lambda}X(s)\|^{2}\leq
c(T)^{2}k^{2}\mathbb{E}\int_{0}^{T}\|C_{\Lambda}X_{n}(s)-Y(s)\|^{2}ds,$

for all $T\geq 0$. It follows that $Y(t)=C_{\Lambda}X(t)$ and

$X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(u(s))dW(s)$

for all $t\in[0,T]$. ∎

######  Theorem 3.4.

Let assumptions (S), (A) and (L) be satisfied. For any initial process $\xi\in
L^{2}(\mathcal{F}_{0},H),$ the semilinear stochastic equation (1.2) has a
unique mild solution $X(\cdot)\in{\mathcal{C}}_{\mathcal{F}}([0,T];H)$ such
that

$\displaystyle X(t)\in D(C_{\Lambda}),\quad\text{a.e}~{}t\geq
0,~{}\mathbb{P}\text{-almost surely},$ $\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(X(s))dW(s).$

###### Proof.

Define the map
$\mathscr{R}:\mathcal{C}_{\mathcal{F}}([0,T];H)\to\mathcal{C}_{\mathcal{F}}([0,T];H)$
by

$\displaystyle(\mathscr{R}u)(t):=X(t,u),$

where $X(t,u)$ is given by (3.3). It suffices to prove that the map
$\mathscr{R}$ has a unique fixed point in
$\mathcal{C}_{\mathcal{F}}([0,T];H)$. To this end, consider
$u,v\in\mathcal{C}_{\mathcal{F}}([0,T];H)$. A similar calculations as in the
proof of Lemma 3.3 lead to

$\displaystyle\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X(s,u)-C_{\Lambda}X(s,v)\|^{2}ds$
$\displaystyle\qquad\leq
2c(T)^{2}k^{2}\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X(s,u)-C_{\Lambda}X(s,v)\|^{2}ds$
$\displaystyle\qquad\qquad+2\gamma(T)^{2}k^{2}\mathbb{E}\int_{0}^{t}\|u(s)-v(s)\|^{2}ds,$

for any $t\in[0,T]$. We have $\lim_{t\to 0}c(t)=0$, then we can choose $T>0$
such that $2c(T)^{2}k^{2}<\frac{1}{2}$, this implies that

$\displaystyle\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X(s,u)-C_{\Lambda}X(s,v)\|^{2}ds\leq
4k^{2}T\gamma(T)^{2}\sup_{t\in[0,T]}\mathbb{E}\|u(t)-v(t)\|^{2}.$

Consequently,

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\|\left(\mathscr{R}u\right)(t)-\left(\mathscr{R}v\right)(t)\|^{2}$
$\displaystyle\qquad\leq
2M^{2}k^{2}e^{2|\beta|T}T\mathbb{E}\int_{0}^{t}\|C_{\Lambda}X(s,u)-C_{\Lambda}X(s,v)\|^{2}ds$
$\displaystyle\qquad\qquad+2M^{2}k^{2}e^{2|\beta|T}\mathbb{E}\int_{0}^{t}\|u(s)-v(s)\|^{2}ds$
$\displaystyle\qquad\leq\zeta_{T}\sup_{t\in[0,T]}\mathbb{E}\|u(t)-v(t)\|^{2},$

where
$\zeta_{T}:=8M^{2}k^{4}e^{2|\beta|T}\gamma(T)^{2}T^{2}+2M^{2}k^{2}e^{2|\beta|T}T$.
We choose $T$ small enough such that $T=max\\{T>0,\quad
2c(T)^{2}k^{2}<\frac{1}{2}~{}~{}\text{and}~{}~{}\zeta_{T}<1\\}$ and
consequently, by the contraction principle, there exists a unique
$v\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that

$v(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s,v))ds+\int_{0}^{t}T(t-s)\mathscr{M}(v(s))dW(s).$

The case of general $T>0$ can be treated by considering the equation in
intervals $[0,T_{0}]$, $[T_{0},2T_{0}]$,$\dots$ with $T_{0}$ such that
$T_{0}=max\\{T_{0}>0,\quad
2c(T_{0})^{2}k^{2}<\frac{1}{2}~{}~{}\text{and}~{}~{}\zeta_{T_{0}}<1\\}$. On
the other hands, from Lemma 3.1, for all
$u\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ there exists a unique
$X(\cdot)\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that (3.3) is satisfied.
Thus, by the uniqueness, there exists a unique
$X(\cdot)\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that

$\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(X(s))dW(s).$

∎

We end this section by giving some examples that satisfy our abstract results.
In the following we give two examples, the first example is devoted to kind of
Kardar-Parizi-Zhang equation. The second is devoted to the Schrödinger
equation with non-local integral term.

