# Relaxed optimal control for the stochastic Landau-Lifshitz-Gilbert equation

Soham Gokhale  Symbiosis Institute of Technology, Pune
<EMAIL_ADDRESS>
###### Abstract.

We consider the stochastic Landau-Lifshitz-Gilbert equation, perturbed by a
real-valued Wiener process. We add an external control to the effective field
as an attempt to drive the magnetization to a desired state and also to
control thermal fluctuations. We use the theory of Young measures to relax the
given control problem along with the associated cost. We consider a control
operator that can depend (possibly non-linearly) on both the control and the
associated solution. Moreover, we consider a fairly general associated cost
functional without any special convexity assumption. We use certain
compactness arguments, along with the Jakubowski version of the Skorohod
Theorem to show that the relaxed problem admits an optimal control.

###### Key words and phrases:

Stochastic Landau Lifshitz Gilbert equation, Ferromagnetism, Optimal control,
Young measures

MSC Subject Classification : 60H15

email<EMAIL_ADDRESS>
##  1\. Introduction

Magnetic data storage devices are some of the most widely used devices to
store and process digital data, for example a magnetic hard drive. One of the
modern day problems is to optimize this storage, that is to have the ability
to store and process large amounts of data on a small device, without
hampering the processing speed. This speed of operation can depend on the
speed of switching of the magnetization. Therefore, a study of this switching
phenomenon becomes key in such applications. A lot of these devices use
ferromagnets (such as iron, oxides of chromium, etc.) as a material. One of
the main points of study in this regard is optimal control. Is it possible to
have an external input (control) that can drive the magnetization from one
state to another? Can we control the thermal fluctuations to some degree, and
reduce the possible damage or loss of storage for the given device? Having
commercial uses, can this be done with optimal cost? We try to answer this
question in some sense.

Weiss initiated the study of the theory of ferromagnetism, (see for example
[6] and references therein). This was further developed by Landau and Lifshitz
[39] and Gilbert [26]. Let $\mathcal{O}\subset\mathbb{R}$ be a bounded
interval. For a temperature below the Curie temperature, the magnetization $m$
satisfies the Landau-Lifshitz-Gilbert (LLG) equation

$\displaystyle\begin{cases}&dm=\left[(m\times
H_{\text{eff}}(m))-\alpha\,m\times(m\times H_{\text{eff}}(m))\right]\,dt\
\text{in}\ \mathcal{O}_{T}=(0,T)\times\mathcal{O},\\\ &\frac{\partial
m}{\partial\nu}=\,0\ \text{on}\ \partial\mathcal{O}_{T},\\\ &m(0,\cdot)=m_{0}\
\text{on}\ \mathcal{O}.\end{cases}$ (1.1)

Here $\times$ denotes the vector product in $\mathbb{R}^{3}$. Also, $\nu$
denotes the outward pointing normal vector. The constant $\alpha>0$ depends on
the gyromagnetic ratio and the damping parameter [14]. $H_{\text{eff}}$
denotes the effective field, which will be described shortly. For
ferromagnetic materials at temperature below the Curie temperature, the
modulus of the magnetization remains constant, giving rise to a constraint
condition (see (1.3)). For simplicity, we assume the constant to be 1.

The solutions of (1.1) correspond to the equilibrium states of the
ferromagnet. We aim to study the phase transition between equilibrium states
that is induced by the thermal fluctuations in the effective field
$H_{\text{eff}}$. In order to do this, we modify the equation (1.1) to include
the random fluctuations affecting the dynamics of $m$. The study of analysis
of noise induced transitions was started by Neel [47] and further studied in
[7, 37], among others. The introduction of a stochastic forcing term in (1.1)
is due to Brown, see [6, 7].

The effective field $H_{\text{eff}}=H_{\text{eff}}(m)=-D\mathcal{E}(m)$ is
deduced from the Landau–Lifshitz energy $\mathcal{E}(m)$, primarily consisting
of three parts: the exchange energy, the anistropy energy and the
magnetostatic energy, see [38] for example. Visintin [57] studied the equation
with contribution from all three energies to the total magnetic energy. For
simplicity, we only the exchange energy has been considered here. Similar
simplifying assumptions can be seen in [4, 8] to name a few. DeSimone [18],
Gioia and James [27] studied the LLG equation (with exchange energy only) as a
limiting case of the models that include the other types of energy. The
exchange energy is given by

$\displaystyle\mathcal{E}(m)=\int_{\mathcal{O}}\frac{A_{\text{exc}}}{2}|\nabla
m|_{\mathbb{R}^{3}}^{2}\,dx,$

where $A_{\text{exc}}$ is a material constant [14]. In the calculations that
follow, we assume that $A_{\text{exc}}=1$. Therefore, we have the effective
field $H_{\text{eff}}=\Delta m$.

Following [8], we perturb the equation by adding Gaussian noise to the
effective field. Further, the noise needs to be invariant under the coordinate
transformation. Hence it is suggested, see [7, 37] for instance, that we
consider the noise in the Stratonovich form. The resulting stochastic
counterpart (stochastic LLG equation) of equation (1.1) is

$\displaystyle\begin{cases}&dm=\left(m\times
H_{\text{eff}}-\alpha\,m\times(m\times H_{\text{eff}})\right)\,dt\\\ &\qquad\
+\bigl{(}m\times h-\alpha\,m\times(m\times h)\bigr{)}\circ\,dW(t)\ \text{in}\
\mathcal{O}_{T}=(0,T)\times\mathcal{O},\\\ &\frac{\partial
m}{\partial\nu}=\,0\ \text{on}\ \partial\mathcal{O}_{T},\\\ &m(0,\cdot)=m_{0}\
\text{on}\ \mathcal{O}.\end{cases}$ (1.2)

Here $h:\mathcal{O}\to\mathbb{R}^{3}$ is a given bounded function. $W$ is a
real valued Wiener process on a given probability space
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$. Also, $\circ\,dW$ denotes the
Stratonovich differential. The equation (1.2) is subject to the constraint
condition

$\left|m(t,x)\right|_{\mathbb{R}^{3}}=1,\ \text{for Leb. a.a.
}x\in\mathcal{O},\ \text{for all }t\in[0,T],\ \mathbb{P}^{\prime}\text{-a.s.}$
(1.3)

######  Remark 1.1.

A natural question here is about what happens when the temperature is above
the Curie temperature. In that case, the model can be replaced by the Landau-
Lifshitz-Bloch (LLB) equation. The model was proposed by Geranin in [24] in
1997. The LLB equation essentially interpolates between the LLG equation at
low temperatures and the Ginzburg-Landau theory of phase transitions. Le in
[40] showed the existence of a weak solution to the LLB equation. The same
author with Brzeźniak and Goldys in [10] showed the existence and uniqueness
of a solution for the stochastic LLB equation, along with the existence of
invariant measures for dimensions 1 and 2. On similar lines, Jiang, Ju and
Wang in [35] showed the existence of a weak martingale solution to the
stochastic LLB equation. In [52], the authors established a large deviation
principle and central limit theorem for the 1 dimensional stochastic Landau-
Lifshitz-Bloch equation. The authors in the preprint [28] show the existence
and pathwise uniqueness of a solution for the stochastic LLB equation
(dimensions $1,2$) driven by a pure jump noise. Further, the authors establish
a Wentzell-Friedlin type large deviations principle for the small noise
asymptotic of the solutions. On a related note, the authors in [29] proved
Wong-Zakai type approximations for the stochastic LLB equation perturbed by a
real-valued Wiener process. In the preprint [30], the authors consider the
stochastic LLB equation. Following the strategy from [20] and [11] and
considering a very general control operator and cost, the authors show that
the problem admits a relaxed optimal control.

As indicated earlier, our aim is to try and optimize the switching of
magnetization by giving some external input. Following the works [12, 19, 20],
we add the control to the effective field $H_{\text{eff}}$ as an attempt to
control the magnetization switching, and also to control the thermal
fluctuations. Instead of just adding the control process, we add operators
$L_{1}(m,u),L_{2}(m,u)$ that depend (possibly non-linearly) on both the
control process $u$ and the corresponding solution process $m$. The resulting
controlled equation is as follows.

$\displaystyle dm(t)=$ $\displaystyle\big{[}m(t)\times\Delta m(t)-\alpha
m(t)\times\left(m(t)\times\Delta m(t)\right)$ $\displaystyle+m(t)\times
L_{1}(m,u)-\alpha m(t)\times\left(m(t)\times L_{2}(m,u)\right)\big{]}\,dt$
$\displaystyle+\left[m(t)\times h-\alpha m(t)\times\left(m(t)\times
h\right)\right]\circ dW(t)$ (1.4)

######  Remark 1.2.

A natural question here would be why we have not considered the control term
just as $L(m,u)$ (say) and why in the specific form above in (1). It is
recommended (see [19, 20]) that the added control should lie in the plane
perpendicular to the solution $m$. For some vector $\mathscr{V}$ (non-zero and
not parallel to $m$), the said tangent plane can be spanned by the vectors
$m\times\mathscr{V}$ and $m\times(m\times\mathscr{V})$. Hence, considering the
said form of the control term, in (1), suffices to study all the controls that
are acting on the tangent plane of the solution $m$.

For the system (1), we define the associated cost $J$. Let
$\left(\Omega,\mathcal{F},\mathbb{P}\right)$ denote a probability space. For a
solution process $m$ corresponding to a control process $u$, the cost
functional $J(m,u)$ is given by

$\displaystyle
J(m,u)=\mathbb{E}\left[\int_{0}^{T}F(m,u)\,dt+\Psi(m(T))\right].$ (1.5)

Here, $F$ is the running cost and $\Psi$ denotes the terminal cost.
$\mathbb{E}$ denotes the expectation corresponding to the space $\Omega$ with
respect to the probability measure $\mathbb{P}$ defined on the
$\sigma$-algebra $\mathcal{F}$ The corresponding control problem is then to
minimize $J$ over the space of all solutions to the problem (1).

In this work, we employ the theory of Young measures, see for instance [3,
61], to obtain a relaxed version of the above control problem. Let us denote
the considered control set by $\mathbb{U}$, which is assumed to be a
metrizable Suslin space. Let $L:L^{2}\times\mathbb{U}\to L^{2}$ denote the
control operator, which depends (possibly non-linearly) on the control
$u\in\mathbb{U}$ as well as the corresponding solution $m$. Let $\lambda$
denote a random Young measure on the space $\mathbb{U}$, see Definition 2.2.
Relaxed Control Problem:

$\displaystyle dm(t)=$ $\displaystyle\big{[}m(t)\times\Delta m(t)-\alpha
m(t)\times\left(m(t)\times\Delta m(t)\right)\big{]}\,dt$
$\displaystyle+\int_{\mathbb{U}}\big{[}m(t)\times L_{1}(m,v)-\alpha
m(t)\times\left(m(t)\times L_{2}(m,v)\right)\,\lambda(dv,dt)\big{]}$
$\displaystyle+\left[m(t)\times h-\alpha m(t)\times\left(m(t)\times
h\right)\right]\circ dW(t).$ (1.6)

Relaxed Cost Functional:
Corresponding to the problem (1), we relax the cost (1.5) as follows. Let
$\pi=\left(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},m,\lambda,W\right)$ be a
weak martingale solution to the relaxed problem (1) (see Definition 3.1). The
corresponding relaxed cost is given by

$\displaystyle\mathscr{J}(\pi)=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{U}}F(m,v)\lambda(dm,dt)+\Psi(m(T))\right].$
(1.7)

Our aim is to show the existence of an optimal control, see Definition 3.5, to
the above problem. That is, we want to show that there exists a weak
martingale solution with $m$ as the process corresponding to the control
process $u$ such that the cost $J$ is minimized.

