dcu Shinji Imahori Faculty of Science and Engineering, Chuo University
1-13-27 Kasuga, Bunkyo, Tokyo 112-8551, Japan
<EMAIL_ADDRESS>

#  A $1+O(1/n)$ APPROXIMATION ALGORITHM FOR TTP(2)

###### Abstract

The traveling tournament problem is a well-known benchmark problem of the
sports scheduling. We propose an approximation algorithm for the traveling
tournament problem with the constraints such that both the number of
consecutive home games and that of consecutive away games are at most two
(called TTP(2)). The approximation ratio of the proposed algorithm is $1+24/n$
for $n$ teams, which is the first 1 + $O(1/n)$ approximation algorithm for
TTP(2).

Keywords: Sports scheduling, Timetable, Traveling tournament problem,
Approximation algorithm.

##  1 INTRODUCTION

Sports scheduling is an attractive research area in these decades
[deWerra2006, Trick2011]. In the field of sports scheduling, the traveling
tournament problem (TTP) is a well-known benchmark problem proposed by
Easton2001. Various approximation algorithms were proposed for the problem in
the last decade [Hoshino2012, Imahori2014, Thielen2012, Yamaguchi2011]. In the
following, some terminology and the problem are introduced.

Given a set $T$ of $n$ teams, where $n\geq 4$ and is even, a game is specified
by an ordered pair of teams. Each team has its home venue. For any pair of
teams $i,j\in T$, $d_{ij}\geq 0$ denotes the distance between the home venues
of $i$ and $j$. Throughout the paper, we assume that triangle inequality
($d_{ij}+d_{jk}\geq d_{ik}$) and symmetry ($d_{ij}=d_{ji}$) hold.

A double round-robin tournament is a set of games in which every team plays
every other team once at its home venue (called home game) and once at the
home venue of the opponent (called away game). Consequently, $2(n-1)$ slots
are necessary to complete a double round-robin tournament with $n$ teams.

Each team stays at its home venue before a tournament and then travels to play
its games at the chosen venues. After a tournament, each team returns to its
home venue if the team plays an away game at the last slot. When a team plays
two consecutive away games, the team goes directly from the venue of the first
opponent to that of another opponent without returning to its home venue. The
traveling distance of a team is defined by the sum of distances $d_{ij}$ if
the team travels from the home venue of $i$ to the home venue of $j$.

The objective of TTP is to minimize the total traveling distance, which is the
sum of traveling distances of $n$ teams. Two types of constraints, called no-
repeater and at-most constraints, should be satisfied. The no-repeater
constraint is that, for any pair of teams $i$ and $j$, two games of $i$ and
$j$ cannot be held in two consecutive slots. The at-most constraint is that,
for a given parameter $k$, no team plays more than $k$ consecutive home games
and more than $k$ consecutive away games. The present paper considers the case
for $k=2$; the problem is called TTP(2), which is defined as follows.

Traveling Tournament Problem for $k=2$ (TTP(2))
Input: A set of teams $T$ and a distance matrix $D=(d_{ij})$.
Output: A double round-robin tournament such that

C1. No team plays more than two consecutive away games,

C2. No team plays more than two consecutive home games,

C3. The no-repeater constraint is satisfied for all teams,

C4. The total distance traveled by the teams is minimized.

For this problem, Thielen2012 proposed two types of approximation algorithms.
The first algorithm is a 1.5 + $O(1/n)$ approximation algorithm. The second
one is a 1 + $O(1/n)$ approximation algorithm, though it works only for the
case with $n=4m$ teams.

In this paper, we propose a 1 + $24/n$ approximation algorithm for the case
with $n=4m+2$ teams. With the algorithm by Thielen2012, we achieve an
approximation ratio 1 + $O(1/n)$ for TTP(2).

##  2 LOWER BOUNDS

In this section, we present the independent lower bound for TTP(2) obtained by
Campbell1976 and another lower bound for analyzing schedules generated by our
approximation algorithm. The basic idea of the independent lower bound is that
the optimal trips for a team can be obtained by computing a minimum weight
perfect matching $M$ in a complete undirected graph $G$ on the set of teams,
where the weight of the edge from team $i$ to $j$ given as the distance
$d_{ij}$ between the home venues of teams $i$ and $j$.

Let $s(i):=\sum_{j\neq i}d_{ij}$ be the sum of weights of the edges between
team $i$ and all the other teams $j$, and let $\Delta:=\sum_{i}s(i)$. Let
$d(M)$ be the weight of a minimum weight perfect matching $M$ in $G$. Then the
traveling distance of team $i$ is at least $s(i)+d(M)$, and the total
traveling distance is at least

$\sum_{i=1}^{n}\left(s(i)+d(M)\right)=\Delta+n\cdot d(M),$ (1)

which is called the independent lower bound. We note that Thielen2012 showed
that this lower bound cannot be reached in general.