######  Example 3.5.

Let $\mathscr{O}\subset\mathbb{R}^{n}$ be an open bounded set and put
$H=L^{2}(\mathscr{O})$. We consider the following nonlinear initial value
problem

$\displaystyle\begin{cases}dX(t,x)=\left[\Delta
X(t,x)+f\left((a.\nabla+c)X(t,x)\right)\right]dt\cr\hskip
85.35826pt+b(X(t,x))dW(t,x),&x\in\mathscr{O},t\in[0,T],\\\
X(t,x)=0,&x\in\partial\mathscr{O},t\in[0,T],\\\
X(0,x)=\xi(x),&x\in\mathscr{O},\xi\in L^{2}(\mathscr{O}),\end{cases}$ (3.5)

where $a\in L^{\infty}(\mathscr{O};\mathbb{C}^{n})$, $c\in
L^{\infty}(\mathscr{O})$ and $W(t)$ is $Q$-Wiener process on
$V=H=L^{2}(\mathscr{O})$. We assume that $f$, $b$ are real valued functions
and globally Lipschitz. Define $A:D(A)\to H$ to be

$\displaystyle A\phi=\Delta\phi,\qquad D(A)=H^{2}(\mathscr{O})\cap
H^{1}_{0}(\mathscr{O}).$

So that $-A$ is a strictly positive densely defined operator on $H$. It is
known that $A$ generates a strongly continuous and diagonalizable semigroup
$(T(t))_{t\geq 0}$ on $H$. We select
$H_{\frac{1}{2}}=D((-A)^{\frac{1}{2}})=H^{1}_{0}(\mathscr{O})$. We now define
the following operators

$\displaystyle\mathscr{C}z=a\cdot\nabla z+cz,\quad z\in H_{\frac{1}{2}},$
$\displaystyle(F\phi)(x)=f(\phi(x)),\quad x\in\mathscr{O},$
$\displaystyle(\mathscr{M}(\phi)u)(x)=b(\phi(x))u(x),\quad
x\in\mathscr{O},\quad u\in H.$

Thus, the system (3.5) becomes

$\displaystyle\begin{cases}dX(t)=\left[AX(t)+F\left(\mathscr{C}X(t)\right)\right]dt+\mathscr{M}(X(t))dW(t),&t\in[0,T],\\\
X(0)=\xi.\end{cases}$

Moreover, it is shown in [37, Example 5.1.4] that $C:=\mathscr{C}$ with domain
$D(C)=D(A)$ satisfies the condition (A). Also as in Remark 2.2 we have
$C_{\Lambda}=\mathscr{C}$. So, for each $\xi\in L^{2}(\mathcal{F}_{0},H)$,
(3.5) has a unique mild solution
$X(\cdot)\in{\mathcal{C}}_{\mathcal{F}}([0,T];H)$ such that

$\displaystyle X(t)\in H_{\frac{1}{2}},\quad\text{a.e}~{}t\geq
0,\mathbb{P}\text{-almost surely},$ $\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(\mathscr{C}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(X(s))dW(s).$

######  Example 3.6.

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$, $n\geq 2$ with
sufficiently smooth boundary
$\partial\Omega=\overline{\Gamma_{0}\cup\Gamma_{1}}$,
$\Gamma_{0}\cap\Gamma_{1}=\emptyset$. We consider the following semilinear
stochastic Schrodinger equation with non-local integral term

$\displaystyle\begin{cases}dX(t,x)=\left[i\Delta
X(t,x)+f\left(i\displaystyle\int_{\Gamma_{1}}\Upsilon(x,y)X(t,y)dy\right)\right]dt\cr\hskip
85.35826pt+b(X(t,x))dW(t,x),&x\in\Omega,t\in[0,T],\\\
X(t,x)=0,&x\in\partial\Gamma_{0},t\in[0,T],\\\ \frac{\partial
X}{\partial\nu}(t,x)=0,&x\in\partial\Gamma_{1},t\in[0,T],\\\
X(0,x)=\xi(x),&x\in\Omega,\xi\in L^{2}(\Omega),\end{cases}$ (3.6)

where $\nu$ is the unit normal vector,
$\Upsilon:\overline{\Omega}\times\partial\Omega\to\mathbb{R}$ is a continuous
function and $W(t)$ is $Q$-Wiener process on $L^{2}(\Omega)$. As usual, we
write the equation in an abstract form. We introduce the space
$H=L^{2}(\Omega)$. We define the operators