######  Remark 1.3.

By Lemma 2.3, one can disintegrate a given random Young measure $\lambda$
using a relaxed control process $\left\\{q_{t}\right\\}_{t\in[0,T]}$ and the
Lebesgue measure $dt$. The original problem, along with the cost functional
can be recovered from the respective relaxed ones by replacing the Random
Young Measure $\lambda(dv,dt)=q_{t}(dv)dt$ by
$\lambda(dv,dt)=\delta_{u(t)}(dv)dt$, where $\delta_{u(t)}$ denotes the Dirac
Delta mass saturated at $u(t)$.

Observe here that no special conditions on the control operator $L$ with
respect to the control process $u$ are assumed, and no special convexity
assumption is made on the cost functional $F$. With that in mind, we extend
the space of admissible controls to a bigger space, called the space of
admissible relaxed controls. This is done by the method of relaxation (Young
[60, 61] and Warga [58, 59]). Ahmed [1], Papageorgiou, et. al. ([2, 49, 50])
use relaxed controls in the evolution equations in Banach spaces, with Polish
space valued controls (see also Lou [41, 42], Karoui, Nguyen, and Picque [21],
Gatarek, Sobczyk [25], Haussmann, Lepeltier [32]).

Regarding the stochastic case, Nagase and Nisio [45, 46] initiated the study
of relaxed control for stochastic partial differential equations. Fleming and
Nisio [22, 23] introduced Young measure technique for finite-dimensional
stochastic systems. Sritharan [54] studied the optimal relaxed control of both
deterministic and stochastic Navier-Stokes equations with linear and non-
linear constitutive relations. He used the Minty-Browder technique (along with
the martingale problem formulation of Stroock and Varadhan for the stochastic
case) to establish the existence of optimal controls. Cutland and Grzesiak
[15, 16] study the existence of an optimal control for 2D (respectively 3D)
Navier-Stokes equations. They use non-standard analysis (see also [56]), along
with relaxed control techniques. Brzeźniak and Serrano [11] study an optimal
relaxed control problem for a class of semilinear stochastic partial
differential equations on Banach spaces. They use compactness properties of
the class of Young measures on metrizable Suslin spaces (the control sets).
Manna and Mukherjee [43] use the technique of relaxed controls , with Suslin
space-valued controls, combine them with the martingale problem formulation in
the sense of Stroock and Varadhan to establish the existence of optimal
controls.

###  1.1. Comparison with some relevant works

  * •

Dunst, Majee, Prohl and Vallet in [20] considered the stochastic LLG equation
driven by a Wiener process, and showed the existence of a relaxed optimal
control. The main differences between that work and this work are as follows.
We have considered the full noise (including the non-linear term), of which
the said paper [20] considers only the linear term. We have also considered a
very general form of the control operator as well as the cost functional.

  * •

The authors in [12] (preprint) consider the stochastic LLG equation in
dimension $d=1$. They show the existence of a unique maximal regular solution,
while considering the full noise. They also show the existence of an optimal
control. The main difference between that work and this is that we consider
the dimensions $d=1,2,3$ here, as opposed to $d=1$ in [12]. Moreover, we do
not tackle the problem of existence of a solution in this work. We have
considered a fairly general control operator and cost functional as compared
to the specific ones in the said work.

  * •

The authors in [30] consider the stochastic Landau-Lifshitz-Bloch equations.
They consider a general (Suslin space valued) control operator, along with a
general cost functional. Relaxing the problem, they first show the existence
of a weak martingale solution to the controlled (relaxed) equation, followed
by showing the existence of a relaxed optimal control to the said problem. In
this paper, we consider a similar approach, but for the stochastic LLG
equation.

##  2\. Some Notations, Preliminaries

We state some of the results that we have used in the proofs. We have mostly
imitated the notations and borrowed results from [11, 13], and [54]. For more
details, we refer the reader to [3, 33, 34, 53, 61] among others.

We assume that $\mathcal{O}\subset\mathbb{R}^{d},\ d=1,2,3$ is a bounded
domain with a smooth boundary. Let $0<T<\infty$ be arbitrary but fixed. Let
$\mathbb{U}$ (the control set) denote a Hausdorff topological space. Let
$\mathcal{B}\left(\mathbb{U}\right)$ denote the Borel $\sigma$-algebra on
$\mathbb{U}$. Let $\mathcal{P}\left(\mathbb{U}\right)$ denote the set of all
probability measures on $\mathcal{B}\left(\mathbb{U}\right)$. The set
$\mathcal{P}\left(\mathbb{U}\right)$ is endowed with the $\sigma$-algebra
generated by the projection maps

$\pi_{C}:\mathcal{P}\left(\mathbb{U}\right)\ni q\mapsto q(C)\in[0,1],\
C\in\mathcal{B}\left(\mathbb{U}\right).$ (2.1)

Let $\left(\Omega,\mathcal{F},\mathbb{F},\mathbb{P}\right)$ denote a filtered
probability space (satisfying the usual hypothesis) with filtration
$\mathbb{F}=\left(\mathcal{F}_{t}\right)_{t\in[0,T]}$. Throughout the work,
let $W$ denote a $\mathbb{R}$-valued Wiener process on the given probability
space. Let $0<T<\infty$ and let $h$ denote the given function.

######  Definition 2.1 (Relaxed Control Process).

A stochastic process $q=\left\\{q_{t}\right\\}_{t\in[0,T]}$ with values in
$\mathcal{P}\left(\mathbb{U}\right)$ is called a relaxed control process on
$\mathbb{U}$ if the map

$[0,T]\times\Omega\ni\left(t,\omega\right)\mapsto
q_{t}\left(\omega,\cdot\right)\in\mathcal{P}\left(\mathbb{U}\right),$ (2.2)

is measurable. In other words, a relaxed control process on $\mathbb{U}$ is a
measurable process with values in $\mathcal{P}\left(\mathbb{U}\right)$.

######  Definition 2.2 (Young Measure).

Let $\lambda$ denote a non-negative $\sigma$-additive measure on
$\mathcal{B}\bigl{(}\mathbb{U}\times[0,T]\bigr{)}$. We say that $\lambda$ is a
Young measure on $\mathbb{U}$ if and only if $\lambda$ satisfies

$\lambda\left(\mathbb{U}\times D\right)=\text{Leb}\left(D\right),\ \forall
D\in\mathcal{B}\bigl{(}[0,T]\bigr{)},$ (2.3)

with Leb denoting the Lebesgue measure on $[0,T]$. The space of all Young
measures on $\mathbb{U}$ will be denoted by
$\mathcal{Y}\left(0,T:\mathbb{U}\right)$.

######  Lemma 2.3 (Disintegration of Young measures, Lemma 2.3, [11]).

Let $\mathbb{U}$ be a Radon space. Let
$\lambda:\Omega\to\mathcal{Y}\left(0,T:\mathbb{U}\right)$ be such that for
every $J\in\mathcal{B}\bigl{(}\mathbb{U}\times[0,T]\bigr{)}$, the mapping

$\Omega\ni\omega\mapsto\lambda\left(\omega\right)\in[0,T],$ (2.4)

is measurable. Then there exists a relaxed control process
$q=\left\\{q_{t}\right\\}_{t\in[0,T]}$ on $\mathbb{U}$ such that

$\lambda\left(\omega,C\times D\right)=\int_{D}q_{t}\left(\omega,C\right)\,dt,\
\forall C\in\mathcal{B}\left(\mathbb{U}\right),\
D\in\mathcal{B}\bigl{(}[0,T]\bigr{)}.$ (2.5)

The formula (2.5) is frequently denoted by

$\lambda\left(du,dt\right)=q_{t}\left(du\right)\,dt.$ (2.6)

Note that the above mapping (2.4) being measurable justifies calling $\lambda$
a random Young measure.

######  Definition 2.4 (Stable Topology).

The stable topology on $\mathcal{Y}(0,T:\mathbb{U})$ is the weakest topology
on $\mathcal{Y}(0,T:\mathbb{U})$ such that the mappings

$\mathcal{Y}(0,T:\mathbb{U})\ni\lambda\mapsto\int_{D}\int_{\mathbb{U}}f(u)\,\lambda(du,dt)\in\mathbb{R},$
(2.7)

is continuous for every $D\in\mathcal{B}([0,T])$ and $f\in
C_{\text{b}}(\mathbb{U})$.

######  Proposition 2.5.

The space of Young measures on a metrizable space (respectively metrizable
Suslin) with the stable topology is metrizable (respectively metrizable
Suslin).

######  Definition 2.6 (Flexible Tighness).

We say that a set $\mathscr{J}\subset\mathcal{Y}(0,T:\mathbb{U})$ is flexibly
tight, if for each $\varepsilon>0$, there exists a measurable set-valued
mapping $[0,T]\ni t\mapsto K_{t}\subset\mathbb{U}$ such that $K_{t}$ is
compact for all $t\in[0,T]$ and

$\sup_{\lambda\in\mathscr{J}}\int_{0}^{T}\int_{\mathbb{U}}\mathbbm{1}_{K_{t}^{c}}(u)\lambda(du,dt)<\varepsilon.$
(2.8)

######  Definition 2.7 (Inf-compactness).

A function $\eta:\mathbb{U}\to[0,\infty]$ is called inf-compact if and only if
for every $R>0$ the level set

$\left\\{\eta\leq R\right\\}=\left\\{u\in\mathbb{U}:\eta(u)\leq R\right\\},$

is compact.

######  Lemma 2.8 (Theorem 2.13, [11]).

The following are equivalent for any
$\mathscr{J}\subset\mathcal{Y}(0,T:\mathbb{U})$.

  1. (1)

$\mathscr{J}$ is flexibly tight.