We introduce another lower bound, which is weaker than the above independent
lower bound but is useful to analyze the solutions we generate in the next
section. Let $d(T)$ be the weight of a minimum spanning tree $T$ in $G$. Then,
for any team $i$, $d(T)\leq s(i)$ holds. Hence the total traveling distance
(i.e., another lower bound) is at least

$\sum_{i=1}^{n}\left(d(T)+d(M)\right)=n\cdot(d(T)+d(M)).$ (2)

It is known that Christofides algorithm [Christofides1976] for the traveling
salesman problem generates a Hamilton cycle $C$ in $G$, whose length is at
most $d(T)+d(M)$.

##  3 ALGORITHM

We propose an approximation algorithm for TTP(2) for the case with $n=4m+2$
teams. Our algorithm is similar to the 1 + $O(1/n)$ approximation algorithm by
Thielen2012 for $n=4m$ teams. A key concept of the algorithm is the use of a
minimum weight perfect matching, a Hamilton cycle computed by Christofides
algorithm, and the circle method to construct a single round-robin tournament.

We first compute a minimum weight perfect matching $M$ and a Hamilton cycle
$C$ in the graph $G$. By using Christofides algorithm, the length of the
Hamilton cycle $C$ is at most $d(M)+d(T)$, where $T$ is a minimum weight
spanning tree in $G$. The $n$ teams are assumed to be numbered such that the
edges $\langle 1,2\rangle,\langle 3,4\rangle,\ldots,\langle n-1,n\rangle$ form
the minimum weight perfect matching $M$ in $G$. Among possible numberings, we
choose a numbering with the following three properties.
(a) $s(n-5)+s(n-4)+\cdots+s(n)\leq 6\Delta/n$,
(b) $t(n-7)+t(n-6)\leq 12\Delta/\\{n(n-6)\\}$,
(c) teams $2,4,\ldots,n-8$ are appeared in the Hamilton
cycle $C$ in this order if the other teams are removed,
where $t(i):=d_{i,n-5}+d_{i,n-4}+\cdots+d_{in}$. We note that the existence of
a numbering with the above property (a) comes from an equation

$\Delta=\sum_{i=1}^{n}s(i)=\sum_{i=1}^{n/2}\\{s(2i-1)+s(2i)\\},$

in fact, we choose three pairs $\langle 2i-1,2i\rangle$ in ascending order of
the values $\\{s(2i-1)+s(2i)\\}$ for $\langle n-5,n-4\rangle$, $\langle
n-3,n-2\rangle$ and $\langle n-1,n\rangle$. We use the following equation for
existence of a numbering with property (b):

$\sum_{i=1}^{(n-6)/2}\\{t(2i-1)+t(2i)\\}=s(n-5)+\cdots+s(n)\leq 6\Delta/n.$

We then construct a double round-robin tournament. As the first phase, we
construct a schedule with $2n-16$ slots. These $2n-16$ slots are divided into
four slots, and we call each of them a block. Thus, there are $n/2-4$ blocks
in the first phase; block 1 contains slots 1, 2, 3, 4, block 2 contains slots
5, 6, 7, 8 and so on. As the second phase, we construct schedules of the last
14 slots. We use the mirroring technique in this phase; a schedule with seven
slots is constructed and the same schedule is copied by changing the venues.
Moreover, in this phase, we construct schedules for teams $1,2,\ldots,n-8$ and
teams $n-7,n-6,\ldots,n$ independently. Fig. 1 shows the whole image of the
schedule we construct.

Figure 1: Whole image of the schedule we construct

In the first phase, we apply a scheme inspired by the circle method, where a
similar idea was also used in Thielen2012. As an example, see Fig. 2 to Fig. 4
for a case with $n=30$ teams. As displayed in these figures, we put two teams
($2i-1,2i$) for $i=1,2,\ldots,n/2$ on one vertex.