$\displaystyle A=i\Delta,\quad D(A)=\left\\{g\in H^{2}(\Omega),\qquad
g_{|\Gamma_{0}}=\left.\frac{\partial
g}{\partial\nu}\right|_{\Gamma_{1}}=0\right\\},$
$\displaystyle\mathscr{C}g=i\int_{\Gamma_{1}}\Upsilon(\cdot,y)g(y)dy,\quad
g\in H^{2}(\Omega),$ $\displaystyle(F\phi)(x)=f(\phi(x)),\quad x\in\Omega,$
$\displaystyle(\mathscr{M}(\phi)u)(x)=b(\phi(x))u(x),\quad x\in\Omega,\quad
u\in H,$

and rewrite (3.6) as

$\displaystyle\begin{cases}dX(t)=\left[AX(t)+F\left(\mathscr{C}X(t)\right)\right]dt+\mathscr{M}(X(t))dW(t),&t\in[0,T],\\\
X(0)=\xi.\end{cases}$

It is well known that the operator $A$ generates an unitary group
$(T(t))_{t\geq 0}$ on $H$. Moreover, it is shown in [27, Example 4.2] that the
operator $C:=\mathscr{C}$ with domain $D(C)=D(A)$ satisfies the condition (A).
Therefore, by Theorem 3.4, the equation (3.6) has unique mild solution
$X(\cdot)\in\mathcal{C}_{\mathcal{F}}([0,T];H)$ such that

$\displaystyle X(t)\in D(C_{\Lambda}),\quad\text{a.e}~{}t\geq
0,\mathbb{P}\text{-almost surely},$ $\displaystyle
X(t)=T(t)\xi+\int_{0}^{t}T(t-s)F(C_{\Lambda}X(s))ds+\int_{0}^{t}T(t-s)\mathscr{M}(X(s))dW(s).$

##  4\. The Feller property of the transition semigroup

We are interested here in the continuous dependence of the mild solution of
(1.2) on the initial data and the boundedness of the moments. Specifically, we
have the following.

######  Theorem 4.1.

Under the assumptions of Theorem 3.4, there exists $C_{T}>0$ such that, for
arbitrary $\xi$, $\eta\in L^{2}(\mathcal{F}_{0},H)$ and $t\in[0,T]$, the
following estimate holds

$\mathbb{E}\|X(t,\xi)-X(t,\nu)\|^{2}\leq C_{T}\mathbb{E}\|\xi-\nu\|^{2}.$

###### Proof.

Let $\xi$, $\eta\in L^{2}(\mathcal{F}_{0},H)$ and set
$M=\sup_{t\in[0,T]}\|T(t)\|$. Let $T_{0}>0$ such that
$3k^{2}c(T_{0})^{2}<\frac{1}{2}$. By the uniqueness of the solution we have
that

$X(t,\xi)=\mathbbm{1}_{[0,T_{0}]}X_{1}(t,\xi)+\mathbbm{1}_{[T_{0},2T_{0}]}X_{2}(t,\xi),\quad\forall
t\in[0,2T_{0}],$

where,

$\displaystyle X_{1}(t,\xi)=T(t)\xi+\int_{0}^{t}T(t-s)$ $\displaystyle
F(C_{\Lambda}X_{1}(\xi,s))ds$
$\displaystyle+\int_{0}^{t}T(t-s)\mathscr{M}(X_{1}(s,\xi))dW(s)$

and

$\displaystyle X_{2}(t,\xi)=T(t-T_{0})X_{1}(T_{0},\xi)$
$\displaystyle+\int_{T_{0}}^{t}T(t-s)F(C_{\Lambda}X_{2}(s,\xi))ds$
$\displaystyle+\int_{T_{0}}^{t}T(t-s)\mathscr{M}\left(X_{2}(s,\xi)\right)dW(s).$

For all $t\in[0,T_{0}]$ we have

$\displaystyle\|C_{\Lambda}X_{1}(s,\xi)-C_{\Lambda}X_{1}(s,\eta)\|^{2}$
$\displaystyle\qquad\leq 3\|C_{\Lambda}T(s)(\xi-\eta)\|^{2}$
$\displaystyle\qquad\qquad+3\left\|C_{\Lambda}\int_{0}^{s}T(s-\sigma)\left[F(C_{\Lambda}X_{1}(\sigma,\xi))-F(C_{\Lambda}X_{1}(\sigma,\eta))\right]d\sigma\right\|^{2}$
$\displaystyle\qquad\qquad+3\left\|C_{\Lambda}\int_{0}^{s}T(s-\sigma)\left[\mathscr{M}(X_{1}(\sigma,\xi))-\mathscr{M}(X_{1}(\sigma,\eta))\right]dW(\sigma)\right\|^{2}.$