  2. (2)

There exists an inf-compact function $\eta$ such that

$\sup_{\lambda\in\mathscr{J}}\int_{0}^{T}\int_{\mathbb{U}}\eta(t,u)\lambda(du,dt)<\infty.$
(2.9)

######  Lemma 2.9 (Theorem 2.16, [11]).

Let $\mathbb{U}$ be a metrizable Suslin space. If $\lambda_{n}\to\lambda$
stably in $\mathcal{Y}(0,T:\mathbb{U})$, then for every $f\in
L^{1}(0,T:C_{\text{b}}(\mathbb{U}))$ we have

$\lim_{n\to\infty}\int_{0}^{T}\int_{\mathbb{U}}f(t,u)\,\lambda_{n}(du,dt)=\int_{0}^{T}\int_{\mathbb{U}}f(t,u)\,\lambda(du,dt).$
(2.10)

##  3\. Formulation of the problem, Statements of the main result

First of all, recall that the equality (specifically the stochastic
differential) in (1) is to be understood in the Stratonovich sense. This can
be written in the Itô sense by adding a correction term as follows. Let us
define an operator $G:L^{2}\to L^{2}$ given by

$\displaystyle G(v)=v\times h-\alpha v\times(v\times h),\ v\in L^{2}.$ (3.1)

With this in mind, the equality (1) is equivalent see [17, 48], to the
equality

$\displaystyle dm(t)=$ $\displaystyle\big{[}m(t)\times\Delta m(t)-\alpha
m(t)\times\left(m(t)\times\Delta m(t)\right)\big{]}\,dt$
$\displaystyle+\int_{\mathbb{U}}\big{[}m(t)\times L_{1}(m,v)-\alpha
m(t)\times\left(m(t)\times L_{2}(m,v)\right)\,\lambda(dv,dt)\big{]}$
$\displaystyle+\frac{1}{2}DG(m(t))\bigl{(}G(m(t))\bigr{)}\,dt+G(m(t))\,dW(t).$
(3.2)

Here $DG$ denotes the Gateaux derivative of $G$.

######  Definition 3.1 (Weak Martingale Solution).

A tuple
$\pi^{\prime}=\left(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{F}^{\prime},\mathbb{P}^{\prime},m^{\prime},\lambda^{\prime},W^{\prime}\right)$
is said to be a weak martingale solution to the problem (1) (equivalently a
relaxed solution to the problem (1) if the following hold:

  1. (1)

$\left(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{F}^{\prime},\mathbb{P}^{\prime}\right)$
is a filtered probability space that satisfies the usual hypotheses.

  2. (2)

$W^{\prime}$ is a Wiener process on the said probability space.

  3. (3)

$\lambda^{\prime}$ is a random Young measure on the space $\mathbb{U}$ such
that the laws of $\lambda^{\prime}$ and $\lambda$ are equal on the space
$\mathcal{Y}(0,T:\mathbb{U})$.

  4. (4)

The process $m^{\prime}:\Omega^{\prime}\times[0,T]\times\mathscr{O}$ is
adapted, taking values in the space $C([0,T]:(H^{1})^{\prime})$, and satisfies
the equality (3) in the weak sense. That is, for $\phi\in W^{1,4}$, the
following equality holds.

$\displaystyle\left\langle m^{\prime}(t),\phi\right\rangle_{L^{2}}=$
$\displaystyle\int_{0}^{t}\left\langle\big{[}m^{\prime}(t)\times\Delta
m^{\prime}(t)-\alpha m^{\prime}(t)\times\left(m^{\prime}(t)\times\Delta
m^{\prime}(t)\right)\big{]},\phi\right\rangle_{L^{2}}\,dt$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{U}}\left\langle\big{[}m^{\prime}(t)\times
L(m^{\prime}(t),v)-\alpha m^{\prime}(t)\times\left(m^{\prime}(t)\times
L(m^{\prime}(t),v)\right)\,\lambda^{\prime}(dv,dt)\big{]},\phi\right\rangle_{L^{2}}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\left\langle
DG\left(m^{\prime}(t)\right)\bigl{(}G\left(m^{\prime}(t)\right)\bigr{)},\phi\right\rangle_{L^{2}}\,dt+\int_{0}^{t}\left\langle
G\left(m^{\prime}(t)\right),\phi\right\rangle_{L^{2}}\,dW(t).$ (3.3)

  5. (5)

The process $m^{\prime}$ satisfies the following bounds for $p\geq 1$:

    1. (a)

$\mathbb{E}\sup_{t\in[0,T]}\left|m^{\prime}(t)\right|_{H^{1}}^{2p}<\infty,$
(3.4)

    2. (b)

$\mathbb{E}\int_{0}^{T}\left|m^{\prime}(t)\times\Delta
m^{\prime}(t)\right|_{L^{2}}^{2p}\,dt<\infty.$ (3.5)

  6. (6)

The process $m^{\prime}$ satisfies the constraint condition (1.3).

An Assumption on the control operators:
We assume that there exist two inf-compact functions $\kappa_{1},\kappa_{2}$
along with constants $C_{1},C_{2}$ such that for some $r\geq 0$,

$\displaystyle L_{1}(m,u)\leq
C_{1}\left|m\right|_{L^{2}}^{r}+C_{1}\kappa_{1}(u,\cdot)+C_{1}\left|m\right|^{r}_{L^{2}}\kappa_{1}(u,\cdot),\
m\in L^{2},u\in\mathbb{U},$ (3.6) $\displaystyle L_{2}(m,u)\leq
C_{2}\left|m\right|_{L^{2}}^{r}+C_{2}\kappa_{2}(u,\cdot)+C_{2}\left|m\right|^{r}_{L^{2}}\kappa_{2}(u,\cdot),\
m\in L^{2},u\in\mathbb{U}.$ (3.7)

######  Remark 3.2.

In the two inequalities above, one can do away with the first two elements on
the right hand sides. This is because they only add length to the
calculations, and the complexity remains the same if we consider only the last
terms on the respective right hand sides. Moreover, we assume $L_{1}=L_{2}$,
and hence also $\kappa_{1}=\kappa_{2}$ and $C_{1}=C_{2}$. Note that this also
does not simplify the problem any more than reducing the length of certain
calculations.

######  Remark 3.3.

For comparing this work with the preprint [12], a main contrast is that even
if we assume $L(m,u)=u$, the inf-compact function cannot be just the $L^{2}$
norm since it may not satisfy the required properties. In that case, we may
need to either assume $\kappa$ to be the $H^{1}$ norm, or maybe assume the
space of admissible controls to be $L^{2}_{\text{w}}(0,T:L^{2})$, which is the
space $L^{2}(0,T:L^{2})$ endowed with the weak topology. For the latter case,
the following embedding is compact.

$L^{2}(0,T:L^{2})\hookrightarrow L^{2}_{\text{w}}(0,T:L^{2}).$

Also, the authors in [12] deals with the control problem in dimension $d=1$,
proving the existence of a unique maximal regular solution. In this work, we
do not prove the existence/non-existence of a solution for the controlled
problem.

We now state some assumptions on the control operator $L$ the cost functional
$F$.

######  Assumption 3.4.

We assume the following.

  1. (1)

We will assume, in the first part of the paper, that $\alpha=0$ for the noise
part. This will keep only the linear terms of the noise part. Later, in
Appendix A, we present the arguments for the case $\alpha\neq 0$. With this in
mind, the equality (3) takes the form

$\displaystyle dm(t)=$ $\displaystyle\big{[}m(t)\times\Delta
m(t)-m(t)\times\left(m(t)\times\Delta m(t)\right)\big{]}\,dt$
$\displaystyle+\big{[}m(t)\times L(m,v)-m(t)\times\left(m(t)\times
L(m,v)\right)\,\lambda(dv,dt)\big{]}$
$\displaystyle+\frac{1}{2}DG(m(t))\bigl{(}G(m(t))\bigr{)}\,dt+G(m(t))\,dW(t),$
(3.8)

with $G(m)=m\times h$ and $DG(m)(G(m))=G^{2}(m)=(m\times h)\times h$. For the
coefficient of the triple product term $\left(m\times\left(m\times\Delta
m\right)\right)$, we have replaced $\alpha$ by $1$ for simplicity.

  2. (2)

Taking a cue from the previous (formal) discussion in Remark 3.2, we assume
that there exists an inf-compact function $\kappa$ and a constant $C>0$ such
that the following holds. Let $r\geq 0$. Then

$\left|L(m,u)\right|_{L^{2}}\leq C\left|m\right|_{L^{2}}^{r}\kappa(\cdot,u).$
(3.9)

Further, we assume that for any random Young measure, we have

$\mathbb{E}\int_{0}^{T}\kappa^{5}(\cdot,v)\lambda(dv,dt)<\infty.$ (3.10)

For higher order estimates, we require the following. For $p\geq 1$,

$\mathbb{E}\int_{0}^{T}\kappa^{5p}(\cdot,v)\lambda(dv,dt)<\infty.$ (3.11)

  3. (3)

The operator $L$ is continuous in the first variable in the following sense.
For any $R>0$, let $K_{R}=\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\leq
R\right\\}$. Let $m_{n}\to m$ in $L^{4}(\Omega:L^{4}(0,T:L^{4}))$ as
$n\to\infty$ and let $\lambda$ denote a random Young measure on $\mathbb{U}$.
Then

$\displaystyle\mathbb{E}\int_{0}^{T}\sup_{v\in
K_{R}}\left|L(m_{n},v)-L(m,v)\right|_{L^{2}}^{2}\,ds\to 0,\ \text{as}\
n\to\infty.$ (3.12)

  4. (4)

The functional $F$ is lower semicontinuous on the space $(H^{1})^{\prime}$.
That is, if $m_{n}\to m$ as $n\to\infty$ in $(H^{1})^{\prime}$ then

$F(m)\leq\liminf_{n\to\infty}F(m_{n}).$ (3.13)

  5. (5)

The terminal cost $\Psi$ is assumed to be continuous with values in the space
$(H^{1})^{\prime}$.

  6. (6)

The terminal time is $0<T<\infty$ and given data $h\in W^{2,\infty}$ is fixed.

  7. (7)

We assume the following coercivity assumption on the cost $F$.

$\displaystyle F(m,v)\geq C\kappa^{5}(m(\cdot),v).$ (3.14)

  8. (8)

We assume that the initial data and the given data have the following
regularity: $m_{0}\in W^{1,2}(\mathscr{O}:\mathbb{S}^{2})$ and $h\in
W^{2,\infty}$. Here, $\mathbb{S}^{2}$ denotes the unit sphere in
$\mathbb{R}^{3}$. That is, we have assumed $\mathbb{S}^{2}$-valued initial
data.

For the control problem, we need to define the space of admissible solutions.
Here, we consider a weak martingale solution $\pi$ to be admissible if the
associated relaxed cost is finite. That is,

$\displaystyle\mathscr{J}(\pi)<\infty.$ (3.15)

Let us denote by $\mathcal{U}_{m_{0},T}$ the space of all admissible solutions
corresponding to the initial data $m_{0}$ and the terminal time $T$.

######  Definition 3.5.

A weak martingale solution $\pi^{*}$ to the problem (1) is said to be an
optimal control for the problem (1) with respect to the cost (1.7) (a relaxed
optimal control for the problem (1) with the cost (1.5)) if

$\mathscr{J}(\pi^{*})=\inf_{\pi\mathcal{U}_{m_{0},T}}\mathscr{J}(\pi).$ (3.16)

######  Theorem 3.6.

Let $d=1,2,3$. Let Assumption 3.4 hold. Then the problem (1), with the cost
(1.7), admits an optimal control.