Figure 2: Games at the first block for a case with 30 teams
Figure 3: Games at the second block for a case with 30 teams
Figure 4: Games at the last block for a case with 30 teams

Teams 1 to $n-8$ are put on black vertices in sequence. Teams ($n-7,n-6$) are
put on the bottom right gray vertex, teams ($n-5,n-4$) are put on the top
right, teams ($n-3,n-2$) are put on the leftmost, and teams ($n-1,n$) are put
on the gray vertex on the leftmost vertical arc. Home away patterns of the
teams are also displayed in these figures. An arc from teams ($i,j$) to
($k,l$) represents four games in one block for each team $i,j,k$ and $l$.
After four games, teams on the black vertices are moved to the next vertices
as shown in Fig. 3, and then play next four games at the second block. This is
repeated for $n/2-4$ times, and Fig. 4 shows the last block. For all blocks,
directions of arcs and teams on the gray vertices are fixed. The home away
pattern for each vertex is also fixed except for the last block; at the last
block all the vertices have HAAH or AHHA patterns. This modification is done
for the compatibility with the second phase. We note that there are $n/2-4$
possibilities for the initial position of teams $(1,2)$. We choose the best
one among $n/2-4$ possibilities, where the importance to choose the best one
will be analyzed in the next section.

slots |  |  |  |
---|---|---|---|---
teams | 1 | 2 | 3 | 4
1 | 3H | 4H | 3A | 4A
2 | 4H | 3H | 4A | 3A
3 | 1A | 2A | 1H | 2H
4 | 2A | 1A | 2H | 1H

Figure 5: Games for an arc with HHAA pattern

slots |  |  |  |
---|---|---|---|---
teams | 1 | 2 | 3 | 4
1 | 3H | 4A | 3A | 4H
2 | 4H | 3A | 4A | 3H
3 | 1A | 2H | 1H | 2A
4 | 2A | 1H | 2H | 1A

Figure 6: Games for an arc with HAAH pattern

slots |  |  |  |
---|---|---|---|---
teams | 1 | 2 | 3 | 4
1 |  $x$H | 3H |  $x$A | 3A
2 | 4H |  $x$H | 4A |  $x$A
3 |  $y$A | 1A |  $y$H | 1H
4 | 2A |  $y$A | 2H |  $y$H
$x$ | 1A | 2A | 1H | 2H
$y$ | 3H | 4H | 3A | 4A

Figure 7: Games for the leftmost vertical arc

slots |  |  |  |
---|---|---|---|---
teams | 1 | 2 | 3 | 4
1 |  $x$H | 3A |  $x$A | 3H
2 | 4H |  $x$A | 4A |  $x$H
3 |  $y$A | 1H |  $y$H | 1A
4 | 2A |  $y$H | 2H |  $y$A
$x$ | 1A | 2H | 1H | 2A
$y$ | 3H | 4A | 3A | 4H

Figure 8: Games for the leftmost vertical arc at the last block

Here, we describe games for each arc in Fig. 5 to Fig. 8. If one vertex has
HHAA and the other has AAHH patterns, each team plays games as displayed in
Fig. 5. Each number in this table corresponds to the opponent and away (home)
game is denoted by A (H). Fig. 6 shows the games for an arc with HAAH and AHHA
patterns, this kind of arcs appear in the top right and the leftmost
horizontal arcs and at the last block. We have an exceptional arc in Figs. 2
to 4, the left most vertical arc with an intermediate gray vertex. This kind
of arc is not necessary for the case with $n=4m$ teams. For six teams on this
arc, we give four games for each team as described in Fig. 7 and Fig. 8.

In the second phase, every team plays the remaining games. Let
$T_{1}:=\\{1,2,\ldots,n-8\\}$ and $T_{2}:=T\setminus T_{1}$. Then, for every
pair of teams $i\in T_{1}$ and $j\in T_{2}$, team $i$ plays with team $j$ for
two games in the first phase. Thus we can construct second phase schedules for
$T_{1}$ and $T_{2}$ independently.

Figure 9: A schedule for teams $1,2,\ldots,n-8$ in the second phase (before
the mirroring)

For teams belonging to $T_{1}$, team $2i-1$ has not played against teams
$2i-4,2i-3,2i-2,2i,2i+1,2i+2,2i+4$, and team $2i$ has not played against teams
$2i-5,2i-3,2i-2,2i-1,2i+1,2i+2,2i+3$, where mod $2n-8$ is applied for team
numbers (more precisely, team $2n-7$ means team 1, team 0 means team $2n-8$,
team $-1$ means team $2n-9$ and so on). Fig. 9 shows how we schedule the
remaining games for teams in $T_{1}$. Every arc in this figure corresponds to
a game at the venue of the team at the head of the arc. Apart from the games
(of Days 1 to 4) in the gray box, the schedule is regularly constructed and
can be repeated as many times as necessary. We note that the number of teams
on the upper column of Fig. 9 becomes odd for any number of teams $n=4m+2$,
and we need a gray box in which the schedule can be irregular. The schedule
shown in Fig. 9 is designed such as (1) no team plays more than two
consecutive home/away games in these seven slots, (2) every team has HA or AH
pattern in Days (slots) 1 and 2, and (3) every team has HA or AH pattern in
Days 6 and 7. By the mirroring technique, we construct the same schedule
except for the venues of games. By concatenating them, the second phase (14
slots) schedule for teams belonging to $T_{1}$ is constructed.