Using Proposition 2.1 and Proposition 2.4 we get

$\displaystyle\int_{0}^{t}\mathbb{E}\|C_{\Lambda}X_{1}(s,\xi)-C_{\Lambda}X_{1}(s,\eta)\|^{2}ds$
$\displaystyle\qquad\leq 3\gamma(T_{0})^{2}\mathbb{E}\|\xi-\eta\|^{2}$
$\displaystyle\qquad\qquad+3k^{2}c(T_{0})^{2}\int_{0}^{t}\mathbb{E}\left\|C_{\Lambda}X_{1}(s,\xi)-C_{\Lambda}X_{1}(s,\eta)\right\|^{2}ds$
$\displaystyle\qquad\qquad+3k^{2}\gamma(T_{0})^{2}\int_{0}^{t}\mathbb{E}\left\|X_{1}(s,\xi)-X_{1}(s,\eta)\right\|^{2}ds.$

Then

$\displaystyle\begin{split}&\int_{0}^{t}\mathbb{E}\|C_{\Lambda}X_{1}(s,\xi)-C_{\Lambda}X_{1}(s,\eta)\|^{2}ds\cr&\qquad\leq
6\gamma(T_{0})^{2}\mathbb{E}\|\xi-\eta\|^{2}+6k^{2}\gamma(T_{0})^{2}\int_{0}^{t}\mathbb{E}\left\|X_{1}(s,\xi)-X_{1}(s,\eta)\right\|^{2}ds,\end{split}$
(4.1)

for all $t\in[0,T_{0}]$. Repeating the same argument as above and using a
change of variables, we obtain

$\displaystyle\int_{T_{0}}^{t}\mathbb{E}\|C_{\Lambda}X_{2}(s,\xi)-C_{\Lambda}X_{2}(s,\eta)\|^{2}ds$
$\displaystyle\qquad\qquad\leq
6\gamma(T_{0})^{2}\mathbb{E}\|X(T_{0},\xi)-X(T_{0},\eta)\|^{2}$
$\displaystyle\qquad\qquad\qquad+6k^{2}\gamma(T_{0})^{2}\int_{T_{0}}^{t}\mathbb{E}\left\|X_{2}(s,\xi)-X_{2}(s,\eta)\right\|^{2}ds,$

for all $t\in[T_{0},2T_{0}]$. Moreover, simple calculations lead to

$\displaystyle\mathbb{E}\|X(T_{0},\xi)-X(T_{0};\eta)\|^{2}$
$\displaystyle\qquad\leq 3M^{2}\mathbb{E}\|\xi-\mu\|^{2}$
$\displaystyle\qquad\qquad+3T_{0}M^{2}k^{2}\mathbb{E}\int_{0}^{T_{0}}\|C_{\Lambda}X_{1}(s,\xi)-C_{\Lambda}X_{1}(s,\eta)\|^{2}ds$
$\displaystyle\qquad\qquad+3M^{2}k^{2}\mathbb{E}\int_{0}^{T_{0}}\|X_{1}(s,\xi)-X_{1}(s,\eta)\|^{2}ds.$

Using (4.1), we get

$\displaystyle\mathbb{E}\|X(T_{0},\xi)-X(T_{0};\eta)\|^{2}$
$\displaystyle\qquad\leq\left(3M^{2}+18T_{0}M^{2}k^{2}\gamma(T_{0})^{2}\right)\mathbb{E}\|\xi-\mu\|^{2}$
$\displaystyle\qquad\qquad+\left(3M^{2}k^{2}+18M^{2}k^{2}T_{0}\gamma(T_{0})^{2}\right)\mathbb{E}\int_{0}^{T_{0}}\|X_{1}(s,\xi)-X_{1}(s,\eta)\|^{2}ds.$

It follows that

$\displaystyle\begin{split}&\int_{T_{0}}^{t}\mathbb{E}\|C_{\Lambda}X_{2}(s,\xi)-C_{\Lambda}X_{2}(s,\eta)\|^{2}ds\cr&\qquad\leq
6\gamma(T_{0})^{2}\left(3M^{2}+18M^{2}k^{2}T_{0}\gamma(T_{0})^{2}\right)\mathbb{E}\|\xi-\eta\|^{2}\\\
&\qquad+6\gamma(T_{0})^{2}\left(3M^{2}k^{2}+18M^{2}k^{2}T_{0}\gamma(T_{0})^{2}\right)\int_{0}^{T_{0}}\mathbb{E}\left\|X_{1}(s,\xi)-X_{1}(s,\eta)\right\|^{2}ds\\\
&\qquad+6k^{2}\gamma(T_{0})^{2}\int_{T_{0}}^{t}\mathbb{E}\left\|X_{2}(s,\xi)-X_{2}(s,\eta)\right\|^{2}ds.\end{split}$
(4.2)

