# On the stochastic nonlinear Schrödinger equations with non-smooth additive
noise

Tadahiro Oh, Oana Pocovnicu, and Yuzhao Wang  Tadahiro Oh
School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Oana Pocovnicu
Department of Mathematics, Heriot-Watt University and The Maxwell Institute
for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom
<EMAIL_ADDRESS>Yuzhao Wang
School of Mathematics, University of Birmingham, Watson Building, Edgbaston,
Birmingham
B15 2TT
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We study the stochastic nonlinear Schrödinger equations with additive
stochastic forcing. By using the dispersive estimate, we present a simple
argument, constructing a unique local-in-time solution with rougher stochastic
forcing than those considered in the literature.

###### Key words and phrases:

stochastic nonlinear Schrödinger equation; well-posedness; dispersive estimate

###### 2010 Mathematics Subject Classification:

35Q55, 60H30

##  1\. Introduction

###  1.1. Stochastic nonlinear Schrödinger equations

We consider the Cauchy problem of the following stochastic nonlinear
Schrödinger equations (SNLS) with additive noise:111Since our interest is
local in time, the defocusing/focusing nature of the equations does not play
any role in this paper. Hence, we simply consider the defocusing equations.

$\begin{cases}i\partial_{t}u=\Delta u-|u|^{p-1}u+\phi\xi\\\
u|_{t=0}=u_{0},\end{cases}\qquad(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d},$
(1.1)

where $\xi(t,x)$ denotes a (Gaussian) space-time white noise on
$\mathbb{R}_{+}\times\mathbb{R}^{d}$ and $\phi$ is a bounded operator on
$L^{2}(\mathbb{R}^{d})$. We say that $u$ is a solution to (1.1) if it
satisfies the following mild formulation (= Duhamel formulation):

$\displaystyle
u(t)=S(t)u_{0}+i\int_{0}^{t}S(t-t^{\prime})|u|^{p-1}u(t^{\prime})dt^{\prime}-i\int_{0}^{t}S(t-t^{\prime})\phi\xi(dt^{\prime}),$
(1.2)

where $S(t)=e^{-it\Delta}$ denotes the linear Schrödinger propagator. The last
term on the right-hand side represents the effect of the stochastic forcing
and is called the stochastic convolution, which we denote by $\Psi$:

$\displaystyle\Psi(t):=-i\int_{0}^{t}S(t-t^{\prime})\phi\xi(dt^{\prime}).$
(1.3)

In the following, we assume that $\phi\in\textit{HS}(L^{2};H^{s})$ for
appropriate values of $s\geq 0$, namely, it is a Hilbert-Schmidt operator from
$L^{2}(\mathbb{R}^{d})$ to $H^{s}(\mathbb{R}^{d})$, guaranteeing that $\Psi\in
C(\mathbb{R}_{+};H^{s}(\mathbb{R}^{d}))$ almost surely [15]. See Section 2 for
a further discussion on the stochastic convolution $\Psi$. Previously, de
Bouard-Debussche [16] studied (1.1) in the energy-subcritical
setting222Namely, $1<p<1+\frac{4}{d-2}$ when $d\geq 3$ and $1<p<\infty$ when
$d=1,2$. This guarantees that the scaling-critical regularity
$s_{\text{crit}}$ defined in (1.6) satisfies $s_{\text{crit}}<1$. and proved
its well-posedness in $H^{1}(\mathbb{R}^{d})$, assuming that
$\phi\in\textit{HS}(L^{2};H^{1})$. Our main goal in this paper is to present a
simple construction of a unique local-in-time solution to (1.1) with a much
rougher noise (and hence a rougher stochastic convolution) than those
considered in [16].

Before discussing the well-posedness issue for SNLS (1.1), let us first go
over the local well-posedness theory for the following deterministic nonlinear
Schrödinger equations (NLS):

$\begin{cases}i\partial_{t}u=\Delta u-|u|^{p-1}u\\\
u|_{t=0}=u_{0},\end{cases}\qquad(t,x)\in\mathbb{R}\times\mathbb{R}^{d}.$ (1.4)

The equation (1.4) enjoys the following dilation symmetry:

$\displaystyle u(t,x)\longmapsto
u^{\lambda}(t,x)=\lambda^{-\frac{2}{p-1}}u(\lambda^{-2}t,\lambda^{-1}x)$ (1.5)

for $\lambda>0$. Namely, if $u$ is a solution to (1.4), then the scaled
function $u^{\lambda}$ is also a solution to (1.4) with the rescaled initial
data. This dilation symmetry induces the following scaling-critical Sobolev
regularity:

$\displaystyle s_{\text{crit}}=\frac{d}{2}-\frac{2}{p-1}$ (1.6)

such that the homogeneous $\dot{H}^{s_{\text{crit}}}(\mathbb{R}^{d})$-norm is
invariant under the dilation symmetry. This critical regularity
$s_{\text{crit}}$ provides a threshold regularity for well-posedness and ill-
posedness of (1.4). Indeed, when $s\geq\max(s_{\text{crit}},0)$, the Cauchy
problem (1.4) is known to be locally well-posed in $H^{s}(\mathbb{R}^{d})$
[34, 9].333When $p$ is not an odd integer, we need to impose an extra
assumption such as $p\geq[s]+1$ due to the non-smoothness of the nonlinearity.
See also Remark 1.4. Note that this condition can be relaxed or eliminated in
some situations. See, for example, [22].  On the other hand, it is known that
NLS (1.4) is ill-posed in the scaling supercritical regime:
$s<s_{\text{crit}}$. See [12, 24, 27].

Let us now introduce two important critical regularities. When
$s_{\text{crit}}=0$, we say that the Cauchy problem (1.4) is mass-critical.
This corresponds to the case $p=1+\frac{4}{d}$. When $s_{\text{crit}}<0$, i.e.
$p<1+\frac{4}{d}$ (and $s_{\text{crit}}>0$, i.e. $p>1+\frac{4}{d}$,
respectively), we say that (1.4) is mass-subcritical (and mass-supercritical,
respectively). When $s_{\text{crit}}=1$, we say that the Cauchy problem (1.4)
is energy-critical. This corresponds to the case $p=1+\frac{4}{d-2}$. When
$s_{\text{crit}}<1$, i.e. $p<1+\frac{4}{d-2}$ (and $s_{\text{crit}}>1$, i.e.
$p>1+\frac{4}{d-2}$, respectively), we say that (1.4) is energy-subcritical
(and energy-supercritical, respectively). In the following, we use the same
terminology for SNLS (1.1).