###### Idea of the proof.

We first give a heuristic idea of the proof. First of all, if we assume $L=0$,
we can conclude that the problem (3) admits a weak martingale solution (see
for instance [8, 20, 28, 30]. We have assumed a fairly general control
operator and cost functional. Therefore we can take the liberty to assume that
there exists at least one weak martingale solution with finite cost. That is,
we can assume that the infimum stated in Theorem 3.6 is finite. If not, just
showing the existence of a weak martingale solution suffices to show the
existence of an optimal control (with possibly infinite cost). Hence, we can
say that $\inf_{\mathcal{U}_{m_{0},T}}\mathscr{J}(\pi)=\Lambda$ (say) is
finite. Therefore, there exists a sequence
$\left\\{\pi_{n}\right\\}_{n\in\mathbb{N}}$ of weak martingale solutions,
whose costs converge to $\Lambda$. From this sequence, we extract a
subsequence and then show that it converges (in some appropriate sense) to
$\pi^{*}$, which is a weak martingale solution to the problem (1) and a
minimizer of the cost (1.7).

∎

##  4\. Proof of Theorem 3.6

The section is comprised of several lemmas, which culminate into the proof of
Theorem 3.6. Since we have assumed that $\Lambda<\infty$, there exists a
sequence $\left\\{\pi_{n}\right\\}_{n\in\mathbb{N}}$ of weak martingale
solutions to the problem (1) such that

$\lim_{n\to\infty}\mathscr{J}(\pi_{n})=\Lambda.$ (4.1)

As a consequence, there exists some positive constant (say $\mathscr{R}$) such
that

$\displaystyle\sup_{n\in\mathbb{N}}\mathscr{J}(\pi_{n})\leq\mathscr{R}.$ (4.2)

In particular,

$\displaystyle\sup_{n\in\mathbb{N}}\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}F(m,v)\,\lambda_{n}(dv,ds)\leq\mathscr{R}.$
(4.3)

Note that all the above solutions are corresponding to the same initial data
$m_{0}$ and terminal time $T$. Also recall that since each of them is a weak
martingale solution, the constraint condition (1.3) is also satisfied for each
process $m_{n},n\in\mathbb{N}$.

###  4.1. Uniform Bounds

We start by showing some uniform estimates for the sequence of processes
$m_{n},n\in\mathbb{N}$.

######  Lemma 4.1.

There exists a constant $C>0$ such that the following holds for every
$n\in\mathbb{N}$.

$\left|m_{n}\right|_{\mathbb{R}^{3}}=1,$ (4.4)
$\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{L^{2}}^{2}\leq C,$ (4.5)
$\mathbb{E}^{n}\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{H^{1}}^{2}\leq C,$ (4.6)
$\mathbb{E}^{n}\int_{0}^{T}\left|m_{n}(t)\times\Delta
m_{n}(t)\right|_{L^{2}}^{2}\,dt\leq C.$ (4.7)

###### Proof of Lemma 4.1.

Firstly, since $\pi_{n}$ is a weak martingale solution, we have

$\displaystyle dm_{n}(t)=$ $\displaystyle\left[m_{n}(t)\times\Delta
m_{n}(t)-m_{n}(t)\times\left(m_{n}(t)\times\Delta
m_{n}(t)\right)+\frac{1}{2}\left(m_{n}(t)\times h\right)\times h\right]\,dt$
$\displaystyle+\int_{\mathbb{U}}\big{[}m_{n}(t)\times
L(m_{n},v)-m_{n}(t)\times\left(m_{n}(t)\times
L(m_{n},v)\right)\,\lambda(dv,dt)\big{]}$
$\displaystyle+\left[\left(m_{n}(t)\times h\right)\right]\,dW_{n}(t).$ (4.8)

Proof of (4.4):
Proof of (4.4) follows from the fact that each of the tuples $\pi_{n}$ is a
weak martingale solution to the problem (1), and hence satisfies the
constraint condition.
Proof of (4.35):
The inequality (4.35) is a consequence of the constraint condition combined
with the bounded domain $\mathscr{O}$ (and hence domain with finite measure).

We now give proof for the inequalities (4.6), (4.7).
Proof of (4.6):
We apply the Itô formula to the function $\phi:H^{1}\to\mathbb{R}$ given by

$\phi(v)=\frac{1}{2}\left|\nabla v\right|_{L^{2}}^{2},\ v\in H^{1}.$ (4.9)

Let $v,w_{1},w_{2}\in H^{1}$. The Gateaux derivative $\phi^{\prime}$ of $\phi$
is given by

$\phi^{\prime}(v)(w_{1})=\left\langle\nabla v,\nabla
w_{1}\right\rangle_{L^{2}}.$ (4.10)

For the second derivative $\phi^{\prime\prime}$, we have the following.

$\phi^{\prime\prime}(v)(w_{1},w_{2})=\left\langle\nabla w_{1},\nabla
w_{2}\right\rangle_{L^{2}}.$ (4.11)

Note that we will have to apply an infinite dimensional version of the Itô
formula, see for example [17, 31, 51]. Applying the Itô formula gives

$\displaystyle\frac{1}{2}\left|\nabla m_{n}(t)\right|_{L^{2}}^{2}=$
$\displaystyle\frac{1}{2}\left|\nabla m_{0}\right|_{L^{2}}^{2}$
$\displaystyle+\int_{0}^{t}\langle\bigg{[}m_{n}(t)\times\Delta
m_{n}(s)-m_{n}(s)\times\left(m_{n}(s)\times\Delta m_{n}(s)\right)$
$\displaystyle\qquad+\frac{1}{2}\left(m_{n}(s)\times h\right)\times
h\bigg{]},-\Delta m_{n}(t)\rangle_{L^{2}}\,ds$
$\displaystyle+\int_{0}^{t}\bigg{\langle}\int_{\mathbb{U}}\bigg{[}m_{n}(s)\times
L(m_{n}(s),v)$ $\displaystyle\qquad-m_{n}(s)\times\left(m_{n}(s)\times
L(m_{n}(s),v)\right)\,\lambda_{n}(dv,ds)\bigg{]},-\Delta
m_{n}(s)\bigg{\rangle}_{L^{2}}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\left|\nabla\left(m_{n}(s)\times
h\right)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\int_{0}^{t}\left\langle\left(m_{n}(s)\times h\right)\times
h+\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+\sum_{i=1}^{4}I_{i}(t).$ (4.12)

Calculations for $I_{1}$:

$\displaystyle I_{1}(t)=$
$\displaystyle\int_{0}^{t}\left\langle\big{[}m_{n}(t)\times\Delta
m_{n}(s)-m_{n}(s)\times\left(m_{n}(s)\times\Delta m_{n}(s)\right),-\Delta
m_{n}(t)\right\rangle_{L^{2}}\,ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h\big{]},-\Delta m_{n}(t)\right\rangle_{L^{2}}\,ds$
$\displaystyle=$ $\displaystyle I_{1,1}(t)+I_{1,2}(t).$ (4.13) $\displaystyle
I_{1,1}(t)$ $\displaystyle=\int_{0}^{t}\left\langle\big{[}m_{n}(t)\times\Delta
m_{n}(s)-m_{n}(s)\times\left(m_{n}(s)\times\Delta
m_{n}(s)\right)\big{]},-\Delta m_{n}(t)\right\rangle_{L^{2}}\,ds$
$\displaystyle=-\alpha\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}\,ds\leq 0.$ (4.14) $\displaystyle I_{1,2}(t)=$
$\displaystyle\frac{1}{2}\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h\big{]},-\Delta m_{n}(t)\right\rangle_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C+C\int_{0}^{t}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds.$ (4.15)

Calculations for $I_{2}$:

$\displaystyle I_{2}(t)=$
$\displaystyle\int_{0}^{t}\bigg{\langle}\int_{\mathbb{U}}\big{[}m_{n}(s)\times
L(m_{n}(s),v)$ $\displaystyle\qquad-m_{n}(s)\times\left(m_{n}(s)\times
L(m_{n}(s),v)\right)\,\lambda_{n}(dv,ds)\big{]},-\Delta
m_{n}(s)\bigg{\rangle}_{L^{2}}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\left\langle\int_{\mathbb{U}}\big{[}m_{n}(s)\times
L(m_{n}(s),v)\,\lambda_{n}(dv,ds)\big{]},-\Delta
m_{n}(s)\right\rangle_{L^{2}}$
$\displaystyle+\int_{0}^{t}\left\langle\int_{\mathbb{U}}\big{[}-m_{n}(s)\times\left(m_{n}(s)\times
L(m_{n}(s),v)\right)\,\lambda_{n}(dv,ds)\big{]},-\Delta
m_{n}(s)\right\rangle_{L^{2}}$ $\displaystyle=$ $\displaystyle
I_{2,1}(t)+I_{2,2}(t).$ (4.16)

Calculation for $I_{2,1}$:

$\displaystyle\left|I_{2,1}(t)\right|=$
$\displaystyle\left|\int_{0}^{t}\left\langle\int_{\mathbb{U}}\big{[}m_{n}(s)\times
L(m_{n}(s),v)\,\lambda_{n}(dv,ds)\big{]},-\Delta
m_{n}(s)\right\rangle_{L^{2}}\right|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{\mathbb{U}}\left|L(m_{n}(s),v)\right|_{L^{2}}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}\,\lambda_{n}(dv,ds)$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds+\frac{1}{2\varepsilon}\int_{0}^{t}\int_{\mathbb{U}}\left|L(m_{n}(s),v)\right|_{L^{2}}^{2}\,\lambda_{n}(dv,ds)$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\left|m_{n}(s)\right|_{L^{2}}^{2r}\kappa^{2}(m_{n}(s),v)\,\lambda_{n}(dv,ds)$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda_{n}(dv,ds).$
(4.17)

Similarly, one can show that

$\displaystyle\left|I_{2,2}(t)\right|\leq\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda(dv,ds).$
(4.18)

Calculations for $I_{3}$:

$\displaystyle\left|I_{3}(t)\right|=$
$\displaystyle\frac{1}{2}\int_{0}^{t}\left|\nabla\left(m_{n}(s)\times
h\right)\right|_{L^{2}}^{2}\,ds$ $\displaystyle\leq$ $\displaystyle
C+C\int_{0}^{t}\left|\nabla m_{n}(s)\right|_{L^{2}}^{2}\,ds.$ (4.19)

Calculations for $I_{4}$:

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\left|I_{4}(t)\right|=$
$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle\left[m_{n}(s)\times
h\right],-\Delta m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left(\int_{0}^{T}\left\langle\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}^{2}\,ds\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle C+C\mathbb{E}\left(\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{4}\,ds\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle C+C\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(s)\right|_{L^{2}}\left(\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}+C+4C\mathbb{E}\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds.$ (4.20)

Combining all the above calculations (except for $I_{4}$), we obtain the
following.