Thus, from (4.1) and (4.2), we have

$\displaystyle\begin{split}&\int_{0}^{t}\mathbb{E}\|C_{\Lambda}X(s,\xi)-C_{\Lambda}X(s,\eta)\|^{2}ds\cr&\qquad\leq
C_{T_{0},1}\mathbb{E}\|\xi-\eta\|^{2}+C_{T_{0},2}\int_{0}^{t}\mathbb{E}\left\|X(s,\xi)-X(s,\eta)\right\|^{2}ds\end{split}$
(4.3)

for all $t\in[0,2T_{0}]$, where $C_{T_{0},1}$, $C_{T_{0},2}$ are constants.
This process can be repeated to obtain (4.3) for all $t\in[0,T]$, $T>0$. On
the other hand, we have

$\displaystyle\mathbb{E}\|X(t,\xi)-X(t,\eta)\|^{2}$ $\displaystyle\leq$
$\displaystyle 3M^{2}\mathbb{E}\|\xi-\eta\|^{2}$
$\displaystyle+3M^{2}k^{2}T\int_{0}^{t}\mathbb{E}\|C_{\Lambda}X(s,\xi)-C_{\Lambda}X(s,\eta)\|^{2}ds$
$\displaystyle+3M^{2}k^{2}\int_{0}^{t}\mathbb{E}\|X(s,\xi)-X(s,\eta)\|^{2}ds,$

for all $t\in[0,T]$. Now, by using the inequality (4.3) we obtain

$\displaystyle\mathbb{E}\|X(t,\xi)-X(t,\eta)\|^{2}$ $\displaystyle\leq$
$\displaystyle C_{T,3}\mathbb{E}\|\xi-\eta\|^{2}$
$\displaystyle+C_{T,4}\int_{0}^{t}\mathbb{E}\|X(s,\xi)-X(s,\eta)\|^{2}ds,$

where $C_{T,3}$, $C_{T,4}$ are constants. Using Gronwall’s lemma, the estimate
of the theorem follows. ∎

With the use of Theorem 3.4 and Theorem 4.1, we can now prove the Feller
property of the associated semigroup associated to the stochastic evolution
equation (1.2). In fact, under the condition of Theorem 3.4, we define

$\displaystyle P_{t}\phi(x):=\mathbb{E}(\phi(X(t,x))),\quad t\in[0,T],\;x\in
H,$

where $\phi(\cdot)$ is bounded and Borel measurable function. In addition, we
select

$\displaystyle\mathcal{C}_{b}(H):=\left\\{\phi:H\to\mathbb{R}:\phi\quad\text{is
bounded and continuous function}\right\\}.$

We have the following result.

######  Proposition 4.2.

Under the hypotheses (S), (A) and (L) we have

$\displaystyle P_{t}\mathcal{C}_{b}(H)\subset\mathcal{C}_{b}(H)$

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

##  5\. Applications to semilinear stochastic equations of neutral type

We consider the following semi-linear stochastic equation of neutral type

$\displaystyle\begin{cases}d(X(t)-\mathscr{D}X_{t})=[A(X(t)-\mathscr{D}X_{t})+F(LX_{t})]dt\cr\hskip
149.37697pt+B(X_{t})dW(t),&t\in[0,T]\\\
\left(X(t)-\mathscr{D}X_{t}\right)_{|_{t=0}}=\xi,\qquad
X_{0}=\phi,\end{cases}$ (5.1)

where $A:D(A)\subset H\to H$ is the generator of a $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ on a Hilbert space $H$, the process $X:[-r,\infty)\to X$
and its history $X_{t}:=X(t+\cdot):[-r,0]\to H$ is defined for any $t\geq 0$
by

$\displaystyle X_{t}(\theta)=\begin{cases}X(t+\theta),&-t\leq\theta\geq
0,\cr\phi(t+\theta),&-r\leq\theta\leq-t.\end{cases}$

The initial conditions $X(0)=\xi\in L^{2}(\mathcal{F}_{0},H)$, $\phi\in
L^{2}_{\mathcal{F}_{0}}([-r,0],H)$ and $W(t)$ is a real standard Wiener
process. Here $F:H\to H$, $B:L^{2}([-r,0],H)\to H$ are nonlinear Lipschitz
continuous and $L,\mathscr{D}:W^{1,2}([-r,0],H)\to H$ are the following
Riemann-Stieltjes integrals

$\displaystyle
L\phi:=\int_{-r}^{0}d\mu(\theta)\phi(\theta),\quad\text{and}\quad\mathscr{D}\phi:=\int_{-r}^{0}d\nu(\theta)\phi(\theta),$
(5.2)

where $\nu,\mu:[-r,0]\to\mathcal{L}(H)$ are of bounded variation and non-
atomic at zero; that is