One of the main ingredients in establishing local well-posedness of (1.4) is
the following Strichartz estimates [33, 36, 19, 23]:

$\|S(t)u_{0}\|_{L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\leq
C_{d,q,r}\|u_{0}\|_{L^{2}_{x}(\mathbb{R}^{d})},$ (1.7)

which holds true for any Schrödinger admissible pair $(q,r)$, satisfying

$\frac{2}{q}+\frac{d}{r}=\frac{d}{2}$ (1.8)

with $2\leq q,r\leq\infty$ and $(q,r,d)\neq(2,\infty,2)$. In [16], de Bouard-
Debussche used the Strichartz estimates to show that the stochastic
convolution $\Psi$ almost surely belongs to a right Strichartz space. As a
result, under the assumption that $\phi\in\textit{HS}(L^{2};H^{1})$, they
proved that SNLS (1.1) is locally well-posed in $H^{1}(\mathbb{R}^{d})$ in the
energy-subcritical case: $1<p<1+\frac{4}{d-2}$ when $d\geq 3$ and $1<p<\infty$
when $d=1,2$.444In [16], they also proved global well-posedness of SNLS (1.1).
The well-posedness issue for SNLS with multiplicative noise was also
considered in the same paper. See also Cheung-Mosincat [10] for analogous
well-posedness results of SNLS with additive and multiplicative noises in the
periodic setting. Now, let $s\geq\max(s_{\text{crit}},0)$. Then, by slightly
modifying the argument in [16] with Lemma 2.1 below, it is easy to see that
SNLS (1.1) is locally well-posed555Once again, an extra assumption such as
$p\geq[s]+1$ is needed when $p$ is not an odd integer. in
$H^{s}(\mathbb{R}^{d})$, provided that $\phi\in\textit{HS}(L^{2};H^{s})$. In
particular, (1.1) is locally well-posed in $L^{2}(\mathbb{R}^{d})$ in the
mass-(sub)critical case, provided that $\phi\in\textit{HS}(L^{2};L^{2})$.
Therefore, we focus our attention on the mass-supercritical case in the
following.

We point out that so far we assumed that the noise had the same spatial
regularity as that of initial data. On the one hand, the aforementioned ill-
posedness results tell us that we can not take (deterministic) initial data
below the scaling-critical regularity $s_{\text{crit}}$. On the other hand, we
are allowed to take different regularities for initial data and the noise.
Indeed, in the following, we treat rough stochastic noises that have
regularities below the scaling critical regularity $s_{\text{crit}}$, while
keeping (deterministic) initial data above the scaling critical regularity.

###  1.2. Main results

In the following, we use $s_{0}$ and $s$ to denote the regularities of initial
data $u_{0}$ and the noise (i.e. $\phi\in\textit{HS}(L^{2};H^{s})$),
respectively. Our main goal is to lower the value of $s$, while keeping
$s_{0}\geq s_{\text{crit}}$. In order to achieve this goal, we work within the
$L_{x}^{r}$-framework, $r>2$, by exploiting the following dispersive estimate:

$\|S(t)u_{0}\|_{L^{r}_{x}(\mathbb{R}^{d})}\leq\frac{C_{r}}{|t|^{\frac{d}{2}-\frac{d}{r}}}\|u_{0}\|_{L^{r^{\prime}}_{x}(\mathbb{R}^{d})}$
(1.9)

for any $2\leq r\leq\infty$ with $\frac{1}{r}+\frac{1}{r^{\prime}}=1$. Another
key ingredient is the space-time integrability of the stochastic convolution.
By a small modification of the argument in [16], we show that, the stochastic
convolution $\Psi$ almost surely belongs to

$\displaystyle L^{q}([0,T];W^{s,r}(\mathbb{R}^{d}))$ (1.10)

for any $1\leq q<\infty$ and finite $r\geq 2$ such that $r\leq\frac{2d}{d-2}$
when $d\geq 3$, provided that $\phi\in\textit{HS}(L^{2};H^{s})$. See Lemma 2.1
below. Note that the pair $(q,r)$ is no longer restricted to be Schrödinger
admissible. In particular, while keeping $r=\frac{2d}{d-2}$ and sending $q$ to
$\infty$, we basically gain almost one spatial derivative.666Recall that
$H^{s+1}(\mathbb{R}^{d})\subset W^{s,\frac{2d}{d-2}}(\mathbb{R}^{d})$. This
allows us to prove the following improved local well-posedness result.777Here,
we use the term “well-posedness” in a loose sense as in [6]. See also Remark
1.2 (ii).

######  Theorem 1.1.

(i) Energy-subcritical case:888As we mentioned before, we assume that $(d,p)$
satisfies the mass-supercritical condition. Let $d\geq 1$ and
$1+\frac{4}{d}<p<\infty$. When $d\geq 3$, assume that $p<1+\frac{4}{d-2}$ in
addition.

(i.a) Let $s_{0}\geq\frac{d}{2}-\frac{d}{p+1}$. Then, given $u_{0}\in
H^{s_{0}}(\mathbb{R}^{d})$, there exists a unique local-in-time solution $u$
to SNLS (1.1), provided that $\phi\in\textit{HS}(L^{2};L^{2})$. Moreover, the
solution $u$ lies in the class:

$\displaystyle\Psi$ $\displaystyle+C([0,T];L^{p+1}(\mathbb{R}^{d}))\cap
C([0,T];L^{2}(\mathbb{R}^{d}))$ $\displaystyle\subset
C([0,T];L^{2}(\mathbb{R}^{d})),$

where $T=T_{\omega}$ is almost surely positive.

(i.b) Let $s_{0}>s_{\textup{crit}}$. Then, given $u_{0}\in
H^{s_{0}}(\mathbb{R}^{d})$, there exists a unique local-in-time solution $u$
to SNLS (1.1), provided that $\phi\in\textit{HS}(L^{2};L^{2})$. Moreover, the
solution $u$ lies in the class:

$\displaystyle\Psi$ $\displaystyle+L^{q}([0,T];L^{p+1}(\mathbb{R}^{d}))\cap
C([0,T];L^{2}(\mathbb{R}^{d}))$ $\displaystyle\subset
C([0,T];L^{2}(\mathbb{R}^{d})),$

where $q=q(d,p)>2$ is finite and $T=T_{\omega}$ is almost surely positive.

(ii) Energy-(super)critical case: Let $d\geq 3$ and $p\geq 1+\frac{4}{d-2}$ be
an odd integer. Fix $s_{0}>s_{\textup{crit}}$ and $s>s_{\textup{crit}}-1$.
Then, given $u_{0}\in H^{s_{0}}(\mathbb{R}^{d})$, there exists a unique local-
in-time solution $u$ to SNLS (1.1), provided that
$\phi\in\textit{HS}(L^{2};H^{s})$. Moreover, the solution $u$ lies in the
class:

$\displaystyle\Psi$
$\displaystyle+C([0,T];W^{s_{1},\frac{2d}{d-2}-\delta}(\mathbb{R}^{d}))\cap
C([0,T];H^{s_{1}}(\mathbb{R}^{d}))$ $\displaystyle\subset
C([0,T];H^{s_{1}}(\mathbb{R}^{d})),$

where $s_{1}=\min(s_{0}-1,s)$, $\delta=\delta(s_{1})>0$ is sufficiently small,
and $T=T_{\omega}$ is almost surely positive.