$\displaystyle\frac{1}{2}\left|\nabla m_{n}(t)\right|_{L^{2}}^{2}\leq$
$\displaystyle C+\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}-\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda(dv,ds)$
$\displaystyle+C\int_{0}^{t}\left|\nabla m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\left|\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h+\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|$ $\displaystyle=$
$\displaystyle C+\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+\left(-1+\varepsilon\right)\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda(dv,ds)+C\int_{0}^{t}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\left|\int_{0}^{t}\left\langle\left[m_{n}(s)\times
h\right],-\Delta m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|.$ (4.21)

The above inequality can be written as

$\displaystyle\frac{1}{2}\left|\nabla m_{n}(t)\right|_{L^{2}}^{2}$
$\displaystyle+\left(1-\varepsilon\right)\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}\,ds$ $\displaystyle\leq$ $\displaystyle
C+\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}ds$
$\displaystyle+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda_{n}(dv,ds)+C\int_{0}^{t}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\left|\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h+\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|.$ (4.22)

Observe that the second term on the left hand side of the above inequality is
non-negative (whenever $\varepsilon<1$), and hence can be neglected without
changing the inequality. Therefore,

$\displaystyle\frac{1}{2}\left|\nabla m_{n}(t)\right|_{L^{2}}^{2}\leq$
$\displaystyle C+\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda_{n}(dv,ds)+C\int_{0}^{t}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\left|\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h+\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|.$ (4.23)

Taking the supremum over $[0,T]$ followed by taking the expectation of both
sides gives

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(t)\right|_{L^{2}}^{2}\leq$ $\displaystyle
C+\frac{1}{2}\mathbb{E}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+C(\varepsilon)\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\kappa^{2}(m_{n}(s),v)\,\lambda_{n}(dv,ds)$
$\displaystyle+C\mathbb{E}\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle\left(m_{n}(s)\times
h\right)\times h+\left[m_{n}(s)\times h\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)\right|$ $\displaystyle\leq$
$\displaystyle C+C\mathbb{E}\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds+\frac{1}{4}\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}.$ (4.24)

Therefore,

$\displaystyle\frac{1}{4}\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(t)\right|_{L^{2}}^{2}\leq$ $\displaystyle
C+C\mathbb{E}\int_{0}^{T}\left|\nabla m_{n}(s)\right|_{L^{2}}^{2}\,ds.$ (4.25)

Multiplying by a suitable constant and using the Gronwall inequality gives the
inequality (4.6).
Proof of inequality (4.7):
We now go back to the inequality (4.1) (with $\varepsilon<1$). Now, the first
term in the resulting inequality is non-negative, and hence can be neglected
for now, while still preserving the inequality. Taking the supremum over
$[0,T]$ of both sides, followed by taking the expectation gives the following
for some constant $C>0$.

$\displaystyle\left(\alpha-\varepsilon\right)\mathbb{E}\int_{0}^{T}\left|m_{n}(s)\times\Delta
m_{n}(s)\right|_{L^{2}}^{2}\,ds\leq$ $\displaystyle
C+C\mathbb{E}\int_{0}^{T}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}\,ds+\frac{1}{4}\mathbb{E}\sup_{t\in[0,T]}\left|\nabla
m_{n}(s)\right|_{L^{2}}^{2}.$ (4.26)

Multiplying by a suitable constant and then using the inequality (4.6) gives
us the required inequality (4.7). This concludes the proof of Lemma 4.1. ∎

######  Lemma 4.2.

Let $p\geq 1$. There exists a constant $C>0$ such that the following holds for
every $n\in\mathbb{N}$.

$\displaystyle\mathbb{E}^{n}\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{L^{2}}^{2p}\leq
C,$ (4.27)
$\mathbb{E}^{n}\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{H^{1}}^{2p}\leq C,$
(4.28) $\mathbb{E}^{n}\int_{0}^{T}\left|m_{n}(t)\times\Delta
m_{n}(t)\right|_{L^{2}}^{2p}\,dt\leq C.$ (4.29)

###### Proof of Lemma 4.2.

We skip the proof for this lemma, and instead, give a very brief outline for
it (see the proof of Theorem 3.5 in [8] for a similar proof. After simplifying
the result of applying the Itô formula, we can the power $p$ of both sides of
equation (4.1), before taking the supremum over $[0,T]$ followed by the
expectation. Note that this will require the assumption (3.11) on the inf-
compact function $\kappa$. ∎

######  Lemma 4.3.

The following bounds hold. There exists a constant $C>0$ such that for all
$p\in[2,\infty)$ and $\gamma\in\left(0,\frac{1}{2}\right)$

$\mathbb{E}^{n}\left|m_{n}\right|_{W^{\gamma,p}(0,T:L^{\frac{6}{5}})}\leq C.$
(4.30)

As a consequence, the following inequality holds (possibly for a different
constant $C>0$).

$\mathbb{E}^{n}\left|m_{n}\right|_{W^{\gamma,p}(0,T:(H^{1})^{\prime})}\leq C.$
(4.31)

###### Proof.

We skip the proof. The idea is to show that each integral on the right hand
side of (3) satisfies the bounds (4.30), (4.31), and therefore each $m_{n}$
satisfies the bounds as well. For similar details, we refer the reader to
Lemma 4.1 in [8]. ∎

######  Lemma 4.4.

The sequence of laws of the sequence
$\left\\{\left(m_{n},\lambda_{n}\right)\right\\}_{n\in\mathbb{N}}$ is tight on
the space $L^{4}(0,T:L^{4})\cap
C(0,T:(H^{1})^{\prime})\times\mathcal{Y}(0,T:\mathbb{U})$.

###### Proof of Lemma 4.4:.

We skip the proof of this result, and refer the reader to Lemma 4.2 in [8],
see also [30], along with using Lemma 2.8. ∎

###  4.2. Use of the Skorohod Theorem

######  Lemma 4.5.

There exists a probability space
$\left(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}}\right)$, along
with $L^{4}(0,T:L^{4})\cap
C(0,T:(H^{1})^{\prime})\times\mathcal{Y}(0,T:\mathbb{U})$-valued random
variables $\left(\tilde{m}_{n},\tilde{\lambda}_{n}\right),n\in\mathbb{N}$,
$\tilde{m},\tilde{\lambda}$ and $C([0,T]:\mathbb{R})$-valued random variables
$\tilde{W}_{n}$, $n\in\mathbb{N}$, $W$ such that the following hold.

  1. (1)

$\tilde{m_{n}}\to\tilde{m}\text{ in }L^{4}(0,T:L^{4})\cap
C(0,T:(H^{1})^{\prime}),\ \tilde{\mathbb{P}}-a.s.$ (4.32)

  2. (2)

$\tilde{W}_{n}\to\tilde{W}\text{ in }C([0,T]:\mathbb{R}),\
\tilde{\mathbb{P}}-a.s.$ (4.33)

  3. (3)

$\tilde{\lambda}_{n}\to\tilde{\lambda}\text{ stably in
}\mathcal{Y}(0,T:\mathbb{U}),\ \tilde{\mathbb{P}}-a.s.$ (4.34)

  4. (4)

The processes $\left(\tilde{m}_{n},\tilde{\lambda}_{n},\tilde{W}_{n}\right)$
have the same laws (on the space $L^{4}(0,T:L^{4})\cap
C(0,T:(H^{1})^{\prime})\times\mathcal{Y}(0,T:\mathbb{U})\times
C([0,T]:\mathbb{R})$) as the processes $\left(m_{n},\lambda_{n},W_{n}\right)$.

###### Proof of Lemma 4.5.

The lemma from the Skorohod Theorem, see for instance Theorem 4.30 in [36]. ∎

Appealing to the Kuratowski Theorem, see Theorem 1.1 in [55], (see also [44]),
we can show that the processes $\tilde{m}_{n}$ and the corresponding $m_{n}$
satisfy the same bounds, especially the bounds given in Lemma 4.1, Lemma 4.2
and Lemma 4.3. This is made formal in the following lemma.

######  Lemma 4.6.

There exists a constant $C>0$ such that the following holds for every
$n\in\mathbb{N}$.

$\sup_{t\in[0,T]}\left|\tilde{m}_{n}(t)\right|_{L^{2}}^{2}\leq C.$ (4.35)
$\tilde{\mathbb{E}}\sup_{t\in[0,T]}\left|\tilde{m}_{n}(t)\right|_{H^{1}}^{2p}\leq
C,$ (4.36)
$\tilde{\mathbb{E}}\int_{0}^{T}\left|\tilde{m}_{n}(t)\times\Delta\tilde{m}_{n}(t)\right|_{L^{2}}^{2p}\,dt\leq
C.$ (4.37)

The process $\tilde{m}$ also satisfies the same bounds. We write the bounds in
the following lemma.

######  Lemma 4.7.

There exists a constant $C>0$ such that the following holds.

$\sup_{t\in[0,T]}\left|\tilde{m}(t)\right|_{L^{2}}^{2}\leq C.$ (4.38)
$\tilde{\mathbb{E}}\sup_{t\in[0,T]}\left|\tilde{m}(t)\right|_{H^{1}}^{2p}\leq
C,$ (4.39)
$\tilde{\mathbb{E}}\int_{0}^{T}\left|\tilde{m}(t)\times\Delta\tilde{m}(t)\right|_{L^{2}}^{2p}\,dt\leq
C.$ (4.40)

###  4.3. Convergence Arguments

From Lemma 4.5, we recall the following convergence.

$\displaystyle\tilde{m}_{n}\to\tilde{m}\text{ in }L^{4}(0,T:L^{4})\cap
C(0,T:(H^{1})^{\prime}),\ \tilde{\mathbb{P}}-a.s.$ (4.41)

By the bounds in Lemma 4.6 , we conclude that the sequence $m_{n}$ is
uniformly integrable in the space $L^{4}(\Omega:L^{4}(0,T:L^{4}))$. Hence, we
have the following convergence.

$\displaystyle\tilde{m}_{n}\to\tilde{m}\text{ in
}L^{4}(\Omega:L^{4}(0,T:L^{4})).$ (4.42)

Note: Henceforth in the section, we suppress the notation $\tilde{\ }$ for the
sake of simplicity of writing. That is, we replace $\tilde{m}_{n}$ by $m_{n}$,
$\tilde{\lambda}_{n}$ by $\lambda_{n}$ and so on. The resulting processes are
not to be confused with the processes from the previous sections.

######  Lemma 4.8.

The following convergences hold.