$\lim_{\epsilon\to 0}|\mu|([-\epsilon,0])=0,\qquad\lim_{\epsilon\to
0}|\nu|([-\epsilon,0])=0.$

######  Definition 5.1.

A pair $(X(t),X(t+\cdot))$ is a solution of (5.1) if there exist extensions
$\tilde{\mathscr{D}}$ and $\tilde{L}$ of $\mathscr{D}$ and $L$, respectively,
such that

$\displaystyle X_{t}\in D(\tilde{\mathscr{D}})\cap D(\tilde{L})\quad\text{for
a.e}~{}t\geq 0,\mathbb{P}-\text{almost surely},\;and$ $\displaystyle
X(t)=\tilde{\mathscr{D}}X_{t}+T(t)\xi+\int^{t}_{0}T(t-s)F(\tilde{L}X_{s})ds+\int^{t}_{0}T(t-s)B(X_{s})dW(s).$

The equation (5.1) was treated by Webb [38, Section 4] in the particular case
$\mathscr{D}\equiv B\equiv 0$. The method employed consists in constructing a
semigroup of nonlinear operators which may be associated with the solutions of
this equation.

In order to reformulate the equation (5.1) to the equation (1.2), we select

$\displaystyle Z(t):=X(t)-\mathscr{D}X_{t},\qquad t\geq 0.$ (5.3)

Then the neutral equation (5.1) becomes boundary valued stochastic equation

$\displaystyle\begin{cases}dZ(t)=[AZ(t)+F(LX_{t})]dt+B(X_{t})dW(t),&t\in[0,T]\\\
Z(0)=\xi,\qquad X_{0}=\phi\\\ X(t)=Z(t)+\mathscr{D}X_{t},&t\geq 0.\end{cases}$
(5.4)

We now introduce product spaces

$\displaystyle\mathcal{H}:=H\times
L^{2}([-r,0],H),\qquad\mathcal{Z}:=D(A)\times W^{1,2}([-r,0],H).$

The space $\mathcal{H}$ is endowed with the following norm

$\displaystyle\left\|(\begin{smallmatrix}x\\\
g\end{smallmatrix})\right\|:=\|x\|_{H}+\|g\|_{L^{2}([-r,0],H)}$

On the other hand, we introduce the operators

$\displaystyle\mathfrak{A}:=\begin{pmatrix}A&0\\\
0&\frac{d}{d\theta}\end{pmatrix},\quad
D(\mathfrak{A}):=\left\\{(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in\mathcal{Z}:\psi(0)=x+\mathscr{D}\psi\right\\}$
$\displaystyle\mathcal{P}:=\begin{pmatrix}0&L\\\
0&0\end{pmatrix}:D(\mathfrak{A})\to\mathcal{H},$ $\displaystyle
Qg=g^{\prime},\qquad D(Q)=\\{g\in W^{1,2}([-r,0],H):g(0)=0\\},$
$\displaystyle\mathcal{F}:\mathcal{H}\to\mathcal{H},\quad\mathcal{F}(\begin{smallmatrix}x\\\
g\end{smallmatrix})=(\begin{smallmatrix}F(x)\\\ 0\end{smallmatrix}),$
$\displaystyle\mathscr{M}:\mathcal{H}\to\mathcal{H},\quad\mathscr{M}(\begin{smallmatrix}x\\\
g\end{smallmatrix})=(\begin{smallmatrix}B(g)\\\ 0\end{smallmatrix}).$

Moreover, we introduce the following new state

$\displaystyle\varrho(t)=\left(\begin{smallmatrix}Z(t)\\\
X_{t}\end{smallmatrix}\right),\qquad t\geq 0.$

With the above notation, the problem (5.4) is reformulated as

$\displaystyle\begin{cases}d\varrho(t)=[\mathfrak{A}\varrho(t)+\mathcal{F}(\mathcal{P}\varrho(t))]dt+\mathscr{M}(\varrho(t))dW(t),&t\in[0,T]\\\
\varrho(0)=(\begin{smallmatrix}\xi\\\ \phi\end{smallmatrix}).\end{cases}$
(5.5)

######  Theorem 5.2.

Let assumption (S) be satisfied. The neutral equation (5.1) admits a unique
solution $(X(t),X(t+\cdot))$ such that

$\displaystyle X_{t}\in D(\mathscr{D}_{0,\Lambda})\cap
D(L_{0,\Lambda})\quad\text{for a.e}~{}t\geq 0,\mathbb{P}-\text{almost
surely},\;and$ $\displaystyle
X(t)=\mathscr{D}_{0,\Lambda}X_{t}+T(t)\xi+\int^{t}_{0}T(t-s)F(L_{0,\Lambda}X_{s})ds+\int^{t}_{0}T(t-s)B(X_{s})dW(s),$

where $L_{0,\Lambda}$ and $\mathscr{D}_{0,\Lambda}$ are the Yosida extensions
of $L$ and $\mathscr{D}$ with respect to $Q$, respectively.