The structure of the mild formulation (1.2) states that any solution $u$ can
be written as999The decomposition (1.11) is often referred to as the Da Prato-
Debussche trick [14] in the field of stochastic parabolic PDEs. Such an idea
also appears in McKean [26] and Bourgain [6] in the context of (deterministic)
dispersive PDEs with random initial data, preceding [14]. See also de Bouard-
Debussche [16] and Burq-Tzvetkov [8].

$\displaystyle u=v+\Psi.$ (1.11)

We then study the following fixed point problem for the residual term
$v:=u-\Psi$:

$\displaystyle
v(t)=S(t)u_{0}+\int_{0}^{t}S(t-t^{\prime})\mathcal{N}(v+\Psi)(t^{\prime})dt^{\prime},$
(1.12)

where $\mathcal{N}(u)=i|u|^{p-1}u$. In [16], de Bouard-Debussche studied the
fixed point problem (1.12) for $v$ in terms of the standard $L_{x}^{2}$-based
theory for NLS (1.4). In particular, the solution $v$ to (1.12) was
constructed in $C([0,T];H^{1}(\mathbb{R}^{d}))$ intersected with an
appropriate Strichartz space. In the following, we instead work in the
$L^{r}_{x}$-framework with $r>2$ and directly solve the fixed point problem in
$C([0,T];W^{s_{1},r}(\mathbb{R}^{d}))$101010For Theorem 1.1 (i.b), we need to
work in $L^{q}([0,T];L^{p+1}(\mathbb{R}^{d}))$. by applying the dispersive
estimate (1.9).

On the one hand, the spatial regularity $s_{1}$ of $v$ in Theorem 1.1, i.e.
$s_{1}=0$ in (i) and $s_{1}=s_{\textup{crit}}-1+\varepsilon$ in (ii) is below
the scaling critical regularity $s_{\text{crit}}$ defined in (1.6) (when
$\varepsilon<1$). On the other hand, given any $1\leq r\leq\infty$, we can
also consider the scaling-critical Sobolev regularity adapted to the
$L^{r}$-based Sobolev spaces:

$\displaystyle s_{\text{crit}}(r)=\frac{d}{r}-\frac{2}{p-1}$

such that the homogeneous $\dot{W}^{s_{\textup{crit}}(r),r}$-norm is invariant
under the dilation symmetry (1.5). Note that we have
$s_{\text{crit}}(r)<s_{\text{crit}}=s_{\text{crit}}(2)$ for $r>2$. For
example, in the energy-(super)critical case, the gain of spatial integrability
(1.10) of the stochastic convolution $\Psi$ allows us to work in the
$L^{r}_{x}$-based Sobolev space111111For a technical reason, we need to take
$r=\frac{2d}{d-2}-\delta$ for some small $\delta>0$ in the proof of Theorem
1.1. with $r=\frac{2d}{d-2}$, thus lowering the critical regularity from
$s_{\text{crit}}=s_{\text{crit}}(2)$ to $s_{\text{crit}}(r)=s_{\text{crit}}-1$
with $r=\frac{2d}{d-2}$. This heuristically explains how the regularity
threshold $s_{\textup{crit}}-1$ appears in Theorem 1.1 (ii). Moreover, note
that, by working only within the $L^{r}$-based Sobolev space with
$s_{1}>s_{\text{crit}}(r)$, we have made the problem subcritical. Indeed, all
the spatial function spaces such as $L^{p+1}(\mathbb{R}^{d})$ appearing in
Theorem 1.1 are subcritical in the sense described above.

######  Remark 1.2.

(i) Our argument for proving Theorem 1.1 is of subcritical nature in the sense
that the local existence time $T$ depends on the $H^{s_{0}}$-norm of initial
data (and a space-time norm of the stochastic convolution). It is possible to
improve Theorem 1.1 (i.b) so that it also holds when $s_{0}=s_{\text{crit}}$
by relying on the critical local well-posedness theory (in terms of initial
data). See Remark 3.3.

(ii) Theorem 1.1 establishes existence of unique solutions to (1.1). Note that
the (spatial) regularity of the noise is rougher than that of the initial data
in Theorem 1.1 (i.a) and (i.b). As such, the solution inherits the rougher
regularity of the noise and it only lies in $C([0,T];L^{2}(\mathbb{R}^{d}))$.
The situation is slightly more subtle in Theorem 1.1 (ii). Also, note that, in
view of the aforementioned ill-posedness results, the map:
$(u_{0},\phi\xi)\mapsto u$ is not continuous, when the noise has spatial
regularity $s<s_{\text{crit}}$. By the use of the Da Prato-Debussche trick,
however, the map sending an enhanced data set $(u_{0},\Psi)$ to a solution $u$
is continuous, where the stochastic convolution $\Psi$ is measured in an
appropriate space-time function norm.

Our next goal is to study the Cauchy problem (1.1) with random initial data
and prove almost sure local well-posedness for (random) initial data of lower
regularities. More precisely, given a function $u_{0}$ on $\mathbb{R}^{d}$, we
consider a randomization of $u_{0}$ adapted to the so-called Wiener
decomposition [35] of the frequency space:
$\mathbb{R}^{d}=\bigcup_{n\in\mathbb{Z}^{d}}Q_{n}$, where $Q_{n}$ is the unit
cube centered at $n\in\mathbb{Z}^{d}$.

Let $\psi\in\mathcal{S}(\mathbb{R}^{d})$ such that

$\operatorname*{supp}\psi\subset[-1,1]^{d}\qquad\text{and}\qquad\sum_{n\in\mathbb{Z}^{d}}\psi(\xi-n)\equiv
1\ \text{ for any }\xi\in\mathbb{R}^{d}.$

Then, given a function $u_{0}$ on $\mathbb{R}^{d}$, we have

$\displaystyle u_{0}=\sum_{n\in\mathbb{Z}^{d}}\psi(D-n)u_{0},$

where $\psi(D-n)$ is defined by
$\psi(D-n)u_{0}(x)=\int_{\mathbb{R}^{d}}\psi(\xi-n)\widehat{u}_{0}(\xi)e^{2\pi
ix\cdot\xi}d\xi$, namely, the Fourier multiplier operator with symbol
$\mathbf{1}_{Q_{n}}$ conveniently smoothed. This decomposition leads to the
following randomization of $u_{0}$ adapted to the Wiener decomposition. Let
$\\{g_{n}\\}_{n\in\mathbb{Z}^{d}}$ be a sequence of independent mean-zero
complex-valued random variables (with independent real and imaginary parts),
satisfying the following exponential moment bound:

$\displaystyle\mathbb{E}\big{[}e^{\gamma_{1}\operatorname*{Re}g_{n}+\gamma_{2}\operatorname*{Im}g_{n}}\big{]}\leq
e^{c(\gamma_{1}^{2}+\gamma_{2}^{2})}$ (1.13)

for all $\gamma_{1},\gamma_{2}\in\mathbb{R}$ and $n\in\mathbb{Z}^{d}$. Note
that (1.13) is satisfied by standard complex-valued Gaussian random variables,
Bernoulli random variables, and any random variables with compactly supported
distributions. We then define the Wiener randomization121212It is also called
the unit-scale randomization in [17]. of $u_{0}$ by

$u_{0}^{\omega}:=\sum_{n\in\mathbb{Z}^{d}}g_{n}(\omega)\psi(D-n)u_{0}.$ (1.14)

Given $u_{0}\in H^{s}(\mathbb{R}^{d})$, it is easy to see that its Wiener
randomization $u_{0}^{\omega}$ in (1.14) lies in $H^{s}(\mathbb{R}^{d})$
almost surely. One can also show that, under some non-degeneracy condition,
there is no smoothing upon randomization in terms of differentiability; see,
for example, Lemma B.1 in [8]. The main point of the randomization (1.14) is
its improved integrability. For example, under the assumption (1.13),
$u_{0}^{\omega}$ almost surely belongs to $W^{s,r}(\mathbb{R}^{d})$ for any
finite $r\geq 2$. Moreover, by restricting our attention to local-in-time
intervals, the random linear solution $S(t)u_{0}^{\omega}$ satisfies the
Strichartz estimate (1.7) for any finite $q,r\geq 2$ almost surely. See Lemma
3.5 below. This gain of space-time integrability allows us to take random
initial data at the same low regularity as the stochastic forcing.

######  Theorem 1.3.

(i) Energy-subcritical case: Let $d$ and $p$ be as in Theorem 1.1 (i). Then,
given $u_{0}\in L^{2}(\mathbb{R}^{d})$, SNLS (1.1) is almost surely locally
well-posed with respect to the Wiener randomization $u_{0}^{\omega}$ defined
in (1.14), provided that $\phi\in\textit{HS}(L^{2};L^{2})$. More precisely,
there exists a unique local-in-time solution $u=u^{\omega}$ to SNLS (1.1) with
$u|_{t=0}=u_{0}^{\omega}$ in the class:

$\displaystyle S(t)u_{0}^{\omega}+\Psi$
$\displaystyle+C([0,T];L^{p+1}(\mathbb{R}^{d}))\cap
C([0,T];L^{2}(\mathbb{R}^{d}))$ $\displaystyle\subset
C([0,T];L^{2}(\mathbb{R}^{d})),$

where $T=T_{\omega}$ is almost surely positive.

(ii) Energy-(super)critical case: Let $d$ and $p$ be as in Theorem 1.1 (ii).
Moreover, let $s=s_{\textup{crit}}-1+\varepsilon$ for some small
$\varepsilon>0$. Then, given $u_{0}\in H^{s}(\mathbb{R}^{d})$, SNLS (1.1) is
almost surely locally well-posed with respect to the Wiener randomization
$u_{0}^{\omega}$ defined in (1.14), provided that
$\phi\in\textit{HS}(L^{2};H^{s})$. More precisely, there exists a unique
local-in-time solution $u=u^{\omega}$ to SNLS (1.1) with
$u|_{t=0}=u_{0}^{\omega}$ in the class:

$\displaystyle S(t)u_{0}^{\omega}+\Psi$
$\displaystyle+C([0,T];W^{s,\frac{2d}{d-2}-\delta}(\mathbb{R}^{d}))\cap
C([0,T];H^{s}(\mathbb{R}^{d}))$ $\displaystyle\subset
C([0,T];H^{s}(\mathbb{R}^{d})),$

where $\delta=\delta(s)>0$ is sufficiently small and $T=T_{\omega}$ is almost
surely positive.

In view of the probabilistic Strichartz estimates (Lemma 3.5), we see that
$\widetilde{\Psi}:=S(t)u_{0}^{\omega}+\Psi$ solving

$\begin{cases}i\partial_{t}\widetilde{\Psi}=\Delta\widetilde{\Psi}+\phi\xi\\\
\widetilde{\Psi}|_{t=0}=u_{0}^{\omega},\end{cases}$ (1.15)

satisfies the same regularity properties, both in terms of differentiability
and integrability, as the stochastic convolution $\Psi$ in (1.3). Then, by
decomposing $u$ as

$\displaystyle u=v+\widetilde{\Psi},$

Theorem 1.3 follows from repeating the argument in the proof of Theorem 1.1.

We conclude this introduction with several remarks.

######  Remark 1.4.

In Theorems 1.1 and 1.3, we assumed that $p$ is an odd integer in the
energy-(super)critical case. One may apply the fractional chain rule [13] and
remove this restriction in certain situations. For conciseness of the
presentation, however, we do not pursue this direction in this paper.

######  Remark 1.5.

In recent years, the well-posedness issue of the deterministic NLS (1.4) with
respect to the random initial data $u_{0}^{\omega}$ in (1.14) has been studied
intensively [1, 2, 7, 28, 3, 18]. The main idea is to study the fixed point
problem for the residual term $v=u-S(t)u_{0}^{\omega}$, utilizing (a variant
of) the Fourier restriction norm method [5, 20, 21] and carrying out rather
tedious case-by-case analysis. In a recent paper [31], the second and third
authors studied the deterministic NLS (1.4) with the random initial data
$u_{0}^{\omega}$ in (1.14) by exploiting the dispersive estimate (1.9). In
particular, they proved Theorem 1.3 above when $\phi=0$, i.e. when there is no
stochastic noise. This argument with the dispersive estimate bypasses case-by-
case analysis, which is closer in sprit to the almost sure local well-
posedness argument for the nonlinear wave equations [8, 25, 30, 29]. See a
survey paper [4] for a further discussion on the subject.

######  Remark 1.6.

In [11], the second author with Cheung studied SNLS (1.1) on $\mathbb{R}^{d}$,
$d\geq 3$, with the cubic nonlinearity ($p=3$). By adapting the argument in
[2] for the deterministic NLS with random initial data, they proved local
well-posedness of (1.1) with stochastic forcing below the scaling-critical
regularity, i.e. $\phi\in\textit{HS}(L^{2};H^{s})$ with $s<s_{\text{crit}}$.
Moreover, their work shows that the residual part $v=u-\Psi$ lies in
$C([0,T];H^{s_{\text{crit}}}(\mathbb{R}^{d}))$.

Notations: Given $T>0$, we set $L^{q}_{T}B_{x}=L^{q}([0,T];B(\mathbb{R}^{d}))$
and $C_{T}B_{x}=C([0,T];B(\mathbb{R}^{d}))$, where $B(\mathbb{R}^{d})$ denotes
a Banach space of functions on $\mathbb{R}^{d}$.