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m_{n}\times
L(m_{n},v),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times L(m,v),\phi\right\rangle_{L^{2}}\,\lambda(dv,ds)\right|=0.$ (4.43)
$\displaystyle\lim_{n\to\infty}\mathbb{E}\bigg{|}$
$\displaystyle\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m_{n}\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{U}}\left\langle m\times\left(m\times
L(m,v)\right),\phi\right\rangle_{L^{2}}\,\lambda(dv,ds)\bigg{|}=0.$ (4.44)

###### Proof.

First, we observe that

$\displaystyle\int_{0}^{T}$ $\displaystyle\int_{\mathbb{U}}\left\langle
m_{n}\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left(m\times L(m,v)\right),\phi\right\rangle_{L^{2}}\,\lambda(dv,ds)$
$\displaystyle=$
$\displaystyle\bigg{(}\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m_{n}\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)\bigg{)}$
$\displaystyle+\bigg{(}\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{U}}\left\langle m\times\left(m\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)\bigg{)}$
$\displaystyle+\bigg{(}\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left(m\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{U}}\left\langle m\times\left(m\times
L(m,v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)\bigg{)}$
$\displaystyle+\bigg{(}\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left(m\times
L(m,v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{U}}\left\langle m\times\left(m\times
L(m,v)\right),\phi\right\rangle_{L^{2}}\,\lambda(dv,ds)\bigg{)}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{4}S_{i}.$ (4.45)

We calculate the terms separately. We show calculations first for $S_{1}$. The
term $S_{2}$ can be handled similarly.
Calculations for $S_{1}$:

$\displaystyle\mathbb{E}\left|S_{1}\right|$
$\displaystyle=\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}\left\langle\left(m_{n}-m\right)\times\left(m_{n}\times
L(m_{n},v)\right),\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)\right|$
$\displaystyle\leq\int_{0}^{T}\int_{\mathbb{U}}\left|m_{n}-m\right|_{L^{4}}\left|m_{n}\right|_{L^{4}}\left|L(m_{n},v)\right|_{L^{2}}\left|\phi\right|_{L^{\infty}}\,\lambda_{n}(dv,ds)$
$\displaystyle\leq\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\left|m_{n}\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}\centerdot$
$\displaystyle\qquad\centerdot\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\left|L(m_{n},v)\right|_{L^{2}}\,\lambda_{n}(dv,ds)\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\left|\phi\right|_{L^{\infty}}^{4}\,dt\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\left|m_{n}(s)\right|_{L^{2}}^{4r}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}\left(\sup_{s\in[0,T]}\left|m_{n}(s)\right|_{L^{2}}^{4r}\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,dt\right)^{\frac{1}{4}}.$
(4.46)

Note that the dot ($\centerdot$) in the third step is used to denote the
product of terms on the left and right. The right hand side of the above
inequality goes to $0$ as $n\to\infty$. The calculations for $S_{2}$ are
similar. Note that we have used the coercivity assumption (3.14), along with
the inequality (4.3) to conclude that

$\sup_{n\in\mathbb{N}}\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)<\infty.$

Calculations for $S_{3}$:

$\displaystyle\mathbb{E}(S_{3})=$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\left\langle
m\times\left[m\times\left(L(m_{n},v)-L(m,v)\right)\right],\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds).$
(4.47)

Let us define a cut-off function $\Phi_{R}:\mathbb{U}\to[0,1]$ as follows.

$\displaystyle\Phi_{R}(v)=\begin{cases}1,\ \text{if }\kappa(\cdot,v)\leq R,\\\
0,\ \text{if }\kappa(\cdot,v)\geq 2R.\end{cases}$ (4.48)

Further, let us assign a temporary notation as follows. Let

$F(t,v)=\left\langle
m(s)\times\left[m(s)\times\left(L(m_{n}(s),v)-L(m(s),v)\right)\right],\phi\right\rangle_{L^{2}}.$
(4.49)

Therefore,

$\displaystyle\mathbb{E}(S_{3})=$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}F(s,v)\,\lambda_{n}(dv,ds)$
$\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)F(s,v)\,\lambda_{n}(dv,ds)+\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}(1-\Phi(v))F(s,v)\,\lambda_{n}(dv,ds)$
$\displaystyle=$ $\displaystyle S_{3,1}+S_{3,2}.$ (4.50)

Calculations for $S_{3,1}$:

$\displaystyle\left|S_{3,1}\right|=$
$\displaystyle\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}\Phi_{R}(v)\left\langle
m\times\left[m\times\left(L(m_{n},v)-L(m,v)\right)\right],\phi\right\rangle_{L^{2}}\,\lambda_{n}(dv,ds)\right|$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi_{R}(v)\left|m\right|_{L^{4}}\left|m\right|_{L^{4}}\left|L(m_{n},v)-L(m,v)\right|_{L^{2}}\left|\phi\right|_{L^{\infty}}\,\lambda_{n}(dv,ds)$
$\displaystyle\leq$
$\displaystyle\left(\mathbb{E}\int_{0}^{T}\left|m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{2}}\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi_{R}(v)\left|L(m_{n},v)-L(m,v)\right|_{L^{2}}^{2}\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}\centerdot$
$\displaystyle\qquad\centerdot\left(\mathbb{E}\int_{0}^{T}\left|\phi\right|_{L^{\infty}}\,ds\right)^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi_{R}(v)\left|L(m_{n},v)-L(m,v)\right|_{L^{2}}^{2}\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\sup_{v\in\mathbb{U}}\Phi_{R}(v)\left|L(m_{n},v)-L(m,v)\right|_{L^{2}}^{2}\,ds\right)^{\frac{1}{2}}.$
(4.51)

By (3) in Assumption 3.4, for every $R>0$, the right hand side of the above
inequality converges to $0$ as $n\to\infty$.
Calculations for $S_{3,2}$:

$\displaystyle\left|S_{3,2}\right|\leq$
$\displaystyle\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}(1-\Phi(v))F(s,v)\,\lambda_{n}(dv,ds)\right|$
$\displaystyle=$
$\displaystyle\mathbb{E}\left|\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}F(s,v)\,\lambda_{n}(dv,ds)\right|$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\left|\left\langle
m\times\left[m\times\left(L(m_{n},v)-L(m,v)\right)\right],\phi\right\rangle_{L^{2}}\right|\,\lambda_{n}(dv,ds)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\left|m\right|_{L^{4}}\left|m\right|_{L^{4}}2\left|L(m_{n},v)+L(m,v)\right|_{L^{2}}\left|\phi\right|_{L^{\infty}}\,\lambda_{n}(dv,ds)$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi_{R}(v)\left|m\right|_{L^{4}}\left(\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\left[\left|m_{n}\right|_{L^{2}}^{4r}+\left|m\right|_{L^{2}}^{4r}\right]\kappa(s,v)^{4}\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}\centerdot$
$\displaystyle\qquad\centerdot\left(\mathbb{E}\int_{0}^{T}\left|\phi\right|_{L^{\infty}}\,ds\right)^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\kappa^{4}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\int_{\left\\{v\in\mathbb{U}:\kappa(\cdot,v)\geq
R\right\\}}\kappa^{5}(s,v)\cdot\kappa^{-1}(s,v)\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{C}{R^{\frac{1}{4}}}\left(\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\kappa^{5}(s,v)\cdot\,\lambda_{n}(dv,ds)\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\frac{C}{R}.$ (4.52)

The right hand side of the above inequality converges to $0$ as $R\to\infty$.
Calculations for $S_{4}$:
As before, let us fix the following notation.

$\hat{F}(s,v)=\left\langle m(s)\times\left(m(s)\times
L(m(s),v)\right),\phi\right\rangle_{L^{2}}.$ (4.53)

Therefore,

$\displaystyle\mathbb{E}\left(S_{4}\right)=$
$\displaystyle\mathbb{E}\left(\int_{0}^{T}\int_{\mathbb{U}}\hat{F}(s,v)\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\hat{F}(s,v)\,\lambda(dv,ds)\right).$
(4.54)

Recall the cut-off function $\Phi$ defined previously in (4.48). Observe that

$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\hat{F}(s,v)\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\hat{F}(s,v)\,\lambda(dv,ds)$
$\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda(dv,ds)$
$\displaystyle+\mathbb{E}\int_{0}^{T}\int_{\mathbb{U}}\left(1-\Phi(v)\right)\hat{F}(s,v)\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\left(1-\Phi(v)\right)\hat{F}(s,v)\,\lambda(dv,ds).$
(4.55)

Claim:

$\displaystyle\Phi(\cdot)\hat{F}\in L^{1}(0,T:C_{b}(\mathbb{U}).$ (4.56)

That is, we want to show that

$\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|\Phi(v)\hat{F}(s,v)\right|\,ds<\infty.$
(4.57)

Towards this aim, we have the following sequence of inequalities.

$\displaystyle\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|\Phi(v)\hat{F}(s,v)\right|\,ds=$
$\displaystyle\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|\left\langle
m\times\left(m\times L(m,v)\right),\phi\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|m\right|_{L^{4}}^{2}\left|L(m,v)\right|_{L^{2}}\left|\phi\right|_{L^{\infty}}\,ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|m\right|_{L^{4}}^{2}\left|m\right|_{L^{2}}^{r}\kappa(s,v)\left|\phi\right|_{L^{\infty}}\,ds$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{0}^{T}\sup_{v\in\mathbb{U}}\left|m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{2}}\left(\int_{0}^{T}\sup_{v\in\mathbb{U}}\kappa^{2}(s,v)\,ds\right)^{\frac{1}{2}}\left|\phi\right|_{L^{\infty}}\,ds.$
(4.58)

The right-hand side of the above inequality is finite almost surely. This
concludes the proof of the claim. Therefore by Lemma 2.9, the following
convergence holds for almost all $t\in[0,T]$.

$\displaystyle\int_{0}^{t}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda_{n}(dv,ds)\
\text{converges
to}\int_{0}^{t}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda(dv,ds)\ \text{as
}n\to\infty.$ (4.59)

Moreover,

$\displaystyle\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda_{n}(dv,ds)\right|^{\frac{4}{3}}\leq$
$\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\left|m\right|_{L^{4}}^{4}\,ds\right)^{\frac{2}{3}}\left(\mathbb{E}\left|\phi\right|_{L^{\infty}}^{4}\right)^{\frac{2}{3}}.$
(4.60)

Therefore by the Vitali Convergence Theorem (see Theorem 4.5.4 in [5]), we
have

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda_{n}(dv,ds)-\int_{0}^{T}\int_{\mathbb{U}}\Phi(v)\hat{F}(s,v)\,\lambda(dv,ds)\right|=0.$
(4.61)

The second part of $S_{4}$ can be handled similarly to the term $S_{3,2}$.
This concludes the proof of the lemma. ∎

######  Lemma 4.9.

The following convergences hold. Let $\phi\in L^{4}(\Omega:L^{4}(0,T:L^{4}))$.
Then

$\displaystyle\lim_{n\to\infty}\mathbb{E}\int_{0}^{T}\left\langle\left(m_{n}\times\Delta
m_{n}\right),\phi\right\rangle_{L^{2}}\,ds=\mathbb{E}\int_{0}^{T}\left\langle\left(m\times\Delta
m\right),\phi\right\rangle_{L^{2}}\,ds.$ (4.62)
$\displaystyle\lim_{n\to\infty}\mathbb{E}\int_{0}^{T}\left\langle
m_{n}\times\left(m_{n}\times\Delta
m_{n}\right),\phi\right\rangle_{L^{2}}\,ds=\mathbb{E}\int_{0}^{T}\left\langle
m\times\left(m\times\Delta m\right),\phi\right\rangle_{L^{2}}\,ds.$ (4.63)

###### Proof.

We skip the proof and refer the reader to Section 4 in [8]. ∎

######  Lemma 4.10.

Let $\phi\in L^{2}(\Omega:L^{2}(0,T:W^{1,4}))$. Then

$\displaystyle\lim_{n\to\infty}\mathbb{E}\int_{0}^{T}\left\langle
G(m_{n})-G(m),\phi\right\rangle_{L^{2}}\,ds=0.$ (4.64)

###### Proof.

We give a short idea of the proof. We have the following sequence of
inequalities.