###### Proof.

It is known that (see e.g. [16], [19]) that the operator $\mathfrak{A}$ is the
generator of a $C_{0}$–semigroup $(\mathfrak{T}(t))_{t\geq 0}$ on
$\mathcal{H}$. On the other hand, as $F:H\to H$ and $B:L^{2}([-r,0],H)\to H$
are Lipschitz functions, then it is so for the functions $\mathcal{F}$ and
$\mathscr{M}$. Hence the condition ${\bf(L)}$ is satisfied for the equation
(5.5). In order to apply Theorem 3.4, it is necessary to show that the
operators $\mathfrak{A}$ and $\mathcal{P}$ satisfy the condition ${\bf(A)}$.
It is known (see e.g. [11, Chap.II]) that $Q$ generates the left shift
semigroup $(S(t))_{t\geq 0}$ on $L^{2}([-r,0],H)$ given by

$\displaystyle(S(t)g)(\theta):=\begin{cases}g(t+\theta),&t+\theta\leq 0,\\\
0,&t+\theta>0,\end{cases},$

for $t\geq 0,$ $\theta\in[-r,0]$, and $g\in L^{2}([-r,0],H)$. On the other
hand, we define the operators

$\displaystyle\mathcal{A}:=\begin{pmatrix}A&0\\\ 0&Q\end{pmatrix},\qquad
D(\mathcal{A})=D(A)\times D(Q),$
$\displaystyle\mathcal{C}:=\left(I\;\;\mathscr{D}\right):D(\mathcal{A})\to H.$

Clearly, $\mathcal{A}$ generates a diagonal strongly continuous semigroup
$(\mathcal{T}(t))_{t\geq 0}$ on $\mathcal{H}$. As
$\mathscr{D}_{0}=\mathscr{D}$ with domain $D(\mathscr{D}_{0})=D(Q)$ is
admissible for $Q,$ it follows that $\mathcal{C}$ is admissible for
$\mathcal{A}$. We denote by $\mathcal{C}_{\Lambda}$ the Yosida extension of
$\mathcal{C}$ with respect to $\mathcal{A}$. Moreover, it is proved in [19,
Theorem 19], that ${\rm Im}(\mathfrak{T}(t))\subset D(\mathcal{C}_{\Lambda})$
for almost every $t\geq 0,$ and for any $\alpha>0$ there exists a constant
$\kappa(\alpha)>0$ such that

$\displaystyle\left\|\mathcal{C}_{\Lambda}\mathfrak{T}(\cdot)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\right\|_{L^{2}([0,\alpha],X)}\leq\kappa(\alpha)\left\|(\begin{smallmatrix}x\\\
g\end{smallmatrix})\right\|,\qquad\forall(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in\mathcal{H}.$ (5.6)

Clearly we have

$\displaystyle\mathcal{C}_{\Lambda}=\left(I\;\;\mathscr{D}_{0,\Lambda}\right),\quad
D(\mathcal{C}_{\Lambda})=H\times D(\mathscr{D}_{0,\Lambda}),$ (5.7)

where $\mathscr{D}_{0,\Lambda}$ is the Yosida extension of $\mathscr{D}$ with
respect to $Q$.

In order to give an expression of the semigroup $\mathfrak{T}$ which can help
us to prove the admissibility of $\mathcal{P}$ for $\mathfrak{A},$ we select

$\displaystyle\left(\Upsilon_{t}u\right)(\theta)=\begin{cases}u(t+\theta),&t+\theta\geq
0,\cr 0,&t+\theta<0\end{cases},$

for any $t\geq 0,\;\theta\in[-r,0]$ and $u\in L^{2}(\mathbb{R}^{+},H)$. By
using [19, Theorem 19], the semigroup $\mathfrak{T}$ satisfies

$\displaystyle\mathfrak{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})=\begin{pmatrix}T(t)x\\\
S(t)g+\Upsilon_{t}\mathcal{C}_{\Lambda}\mathfrak{T}(\cdot)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\end{pmatrix}$ (5.8)

for any $t\geq 0$ and $(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in\mathcal{H}$. We select,