##  2\. On the stochastic convolution

In this section, we study the regularity properties of the stochastic
convolution $\Psi$ in (1.3). Let us first recall the definition of a
cylindrical Wiener process $W$ on $L^{2}(\mathbb{R}^{d})$. Fix an orthonormal
basis $\\{e_{n}\\}_{n\in\mathbb{N}}$ of $L^{2}(\mathbb{R}^{d})$. Then, a
cylindrical Wiener process $W$ on $L^{2}(\mathbb{R}^{d})$ is defined by the
following random Fourier series:

$\displaystyle W(t)$ $\displaystyle=\sum_{n\in\mathbb{N}}\beta_{n}(t)e_{n},$

where $\\{\beta_{n}\\}_{n\in\mathbb{N}}$ is a family of mutually independent
complex-valued Brownian motions. In terms of the cylindrical Wiener process
$W$, we can express the stochastic convolution $\Psi$ in (1.3) as

$\displaystyle\Psi(t)=-i\int_{0}^{t}S(t-t^{\prime})\phi
dW(t^{\prime})=-i\sum_{n\in\mathbb{N}}\int_{0}^{t}S(t-t^{\prime})\phi(e_{n})d\beta_{n}(t^{\prime}).$
(2.1)

By slightly modifying the argument in [16], we have the following lemma.

######  Lemma 2.1.

Suppose that $\phi\in\textit{HS}(L^{2};H^{s})$ for some $s\in\mathbb{R}$.
Then, the following statements hold almost surely:

  * (i)

$\Psi\in C(\mathbb{R}_{+};H^{s}(\mathbb{R}^{d}))$,

  * (ii)

Given any $1\leq q<\infty$ and finite $r\geq 2$ such that
$r\leq\frac{2d}{d-2}$ when $d\geq 3$, we have $\Psi\in
L^{q}([0,T];W^{s,r}(\mathbb{R}^{d}))$ for any $T>0$.

Compare Part (ii) of Lemma 2.1 with [16], where $(q,r)$ was restricted to be
Schrödinger admissible. It is this gain of integrability in time which allows
us to prove Theorems 1.1 and 1.3.

###### Proof.

For (i), see [15]. Set $\langle\nabla\rangle=\sqrt{1-\Delta}$. Given
$\phi\in\textit{HS}(L^{2};H^{s})$, let
$\\{\phi_{k}\\}_{k\in\mathbb{N}}\subset\textit{HS}(L^{2};H^{s+\sigma})$,
$\sigma>\frac{d}{2}$, such that $\phi_{k}$ converges to $\phi$ in
$\textit{HS}(L^{2};H^{s})$. Then, letting $\Psi_{k}$ denote the stochastic
convolution in (2.1) with $\phi$ replaced by $\phi_{k}$, we see that
$\Psi_{k}$ converges to $\Psi$ in $C(\mathbb{R}_{+};H^{s}(\mathbb{R}^{d}))$
and that $\langle\nabla\rangle^{s}\Psi_{k}\in
C(\mathbb{R}_{+};H^{\sigma}(\mathbb{R}^{d}))\subset
C(\mathbb{R}_{+};C(\mathbb{R}^{d}))$, where the inclusion follows from
Sobolev’s embedding theorem.

Fix $(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$. Then, from (2.1), we see
that, as a linear combination of independent Wiener integrals,
$\langle\nabla\rangle^{s}\Psi_{k}(t,x)$ is a mean-zero complex-valued Gaussian
random variable with variance
$\sigma_{k}(t,x)=\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{2}(\Omega)}^{2}$.
Recall that, for a mean-zero complex-valued Gaussian random variable $g$ with
variance $\sigma$, we have

$\displaystyle\mathbb{E}\big{[}|g|^{2j}\big{]}=j!\cdot\sigma^{j}.$ (2.2)

Given $\rho\geq 2$, let $\widetilde{\rho}$ denote the smallest even integer
such that $\widetilde{\rho}\geq\rho$. Then, by Hölder’s inequality and (2.2),
we have131313In fact, the following estimate holds true:
$\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{\rho}(\Omega)}\leq\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{2}(\Omega)}$
Namely, there is no constant depending on $\rho\geq 2$. See [32, Theorem
I.22]. For our purpose, however, the elementary argument in (2.3) suffices.

$\displaystyle\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{\rho}(\Omega)}$
$\displaystyle\leq\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{\widetilde{\rho}}(\Omega)}$
$\displaystyle=C_{\rho}\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{2}(\Omega)}$
$\displaystyle\sim\bigg{\|}\bigg{(}\int_{0}^{t}|S(t-t^{\prime})\langle\nabla\rangle^{s}\phi_{k}(e_{n})(x)|^{2}dt^{\prime}\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\ell^{2}_{n}}$
$\displaystyle=\|S(\tau)\langle\nabla\rangle^{s}\phi_{k}(e_{n})(x)\|_{\ell^{2}_{n}L^{2}_{\tau}([0,t])}$
(2.3)

for any $\rho\geq 2$ and $(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{d}$.

Now, fix $1\leq q<\infty$ and finite $r\geq 2$ such that $r\leq\frac{2d}{d-2}$
when $d\geq 3$. Let $\widetilde{q}=\widetilde{q}(d,r)\geq 2$ be the unique
index such that $(\widetilde{q},r)$ is Schrödinger admissible, satisfying
(1.8). Then, for $\rho\geq\max(q,r)$, it follows from Minkowski’s integral
inequality and (2.3) that

$\displaystyle\big{\|}\|\Psi_{k}\|_{L^{q}_{T}W^{s,r}_{x}}\big{\|}_{L^{\rho}(\Omega)}$
$\displaystyle\leq\big{\|}\|\langle\nabla\rangle^{s}\Psi_{k}(t,x)\|_{L^{\rho}(\Omega)}\big{\|}_{L^{q}_{T}L^{r}_{x}}$
$\displaystyle\leq
C_{\rho}\big{\|}\|S(\tau)\langle\nabla\rangle^{s}\phi_{k}(e_{n})(x)\|_{\ell^{2}_{n}L^{2}_{\tau}([0,t])}\big{\|}_{L^{q}_{T}L^{r}_{x}}$
By Minkowski’s integral inequality (with $r\geq 2$), Hölder’s inequality in
time, and then applying the Strichartz estimate (1.7), $\displaystyle\leq
T^{\frac{1}{q}}\big{\|}\|S(\tau)\langle\nabla\rangle^{s}\phi_{k}(e_{n})\|_{L^{2}_{\tau}([0,T];L^{r}_{x})}\big{\|}_{\ell^{2}_{n}}$
$\displaystyle\leq
T^{\theta}\big{\|}\|S(\tau)\langle\nabla\rangle^{s}\phi_{k}(e_{n})\|_{L^{\widetilde{q}}_{\tau}([0,T];L^{r}_{x})}\big{\|}_{\ell^{2}_{n}}$
$\displaystyle\leq
T^{\theta}\big{\|}\|\phi_{k}(e_{n})\|_{H^{s}_{x}}\big{\|}_{\ell^{2}_{n}}$
$\displaystyle=T^{\theta}\|\phi_{k}\|_{\textit{HS}(L^{2};H^{s})}<\infty$