$\displaystyle\left|\mathbb{E}\int_{0}^{T}\left\langle
G(m_{n})-G(m),\phi\right\rangle_{L^{2}}\,ds\right|$ $\displaystyle=$
$\displaystyle\left|\mathbb{E}\int_{0}^{T}\left\langle m_{n}-m,\phi\times
h\right\rangle_{L^{2}}\,ds\right|$ $\displaystyle\leq$
$\displaystyle\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\left|\phi\right|_{L^{2}}^{2}\,ds\right)^{\frac{1}{2}}\left(\mathbb{E}\int_{0}^{T}\left|h\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}.$
(4.65)

The right hand side of the above inequality converges to $0$ as $n\to\infty$
due to the convergence (4.42). ∎

######  Lemma 4.11.

The process $m$ satisfies the equality in (3), in the sense given in
Definition 3.1.

###### Proof.

For a similar result, we refer the reader to Section 5 in [8]. ∎

###  4.4. Verification of the constraint condition

So far, we have shown that the process $m$ satisfies the equality (3) and
satisfies the required bounds. We are yet to show that the constraint
condition is satisfied. Towards that, we consider the function
$v\mapsto\frac{1}{2}\left|v\right|_{L^{2}}^{2}$, and apply the Itô formula
from the work of Pardoux [51]. That all the necessary conditions (see Theorem
1.2 in [51]) are satisfied can be verified using the bounds in Lemma 4.7.
Applying the Itô formula, we get the following equality. Let $\phi\in
C_{c}^{\infty}(\mathscr{O})$. Consider the function
$v\mapsto\frac{1}{2}\left\langle\phi v,v\right\rangle_{L^{2}}$.

$\displaystyle\frac{1}{2}\left\langle m(t),\phi m(t)\right\rangle_{L^{2}}=$
$\displaystyle\frac{1}{2}\left\langle m_{0},\phi m_{0}\right\rangle_{L^{2}}$
$\displaystyle+\int_{0}^{t}\left\langle\big{[}m(t)\times\Delta m(s)-\alpha
m(s)\times\left(m(s)\times\Delta m(s)\right)\big{]},\phi
m(t)\right\rangle_{L^{2}}\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{U}}\bigg{\langle}\big{[}m(s)\times
L(m(s),v)$ $\displaystyle\qquad-\alpha m(s)\times\left(m(s)\times
L(m(s),v)\right)\big{]},\phi m(s)\bigg{\rangle}_{L^{2}}\,\lambda(dv,ds)$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\left|\left(m(s)\times
h\right)\right|_{L^{2}}^{2}\,ds$
$\displaystyle+\int_{0}^{t}\left\langle\left[m_{n}(s)\times
h-\alpha\left(m_{n}(s)\times h\right)\times h\right],\phi
m(s)\right\rangle_{L^{2}}\,dW_{n}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\langle m_{0},\phi
m_{0}\right\rangle_{L^{2}}+\sum_{i=1}^{4}J_{i}(t).$ (4.66)

We now observe that

$\displaystyle J_{i}(t)=0.$ (4.67)

Therefore, the resulting equality is

$\displaystyle\left\langle m(t),\phi m(t)\right\rangle_{L^{2}}=\left\langle
m_{0},\phi m_{0}\right\rangle_{L^{2}}.$ (4.68)

That is,

$\displaystyle\int_{\mathscr{O}}\left\langle
m_{n}(t,x),\phi(x)m_{n}(t,x)\right\rangle_{\mathbb{R}^{3}}\,dx=\int_{\mathscr{O}}\left\langle
m_{0}(x),\phi(x)m_{0}(x)\right\rangle_{\mathbb{R}^{3}}\,dx.$ (4.69)

Therefore,

$\displaystyle\left|m(t,x)\right|_{\mathbb{{R}}^{3}}^{2}=\left|m_{0}(t,x)\right|_{\mathbb{{R}}^{3}}^{2}=1,\
Leb.\ a.a.\ x\in\mathscr{O}.$ (4.70)

This completes the verification of the constraint condition.

###  4.5. Minimizing the cost

We now come back to the notation $\tilde{\pi}$ for denoting the elements
obtained after using the Skorohod Theorem (see Lemma 4.5. In the previous
sections, we showed that the tuple $\tilde{\pi}$ is a weak martingale solution
to the problem (1). We now show that this obtained solution also minimizes the
cost (1.7). The aim of this section is therefore, to show that

$\displaystyle\mathscr{J}(\tilde{\pi})=\Lambda.$ (4.71)

By Lemma 4.5, we know that the processes $(m_{n},\lambda_{n})$ and
$(\tilde{m}_{n},\tilde{\lambda}_{n})$ have the same laws on the space
$L^{4}(0,T:L^{4})\cap
C([0,T]:(H^{1})^{\prime})\times\mathcal{Y}(0,T:\mathbb{U})$. Therefore

$\displaystyle\liminf_{n\to\infty}\mathbb{E}^{n}\int_{0}^{T}\int_{\mathbb{U}}F(m_{n},v)\lambda_{n}(dv,ds)\geq\liminf_{n\to\infty}\tilde{\mathbb{E}}^{n}\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m}_{n},v)\tilde{\lambda}_{n}(dv,ds).$
(4.72)

We now recall the assumption that the cost (running) $F$ is lower
semicontinuous. We have that

$\displaystyle m_{n}\to m\text{ in }L^{2}(0,T:L^{2})\ \text{and}\
\lambda_{n}\to\lambda\ \text{stably in}\ \mathcal{Y}(0,T:\mathbb{U}).$ (4.73)

Combining the lower semicontinuity of $F$ with Fatou’s Lemma, we have the
following inequality.

$\displaystyle\int_{0}^{T}\int_{\mathbb{U}}F(m,v)\lambda(dv,ds)\leq\liminf_{n\to\infty}\int_{0}^{T}\int_{\mathbb{U}}F(m_{n},v)\lambda_{n}(dv,ds).$
(4.74)

Hence

$\displaystyle\tilde{\mathbb{E}}\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m},v)\lambda(dv,ds)\leq\liminf_{n\to\infty}\tilde{\mathbb{E}}\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m}_{n},v)\tilde{\lambda}_{n}(dv,ds).$
(4.75)

Therefore,

$\displaystyle\tilde{\mathbb{E}}\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m},v)\tilde{\lambda}(dv,ds)$
$\displaystyle\leq\liminf_{n\to\infty}\tilde{\mathbb{E}}\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m}_{n},v)\tilde{\lambda}_{n}(dv,ds)$
$\displaystyle\leq\liminf_{n\to\infty}\mathbb{E}^{n}\int_{0}^{T}\int_{\mathbb{U}}F(m_{n},v)\lambda_{n}(dv,ds).$
(4.76)

Recall from Assumption 3.4 that we have assumed $\Psi$ to be continuous and
$\tilde{m}_{n}\to\tilde{m}$ in $C(0,T:(H^{1})^{\prime}$. Therefore

$\displaystyle\tilde{\mathbb{E}}\Psi(\tilde{m}(T))\leq\liminf_{n\to\infty}\tilde{\mathbb{E}}\Psi(\tilde{m}_{n}(T)).$
(4.77)

Since $\pi_{n}$ is a minimizing sequence and $\Lambda$ is the infimum, we have

$\displaystyle\Lambda\leq\tilde{\mathbb{E}}\left(\int_{0}^{T}\int_{\mathbb{U}}F(\tilde{m},v)\tilde{\lambda}(dv,ds)+\Psi(\tilde{m}_{n}(T))\right)\leq\Lambda.$
(4.78)

Hence the infumum is attained at $\tilde{\pi}$.

Acknowledgement: The author would like to thank Prof. Utpal Manna, Indian
Institute of Science Education and Research, Thiruvananthapuram, for helpful
discussions and insights.

##  Appendix A Result with full noise

We begin with a remark.

######  Remark A.1.

For the previous part, we stated that with $L=0$, we at least have the
existence of a weak martingale solution. For the problem (3) with full noise,
we cannot say so for $d=2,3$ since the existence is not yet known, to the best
of our knowledge. It is known that the problem admits a strong solution for
$d=1$, see [9, 20, 12]. Therefore, we cannot simply say that $L=0$ admits a
solution. We have considered a very general cost functional in (1.5) (and
hence also (1.7)). As before, we take the liberty here to assume that there
exists at least one solution with finite cost.

We now state the equation that has been considered.

$\displaystyle dm_{n}(t)=$ $\displaystyle\big{[}m_{n}(t)\times\Delta
m_{n}(t)-\alpha m_{n}(t)\times\left(m_{n}(t)\times\Delta
m_{n}(t)\right)\big{]}\,dt$
$\displaystyle+\int_{\mathbb{U}}\big{[}m_{n}(t)\times
L(m_{n},v)-\alpha\,m_{n}(t)\times\left(m_{n}(t)\times
L(m_{n},v)\right)\,\lambda_{n}(dv,dt)\big{]}$
$\displaystyle+\frac{1}{2}DG\left(m_{n}(t)\right)\bigl{(}G\left(m_{n}(t)\right)\bigr{)}\,dt+G\left(m_{n}(t)\right)\,dW_{n}(t).$
(A.1)

with $G(m_{n})=m_{n}\times h-\alpha\,m_{n}\times\left(m_{n}\times h\right)$
and

$\displaystyle DG_{n}\big{(}m_{n}\big{)}\bigl{(}G_{n}(m_{n})\bigr{)}=$
$\displaystyle(m_{n}\times h)\times
h-\alpha\,\left(\big{(}m_{n}\times(m_{n}\times h)\big{)}\times h\right)$
$\displaystyle-\alpha\,\left(m_{n}\times h\right)\times(m_{n}\times h))$
$\displaystyle+m_{n}\times\big{(}(m_{n}\times h)\times h\big{)}$
$\displaystyle-\alpha\,\big{(}m_{n}\times(m_{n}(s)\times
h)\big{)}\times(m_{n}\times h)$
$\displaystyle-\alpha\,\bigg{(}m_{n}\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{6}K_{i}.$ (A.2)

######  Theorem A.2.

Let $d=1,2,3$. Let Assumption 3.4 hold with $\alpha\neq 0$. Then the problem
(1), with the cost (1.7), admits an optimal control.