$\displaystyle L_{0}:=L_{|D(Q)}.$

Denote by $L_{0,\Lambda}$ the Yosida extension of $L_{0}$ with respect to $Q$.
According to [17, Theorem 3], $L_{0}$ is an admissible operator for $Q,$ ${\rm
Im}(\Upsilon_{t})\subset D(L_{0,\Lambda})$ and

$\displaystyle\|L_{0,\Lambda}\Upsilon_{t}u\|_{L^{2}([0,\alpha],H)}\leq
c_{\alpha}\|u\|_{L^{2}([0,t],H)},$ (5.9)

for any $\alpha>0,$ $u\in L^{2}_{loc}(\mathbb{R}^{+},H)$ and some constant
$c_{\alpha}>0$. On the other hand, by using [18, Lemma 3.6], we have
$W^{1,2}([-r,0],H)\subset D(L_{0,\Lambda})$ and $L_{0,\Lambda}\equiv L$ on
$W^{1,2}([-r,0],H)$. Thus, for any $(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in D(\mathfrak{A}),$

$\displaystyle\mathcal{P}\mathfrak{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})$ $\displaystyle=\begin{pmatrix}0&L_{0,\Lambda}\\\
0&0\end{pmatrix}\mathfrak{T}(t)(\begin{smallmatrix}x\\\ g\end{smallmatrix})$
$\displaystyle=\begin{pmatrix}L_{0,\Lambda}S(t)g+L_{0,\Lambda}\Upsilon_{t}\mathcal{C}_{\Lambda}\mathfrak{T}(\cdot)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\\\ 0\end{pmatrix}.$

Thus the admissibility of $\mathcal{P}$ for $\mathfrak{A}$ now follows from
the admissibility of $L_{0}$ for $Q$, and the inequalities (5.6) and (5.9).
Thus the condition (A) holds as well. Now by Theorem 3.4, the semi-linear
stochastic equation (5.5) (hence the neutral stochastic equation (5.1)) has a
unique mild solution $\varrho\in\mathcal{C}_{\mathcal{F}}([0,T];\mathcal{H})$
such that

$\displaystyle\begin{split}\varrho(t)=\mathfrak{T}(t)(\begin{smallmatrix}\xi\\\
\phi\end{smallmatrix})+\int^{t}_{0}\mathfrak{T}(t-s)&\mathcal{F}(\mathcal{P}_{\Lambda,\mathfrak{A}}\varrho(s))ds\cr&+\int^{t}_{0}\mathfrak{T}(t-s)\mathscr{M}(\varrho(s))dW(s)\end{split}$
(5.10)

for any $t\geq 0$, $\xi\in L^{2}(\mathcal{F}_{0},H)$ and $\phi\in
L^{2}_{\mathcal{F}_{0}}([-r,0],H)$.

Let us now give an explicit expression to the first component of $\varrho(t)$.
In fact, from (5.6) we $\mathcal{C}_{\Lambda}$ is an admissible operator for
$\mathfrak{A}$. Thus by using (5.10), Proposition 2.1 and Proposition 2.4, we
obtain

$\displaystyle\left(\begin{smallmatrix}Z(t)\\\
X_{t}\end{smallmatrix}\right)=\varrho(t)\in D(\mathcal{C}_{\Lambda})$

for almost every $t\geq 0$ and $\mathbb{P}$ almost surely. On the other hand,
the relation (5.3) is a feedback low associated with the system (5.1) and can
be reformulated as

$\displaystyle
X(t)=\mathcal{C}_{\Lambda}\varrho(t)=Z(t)+\mathscr{D}_{0,\Lambda}X_{t}$ (5.11)

for a.e. $t\geq 0$. Moreover, by using similar arguments as in [19], one can
see that $\Xi:=H\times[D(L_{0,\Lambda})\cap D(\mathscr{D}_{0,\Lambda}]\subset
D(\mathcal{P}_{\Lambda,\mathfrak{A}})$ and

$\displaystyle\mathcal{P}_{\Lambda,\mathfrak{A}}=\begin{pmatrix}0&L_{0,\Lambda}\\\
0&0\end{pmatrix}\quad\text{on}\quad\Xi.$ (5.12)

Now by combining (5.8), (5.11), (5.12) and (5.10), the first component of
$\varrho(t)$ is given by

$\displaystyle Z(t)$ $\displaystyle=X(t)-\mathscr{D}_{0,\Lambda}X_{t}$
$\displaystyle=T(t)\xi+\int^{t}_{0}T(t-s)F(L_{0,\Lambda}X_{s})ds+\int^{t}_{0}T(t-s)B(X_{s})dW(s).$

∎

######  Remark 5.3.

We mention that the case $\mathscr{D}\equiv 0$ corresponds to the standard
delay equations.

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