for some $\theta=\theta(q,\widetilde{q})>0$. Similarly, we have

$\displaystyle\big{\|}\|\Psi_{k}-\Psi_{j}\|_{L^{q}_{T}W^{s,r}_{x}}\big{\|}_{L^{\rho}(\Omega)}\leq
CT^{\theta}\|\phi_{k}-\phi_{j}\|_{\textit{HS}(L^{2};H^{s})}\longrightarrow 0,$

as $k,j\to\infty$. Namely, $\\{\Psi_{k}\\}_{k\in\mathbb{N}}$ is a Cauchy
sequence in $L^{\rho}(\Omega;L^{q}([0,T];W^{s,r}(\mathbb{R}^{d})))$. By the
uniqueness of the limit, we conclude that $\\{\Psi_{k}\\}_{k\in\mathbb{N}}$
converges to $\Psi$ in
$L^{\rho}(\Omega;L^{q}([0,T];W^{s,r}(\mathbb{R}^{d})))$. In particular, we
have

$\displaystyle\big{\|}\|\Psi\|_{L^{q}_{T}W^{s,r}_{x}}\big{\|}_{L^{\rho}(\Omega)}\leq
CT^{\theta}\|\phi\|_{\textit{HS}(L^{2};H^{s})}<\infty.$

This proves (ii). ∎

##  3\. Proof of Theorems 1.1 and 1.3

In this section, we present the proofs of our main results (Theorems 1.1 and
1.3). We first recall the following nonhomogeneous Strichartz estimate; let
$(q,r)$ and $(\widetilde{q},\widetilde{r})$ be Schrödinger admissible. Then,
we have

$\displaystyle\bigg{\|}\int_{0}^{t}S(t-t^{\prime})F(t^{\prime})dt^{\prime}\bigg{\|}_{L^{q}_{t}L^{r}_{x}}\lesssim\|F\|_{L^{\widetilde{q}^{\prime}}_{t}L^{\widetilde{r}^{\prime}}_{x}},$
(3.1)

where ${\widetilde{q}}^{\prime}$ and $\widetilde{r}^{\prime}$ denote the
Hölder conjugates of $\widetilde{q}$ and $\widetilde{r}$, respectively.

###  3.1. Proof of Theorem 1.1

Let $s_{0},s\in\mathbb{R}$ to be specified later. Given $u_{0}\in
H^{s_{0}}(\mathbb{R}^{d})$ and $\phi\in\textit{HS}(L^{2};H^{s})$, we define
$\Gamma=\Gamma_{u_{0},\phi,\,\xi}$ by

$\Gamma
v(t):=S(t)u_{0}+\int_{0}^{t}S(t-t^{\prime})\mathcal{N}(v+\Psi)(t^{\prime})dt^{\prime}.$

Then, we have the following nonlinear estimates.

######  Proposition 3.1.

Let $d$ and $p$ be as in Theorem 1.1. We set

  * (i.a)

$s_{0}\geq\frac{d}{2}-\frac{d}{p+1}$, $s_{1}=0$, $r=p+1$ in the energy-
subcritical case,

  * (ii)

$s_{0}-1\geq s_{1}>s_{\textup{crit}}-1$ and $r=\frac{2d}{d-2}-\delta$ for some
small $\delta=\delta(s_{1})>0$ in the energy-(super)critical case.

Then, the following estimates hold for some $q\gg 1$:

$\displaystyle\|\Gamma v\|_{C_{T}W^{s_{1},r}_{x}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\Big{(}\|v\|_{C_{T}W_{x}^{s_{1},r}}^{p}+\|\Psi\|_{L^{q}_{T}W_{x}^{s_{1},r}}^{p}\Big{)},$
$\displaystyle\|\Gamma v_{1}-\Gamma v_{2}\|_{C_{T}W^{s_{1},r}_{x}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v_{1}\|_{C_{T}W_{x}^{s_{1},r}}^{p-1}+\|v_{2}\|_{C_{T}W_{x}^{s_{1},r}}^{p-1}$
$\displaystyle\hphantom{XXXXXXXXX}+\|\Psi\|_{L^{q}_{T}W_{x}^{s_{1},r}}^{p-1}\Big{)}\|v_{1}-v_{2}\|_{C_{T}W_{x}^{s_{1},r}}$
(3.2)

for all $v,v_{1},v_{2}\in C([0,T];W^{s_{1},r}(\mathbb{R}^{d}))$ and $T>0$.
Moreover, we have $\Gamma v\in C([0,T];H^{s_{1}}(\mathbb{R}^{d}))$ for all
$v\in C([0,T];W^{s_{1},r}(\mathbb{R}^{d}))$, $\Psi\in
L^{q}([0,T];W^{s_{1},r}(\mathbb{R}^{d}))$, and $T>0$.

Once we prove Proposition 3.1, Theorem 1.1 (i.a) and (ii) follow from a
standard contraction argument with Lemma 2.1 (with
$s=s_{\textup{crit}}-1+\varepsilon$ for Theorem 1.1 (ii)).

###### Proof.

We first consider the energy-subcritical case (i.a). By Sobolev’s inequality
$H^{s_{0}}(\mathbb{R}^{d})\subset L^{p+1}(\mathbb{R}^{d})$ and the dispersive
estimate (1.9), we have

$\displaystyle\|\Gamma v\|_{C_{T}L_{x}^{p+1}}$
$\displaystyle\lesssim\|S(t)u_{0}\|_{C_{T}H^{s_{0}}}+\sup_{t\in[0,T]}\int_{0}^{t}\frac{1}{|t-t^{\prime}|^{\frac{d}{2}-\frac{d}{p+1}}}\|\mathcal{N}(v+\Psi)(t^{\prime})\|_{L_{x}^{\frac{p+1}{p}}}dt^{\prime}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\|v+\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\Big{(}\|v\|_{C_{T}L_{x}^{p+1}}^{p}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\Big{)}$
(3.3)

for some $q\gg 1$ and small $\theta>0$, provided that
${s_{0}}\geq\frac{d}{2}-\frac{d}{p+1}$ and $\frac{d}{2}-\frac{d}{p+1}<1$,
namely $p<1+\frac{4}{d-2}$. When $p$ is an odd integer, a similar computation
yields the following difference estimate:

$\displaystyle\|\Gamma v_{1}-\Gamma v_{2}\|_{C_{T}L_{x}^{p+1}}$
$\displaystyle\lesssim
T^{\theta}\|\mathcal{N}(v_{1}+\Psi)-\mathcal{N}(v_{2}+\Psi)\|_{L^{q}_{T}L_{x}^{\frac{p+1}{p}}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v_{1}\|_{C_{T}L_{x}^{p+1}}^{p-1}+\|v_{2}\|_{C_{T}L_{x}^{p+1}}^{p-1}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p-1}\Big{)}\|v_{1}-v_{2}\|_{C_{T}L_{x}^{p+1}}.$
(3.4)