###### Idea of the proof of Theorem A.2.

We do not give all the proof arguments here. We give the arguments for the
additional correction term that arises due to the non-linear part of the
noise. We treat the proof in the sense that we are adding an extra term
$DG_{n}\big{(}m_{n}\big{)}\bigl{(}G_{n}(m_{n})\bigr{)}$ to the calculations.
We therefore discuss only about the calculations corresponding to the said
term. The rest of the section is dedicated to this discussion. ∎

######  Remark A.3.

We state here some observations about the correction term (A)

  1. (1)

The term $K_{1}$ is linear in $m_{n}$.

  2. (2)

The terms $K_{i},i=2,3,4$ are quadratic (in $m_{n}$).

  3. (3)

The terms $K_{i},i=5,6$ are of third order (in $m_{n}$).

###  A.1. Some calculations for the Correction term:

In this subsection, we show some calculations for proving uniform estimates,
assuming that the full noise term has been considered. The following lemma is
an analogue of Lemma 4.1.

######  Lemma A.4.

There exists a constant $C>0$ such that the following holds for every
$n\in\mathbb{N}$.

$\left|m_{n}\right|_{\mathbb{R}^{3}}=1,$ (A.3)
$\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{L^{2}}^{2}\leq C,$ (A.4)
$\mathbb{E}^{n}\sup_{t\in[0,T]}\left|m_{n}(t)\right|_{H^{1}}^{2}\leq C,$ (A.5)
$\mathbb{E}^{n}\int_{0}^{T}\left|m_{n}(t)\times\Delta
m_{n}(t)\right|_{L^{2}}^{2}\,dt\leq C.$ (A.6)

###### Proof.

The proof is in line with the proof of Lemma 4.1. We do not give a complete
proof of the result here. Instead, we give the arguments that if added to the
proof of Lemma 4.1, will complete the result. That is, we only show some
calculations related to the correction term ($DG(m_{n})(G(m_{n}))$). The
constraint condition holds because each $m_{n}$ is a part of a weak martingale
solution to the problem (1). The inequality (A.4) can be shown again, by using
the constraint condition. For the proof of inequality (4.6), we recall the
equality (4.1). We will, as done in the proof of (4.36), apply the Itô formula
to the function

$v\mapsto\frac{1}{2}\left|\nabla v\right|_{L^{2}}^{2}.$

The resulting equation will be as follows.

$\displaystyle\frac{1}{2}\left|\nabla m_{n}(t)\right|_{L^{2}}^{2}=$
$\displaystyle\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+\int_{0}^{t}\big{\langle}\big{[}m_{n}(t)\times\Delta
m_{n}(s)$ $\displaystyle\qquad-\alpha m_{n}(s)\times\left(m_{n}(s)\times\Delta
m_{n}(s)\right)\big{]},-\Delta m_{n}(t)\big{\rangle}_{L^{2}}\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{U}}\bigg{\langle}\big{[}m_{n}(s)\times
L(m_{n}(s),v)$ $\displaystyle\qquad-\alpha m_{n}(s)\times\left(m_{n}(s)\times
L(m_{n}(s),v)\right)\big{]},-\Delta
m_{n}(s)\bigg{\rangle}_{L^{2}}\,\lambda(dv,ds)$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\left|\nabla
G\left(m_{n}(s)\right)\right|_{L^{2}}^{2}\,ds+\frac{1}{2}\int_{0}^{t}\left\langle
DG\left(m_{n}(s)\right)\left[G\left(m_{n}(s)\right)\right],-\Delta
m_{n}(s)\right\rangle_{L^{2}}\,ds$ $\displaystyle+\int_{0}^{t}\left\langle
G\left(m_{n}(s)\right),-\Delta m_{n}(s)\right\rangle_{L^{2}}\,dW_{n}(s)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left|\nabla
m_{0}\right|_{L^{2}}^{2}+\sum_{i=1}^{5}I_{i}(t).$ (A.7)

Since the constraint condition holds for each $m_{n}$, we do not show all the
calculations for the uniform energy bounds. We just brief the idea, which is
as follows. The calculations for $I_{1},I_{2}$ are the same as the
corresponding integrals ($I_{1,1},I_{2}$) in the proof of Lemma 4.1. We now
show the calculations for the term $I_{4}$. The crux of the rest of the proof
stays the same.
Calculations for $I_{4}$:
Recall the term $DG(m_{n})(G(m_{n}))$ from (A). We do not show the
calculations for all the individual terms on the right hand side of (A). We
show detailed calculations for one of the terms $K_{6}$. The others can be
done in a similar spirit.

$\displaystyle\left|\int_{0}^{t}K_{6}(s)\,ds\right|\leq$
$\displaystyle\left|\alpha\right|\int_{0}^{t}\left|\left\langle\left(m_{n}\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\right),\Delta m_{n}\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle=$
$\displaystyle\left|\alpha\right|\int_{0}^{t}\left|\left\langle\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)},m_{n}\times\Delta
m_{n}\right\rangle_{L^{2}}\right|\,ds$ $\displaystyle\leq$
$\displaystyle\left|\alpha\right|\int_{0}^{t}\left|\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\right|_{L^{2}}\left|m_{n}\times\Delta
m_{n}\right|_{L^{2}}\,ds$ $\displaystyle\leq$
$\displaystyle\left|\alpha\right|\int_{0}^{t}\left|m_{n}\right|_{L^{\infty}}^{2}\left|h\right|_{L^{\infty}}^{2}\left|m_{n}\times\Delta
m_{n}\right|_{L^{2}}\,ds$ $\displaystyle\leq$
$\displaystyle\frac{2\left|\alpha\right|^{2}}{\varepsilon}\int_{0}^{t}\left|m_{n}\right|_{L^{\infty}}^{4}\left|h\right|_{L^{\infty}}^{4}\,ds+\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}\times\Delta
m_{n}\right|_{L^{2}}^{2}\,ds$ $\displaystyle\leq$ $\displaystyle
C(\varepsilon,h)+\frac{\varepsilon}{2}\int_{0}^{t}\left|m_{n}\times\Delta
m_{n}\right|_{L^{2}}^{2}\,ds.$ (A.8)

We recall here the equations (4.1)-(4.1). The choice of $\varepsilon$ is made
in the same spirit as done there. The rest of the proof is the same as that of
Lemma 4.1. ∎

###  A.2. Calculations for convergence:

Continuing in the format of Section 4.3, we have the following lemma. Note
that all the results from the previous section will continue to hold. That is,
with the calculations in Section A.1, the bounds established in Lemmas 4.2,
4.3 will hold. Therefore, following Section 4.2, we can use the Skorohod
Theorem to obtain a pointwise convergent sequence of processes (possibly on a
different probability space) with the same laws as those of the given
processes. We again use $m_{n}$ to denote the newly obtained processes. All we
give here are some of the calculations and results corresponding to the
correction term $DG(m_{n})\left(G(m_{n})\right)$.

###### A general idea of the calculations:.

We state some general idea. The terms in the correction term are polynomials
in $m_{n}$ (see Remark A.3). We add and subtract suitable terms, and then use
the Hölder inequality in order to use the previously established bounds and
convergences. ∎

######  Lemma A.5.

Let $\phi\in L^{4}(\Omega:L^{4}(0,T:W^{1,4}))$. We have the following
convergence.

$\displaystyle\lim_{n\to\infty}\mathbb{E}\int_{0}^{t}\left\langle
DG(m_{n})(G(m_{n})),\phi\right\rangle_{L^{2}}\,ds=\mathbb{E}\int_{0}^{t}\left\langle
DG(m)(G(m)),\phi\right\rangle_{L^{2}}\,ds.$ (A.9)

###### Proof.

We again show calculations for one of the terms from (A) for the convergence
arguments.

$\displaystyle\bigg{(}m_{n}\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}-\bigg{(}m\times\big{(}(m\times\big{(}m\times
h)\big{)}\times h\big{)}\bigg{)}$ $\displaystyle=$
$\displaystyle\bigg{(}m_{n}\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times
h\big{)}\bigg{)}-\bigg{(}m\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}$
$\displaystyle+\bigg{(}m\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times
h\big{)}\bigg{)}-\bigg{(}m\times\big{(}(m\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}$
$\displaystyle+\bigg{(}m\times\big{(}(m\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}-\bigg{(}m\times\big{(}(m\times\big{(}m\times
h)\big{)}\times h\big{)}\bigg{)}$ $\displaystyle=$
$\displaystyle\left(m_{n}-m\right)\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}$
$\displaystyle+m\times\big{(}\big{(}m_{n}-m\big{)}\times\big{(}m_{n}\times
h)\times h\big{)}$
$\displaystyle+m\times\big{(}(m\times\big{(}\left(m_{n}-m\right)\times
h)\big{)}\times h\big{)}.$ (A.10)

Therefore,

$\displaystyle\mathbb{E}\int_{0}^{T}\left|\left\langle\bigg{(}m_{n}\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)}\bigg{)}-\bigg{(}m\times\big{(}(m\times\big{(}m\times
h)\big{)}\times h\big{)}\bigg{)},\phi\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{T}\left|\left\langle\left(m_{n}-m\right)\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)},\phi\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle+\mathbb{E}\int_{0}^{T}\left|\left\langle
m\times\big{(}\big{(}m_{n}-m\big{)}\times\big{(}m_{n}\times h)\times
h\big{)},\phi\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle+\mathbb{E}\int_{0}^{T}\left|\left\langle
m\times\big{(}(m\times\big{(}\left(m_{n}-m\right)\times h)\big{)}\times
h\big{)},\phi\right\rangle_{L^{2}}\right|\,ds$ $\displaystyle=$ $\displaystyle
I_{1}+I_{2}+I_{3}.$ (A.11)

We show calculations for one term. Others can be done similarly.

$\displaystyle\mathbb{E}\int_{0}^{T}\left|\left\langle\left(m_{n}-m\right)\times\big{(}(m_{n}\times\big{(}m_{n}\times
h)\big{)}\times h\big{)},\phi\right\rangle_{L^{2}}\right|\,ds$
$\displaystyle\leq$
$\displaystyle\left|h\right|_{L^{\infty}}\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}\left(\mathbb{E}\int_{0}^{T}\left|m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{2}}\left(\mathbb{E}\int_{0}^{T}\left|\phi\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle
C\left(\mathbb{E}\int_{0}^{T}\left|m_{n}-m\right|_{L^{4}}^{4}\,ds\right)^{\frac{1}{4}}.$
(A.12)

The right-hand side of the above inequality converges to $0$ as $n\to\infty$.
For the remaining convergence arguments, we refer the reader to Section 5 in
[8], see also [9], the preprint [12]. Even though the second and the third
stated references consider the equation only in dimension $d=1$, the crux of
the argument can be applied to this case as well. ∎

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