Next, we consider the case when $p>1$ is not an odd integer. By the mean value
theorem, we have

$\displaystyle\mathcal{N}(v_{1}+\Psi)-\mathcal{N}(v_{2}+\Psi)=\int_{0}^{1}\Big{\\{}$
$\displaystyle\partial_{z}\mathcal{N}(v_{2}+\Psi+\theta(v_{1}-v_{2}))(v_{1}-v_{2})$
$\displaystyle+\partial_{\overline{z}}\mathcal{N}(v_{2}+\Psi+\theta(v_{1}-v_{2}))(\overline{v_{1}-v_{2}})\Big{\\}}d\theta.$
(3.5)

With $\mathcal{N}(z)=i|z|^{p-1}z$, we have

$\displaystyle\textstyle\partial_{z}\mathcal{N}(z)=i\frac{p+1}{2}|z|^{p-1}\qquad\text{and}\qquad\partial_{\overline{z}}\mathcal{N}(z)=i\frac{p-1}{2}|z|^{p-1}\frac{z^{2}}{|z|^{2}}.$
(3.6)

Then, by repeating the computation above with (3.5) and (3.6), we obtain
(3.4). Given $p+1<\frac{2d}{d-2}$, let $(\widetilde{q},p+1)$ be Schrödinger
admissible. Then, it follows from (3.1) that

$\displaystyle\|\Gamma v-S(t)u_{0}\|_{C_{T}L_{x}^{2}}$
$\displaystyle\lesssim\|\mathcal{N}(v+\Psi)\|_{L^{\widetilde{q}^{\prime}}_{T}L_{x}^{\frac{p+1}{p}}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v\|_{C_{T}L_{x}^{p+1}}^{p}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\Big{)}$

for some $q\gg 1$ and small $\theta>0$. Hence, we conclude that $\Gamma v\in
C([0,T];L^{2}(\mathbb{R}^{d}))$ for all $v\in
C([0,T];L^{p+1}(\mathbb{R}^{d}))$ and $T>0$.

Next, we consider the energy-(super)critical case (ii): $p\geq
1+\frac{4}{d-2}$ when $d\geq 3$. Since $s_{0}-1\geq
s_{1}>s_{\text{crit}}-1=\frac{d-2}{2}-\frac{2}{p-1}$, we can choose $\delta>0$
sufficiently small such that

$\displaystyle H^{s_{0}}(\mathbb{R}^{d})\subset
W^{s_{1},r}(\mathbb{R}^{d})\subset L^{\frac{(p-1)r}{r-2}}(\mathbb{R}^{d}),$
(3.7)

where $r=\frac{2d}{d-2}-\delta$. Since $p$ is an odd integer, the nonlinearity
$\mathcal{N}(u)$ is algebraic and hence we can apply the fractional Leibniz
rule. Then, proceeding with the dispersive estimate (1.9), the fractional
Leibniz rule, and (3.7), we have

$\displaystyle\|\Gamma v\|_{C_{T}W^{s_{1},r}_{x}}$
$\displaystyle\lesssim\|S(t)u_{0}\|_{C_{T}H^{s_{0}}}+T^{\theta}\|\langle\nabla\rangle^{s_{1}}\mathcal{N}(v+\Psi)\|_{L^{\frac{q}{p}}_{T}L_{x}^{r^{\prime}}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\|\langle\nabla\rangle^{s_{1}}(v+\Psi)\|_{L^{q}_{T}L_{x}^{r}}\|v+\Psi\|_{L^{q}_{T}L_{x}^{\frac{(p-1)r}{r-2}}}^{p-1}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\|v+\Psi\|_{L^{q}_{T}W_{x}^{s_{1},r}}^{p}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\Big{(}\|v\|_{C_{T}W_{x}^{s_{1},r}}^{p}+\|\Psi\|_{L^{q}_{T}W_{x}^{s_{1},r}}^{p}\Big{)}$
(3.8)

for some $q\gg 1$ and small $\theta>0$. The difference estimate (3.2) follows
in a similar manner. Given $r=\frac{2d}{d-2}-\delta$, let $(\widetilde{q},r)$
be Schrödinger admissible. Then, proceeding as in (3.8) with (3.1), we have

$\displaystyle\|\Gamma v-S(t)u_{0}\|_{C_{T}H^{s_{1}}_{x}}$
$\displaystyle\lesssim\|\langle\nabla\rangle^{s_{1}}\mathcal{N}(v+\Psi)\|_{L^{\widetilde{q}^{\prime}}_{T}L_{x}^{r^{\prime}}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v\|_{C_{T}W_{x}^{s_{1},r}}^{p}+\|\Psi\|_{L^{q}_{T}W_{x}^{s_{1},r}}^{p}\Big{)}$

for some $q\gg 1$ and small $\theta>0$. This shows $\Gamma v\in
C([0,T];H^{s_{1}}(\mathbb{R}^{d}))$ for all $v\in
C([0,T];W^{s_{1},r}(\mathbb{R}^{d}))$ and $T>0$. ∎

Similarly, Theorem 1.1 (i.b) follows from the following proposition. In order
to control the linear solution at a lower regularity, we apply the Strichartz
estimate (1.7).

######  Proposition 3.2.

Let $d$ and $p$ be as in Theorem 1.1 (i.b) and $s_{0}>s_{\textup{crit}}$.
Then, the following estimates hold for some $q\gg 1$:

$\displaystyle\|\Gamma v\|_{L^{q}_{T}L^{p+1}_{x}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\Big{(}\|v\|_{L^{q}_{T}L^{p+1}_{x}}^{p}+\|\Psi\|_{L^{q}_{T}L^{p+1}_{x}}^{p}\Big{)},$
$\displaystyle\|\Gamma v_{1}-\Gamma v_{2}\|_{L^{q}_{T}L^{p+1}_{x}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v_{1}\|_{L^{q}_{T}L^{p+1}_{x}}^{p-1}+\|v_{2}\|_{L^{q}_{T}L^{p+1}_{x}}^{p-1}+\|\Psi\|_{L^{q}_{T}L^{p+1}_{x}}^{p-1}\Big{)}\|v_{1}-v_{2}\|_{L^{q}_{T}L^{p+1}_{x}}$
(3.9)

for all $v,v_{1},v_{2}\in L^{q}([0,T];L^{p+1}(\mathbb{R}^{d}))$ and $T>0$.
Moreover, we have $\Gamma v\in C([0,T];L^{2}(\mathbb{R}^{d}))$ for all $v\in
L^{q}([0,T];L^{p+1}(\mathbb{R}^{d}))$ and $T>0$.

###### Proof.

Given $(d,p)$, fix $q\geq 2$ such that
$\frac{1}{q}+1=\Big{(}\frac{d}{2}-\frac{d}{p+1}+\varepsilon\Big{)}+\frac{p}{q}$
for some small $\varepsilon>0$. Furthermore, let $r\geq 2$ such that $(q,r)$
is Schrödinger admissible. Then, by Sobolev’s inequality, we have

$\displaystyle W^{s_{0},r}(\mathbb{R}^{d})\subset L^{p+1}(\mathbb{R}^{d}).$
(3.10)

By proceeding as in (3.3) with (3.10), the dispersive estimate (1.9), the
Strichartz estimate (1.7), and Young’s inequality, we have

$\displaystyle\|\Gamma v\|_{L^{q}_{T}L_{x}^{p+1}}$
$\displaystyle\lesssim\|S(t)u_{0}\|_{L^{q}_{T}W^{s_{0},r}}+\bigg{\|}\int_{0}^{t}\frac{1}{|t-t^{\prime}|^{\frac{d}{2}-\frac{d}{p+1}}}\|\mathcal{N}(v+\Psi)(t^{\prime})\|_{L_{x}^{\frac{p+1}{p}}}dt^{\prime}\bigg{\|}_{L^{q}_{T}}$
$\displaystyle\lesssim\|u_{0}\|_{H^{s_{0}}}+T^{\theta}\Big{(}\|v\|_{L^{q}_{T}L_{x}^{p+1}}^{p}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\Big{)}$
(3.11)

for some $\theta>0$. The difference estimate (3.9) follows in a similar
manner. Given $p+1<\frac{2d}{d-2}$, let $(\widetilde{q},p+1)$ be Schrödinger
admissible. Then, from the mass-supercritical condition: $p>1+\frac{4}{d}$, we
see that $q>p\,\widetilde{q}^{\prime}$. Hence, it follows from (3.1) that

$\displaystyle\|\Gamma v-S(t)u_{0}\|_{C_{T}L_{x}^{2}}$
$\displaystyle\lesssim\|\mathcal{N}(v+\Psi)\|_{L^{\widetilde{q}^{\prime}}_{T}L_{x}^{\frac{p+1}{p}}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v\|_{L^{q}_{T}L_{x}^{p+1}}^{p}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\Big{)}$

for some small $\theta>0$. ∎

######  Remark 3.3.

In (3.11), it is possible to apply Hardy-Littlewood-Sobolev’s inequality
instead of Young’s inequality since $q<\infty$. Namely, proceeding with
$\varepsilon=0$, Hardy-Littlewood-Sobolev’s inequality gives

$\displaystyle\|\Gamma v\|_{L^{q}_{T}L_{x}^{p+1}}$
$\displaystyle\lesssim\Big{(}\|S(t)u_{0}\|_{L^{q}_{T}W^{s_{0},r}}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\Big{)}+\|v\|_{L^{q}_{T}L_{x}^{p+1}}^{p}.$

The difference estimate (3.9) also holds without the $T^{\theta}$-factor.
Then, we can carry out a contraction argument by making $T=T_{\omega}>0$
sufficiently small such that

$\|S(t)u_{0}\|_{L^{q}_{T}W^{s_{0},r}}+\|\Psi\|_{L^{q}_{T}L_{x}^{p+1}}^{p}\ll
1,$

as in the mass-critical local well-posedness theory for NLS (1.4), and prove
local well-posedness of (1.1) even when $s_{0}=s_{\text{crit}}$. As the
argument is standard, we omit details. Note that we measure the stochastic
convolution $\Psi$ only with the subcritical $L^{p+1}_{x}$-norm.

###  3.2. Proof of Theorem 1.3

Given $u_{0}$ on $\mathbb{R}^{d}$, let $u_{0}^{\omega}$ be its Wiener
randomization defined in (1.14). Then, we define
$\widetilde{\Gamma}=\widetilde{\Gamma}_{u_{0}^{\omega},\phi,\,\xi}$ by

$\widetilde{\Gamma}v(t):=\int_{0}^{t}S(t-t^{\prime})\mathcal{N}(v+\widetilde{\Psi})(t^{\prime})dt^{\prime},$

where $\widetilde{\Psi}$ is the stochastic convolution defined in (1.15) such
that $\widetilde{\Psi}|_{t=0}=u_{0}^{\omega}$. Then, by proceeding as in the
proof of Proposition 3.1, we obtain the following nonlinear estimates.

######  Proposition 3.4.

Let $d$ and $p$ be as in Theorem 1.1. We set

  * (i)

$s=0$ and $r=p+1$ in the energy-subcritical case,

  * (ii)

$s>s_{\textup{crit}}-1$ and $r=\frac{2d}{d-2}-\delta$ for some small
$\delta=\delta(s)>0$ in the energy-(super)critical case.

Then, the following estimates hold for some $q\gg 1$:

$\displaystyle\|\widetilde{\Gamma}v\|_{C_{T}W^{s,r}_{x}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v\|_{C_{T}W_{x}^{s,r}}^{p}+\|\widetilde{\Psi}\|_{L^{q}_{T}W_{x}^{s,r}}^{p}\Big{)},$
$\displaystyle\|\widetilde{\Gamma}v_{1}-\widetilde{\Gamma}v_{2}\|_{C_{T}W^{s,r}_{x}}$
$\displaystyle\lesssim
T^{\theta}\Big{(}\|v_{1}\|_{C_{T}W_{x}^{s,r}}^{p-1}+\|v_{2}\|_{C_{T}W_{x}^{s,r}}^{p-1}+\|\widetilde{\Psi}\|_{L^{q}_{T}W_{x}^{s,r}}^{p-1}\Big{)}\|v_{1}-v_{2}\|_{C_{T}W_{x}^{s,r}}$

for all $v,v_{1},v_{2}\in C([0,T];W^{s,r}(\mathbb{R}^{d}))$.

Next, let us state the following probabilistic Strichartz estimates. See [1]
for the proof.

######  Lemma 3.5.

Let $s\in\mathbb{R}$. Given $u_{0}$ on $H^{s}(\mathbb{R}^{d})$, let
$u_{0}^{\omega}$ be its randomization defined in (1.14), satisfying (1.13).
Then, the following statements hold almost surely:

  * (i)

$S(t)u_{0}^{\omega}\in C(\mathbb{R};H^{s}(\mathbb{R}^{d}))$,

  * (ii)

Given finite $q,r\geq 2$, we have $S(t)u_{0}^{\omega}\in
L^{q}([0,T];W^{s,r}(\mathbb{R}^{d}))$ for any $T>0$.

In particular, Lemmas 2.1 and 3.5 state that the new stochastic convolution
$\widetilde{\Psi}=S(t)u_{0}^{\omega}+\Psi$ also satisfies the conclusion of
Lemma 2.1. Therefore, together with this observation, Proposition 3.4 implies
Theorem 1.3.

######  Acknowledgment.

T. O. was supported by the European Research Council (grant no. 637995
“ProbDynDispEq”).

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