A schedule for teams in $T_{2}$ is constructed by a single round-robin
tournament for eight teams with the following properties: (1) no team plays
more than two consecutive home/away games, (2) every team has HA or AH pattern
in the first two slots, and (3) every team has HH or AA pattern in the first
and the last slots. One example of such single round-robin tournaments is
displayed in Fig. 10. We construct 14 slots schedule for teams in $T_{2}$ by
constructing the same (except for the venues) schedule with the mirroring
technique and concatenating them.

slots |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
teams | 1 | 2 | 3 | 4 | 5 | 6 | 7
1 | 3H | 4A | 5H | 2A | 6A | 8H | 7H
2 | 4H | 3A | 6A | 1H | 5H | 7A | 8H
3 | 1A | 2H | 7H | 4A | 8A | 6H | 5A
4 | 2A | 1H | 8A | 3H | 7H | 5A | 6A
5 | 7H | 8A | 1A | 6H | 2A | 4H | 3H
6 | 8H | 7A | 2H | 5A | 1H | 3A | 4H
7 | 5A | 6H | 3A | 8H | 4A | 2H | 1A
8 | 6A | 5H | 4H | 7A | 3H | 1A | 2A
Figure 10: A schedule for teams $n-7,n-6,\ldots,n$ in the second phase (before
the mirroring)

##  4 ANALYSIS OF ALGORITHM

In this section, we show the feasibility of the schedule we constructed in
Section 3, and estimate the approximation ratio of the proposed algorithm.

### Feasibility

We first show that the constructed schedule in the previous section is a
double round-robin tournament. It is clear that every team plays exactly one
game for each slot. We check whether every team $i$ plays with every other
team $j$ once at its home and once at $j$’s home venues. In the first phase,
we use (a modified version of) the circle method, which constructs a single
round-robin tournament (instead of the rightmost part and the vertical arc
with an intermediate vertex). For each arc without an intermediate vertex, we
assign games as shown in Figs. 5 and 6, which means that teams $i$ and $j$
have two games at $i$’s and $j$’s home venues if they are put on the opposite
vertices of an arc. For the arc with an intermediate vertex (see Figs. 7 and
8), some pairs of teams play exactly two games as home and away games. In the
second phase, teams $i$ and $j$ who have not played each other play games at
$i$’s and $j$’s home venues. Thus the schedule is a double round-robin
tournament.

Next we consider the no-repeater constraint. If team $i$ plays with team $j$
in the first phase, two games between $i$ and $j$ appear in an identical
block. The no-repeater constraint is satisfied for games in a block as shown
in Figs. 5 to 8. If team $i$ plays with team $j$ in the second phase, two
games between $i$ and $j$ do not appear in a consecutive two slots since we
apply the mirroring technique to single round-robin tournaments in Figs. 9 and
10.

Finally we check whether the schedule satisfies the at-most constraint. In
each block of the first phase, every team plays two home games and two away
games. If a team plays home (away) game at the last slot of a block, the team
plays away (home) game at the first slot of the next block. Thus, all teams
have at most two consecutive home/away games. We note that, in the first
phase, the reason why we put two gray vertices on the rightmost positions (two
vertices for teams 23 to 26 for the case with $n=30$) is that the teams on the
rightmost black vertices cannot play games each other under both of the no-
repeater and at-most constraints. The reason why we need an intermediate gray
vertex (for teams 29 and 30) in Fig. 2 to Fig. 4 is that the number of teams
$n=4m+2$ cannot be divided by four. Every team has HA or AH pattern at the end
of the first phase. This property is also held at the beginning of the second
phase. Hence, three consecutive home/away games cannot appear at the junction
of two phases. In the second phase, as shown in Fig. 9 and Fig. 10, no team
have more than two consecutive home/away games. Therefore, the constructed
double round-robin tournament satisfies the at-most constraint.

### Approximation ratio

We then estimate the total traveling distances of the teams in our double
round-robin tournament and evaluate the approximation ratio. When a team plays
two consecutive away games, as stated in Section 1, the team is able to go
directly from the venue of the first opponent to that of another opponent
without returning to its home venue (with this shortcut, the team can shorten
its traveling distance). However, in this analysis, we consider that the team
returns to its home venue unless two consecutive away games are within a block
in the first phase.

In the first phase, except for the last block, we consider three types of arcs
as shown in Figs. 5, 6 and 7. Games for the arc in Fig. 5 are the ideal; the
total traveling distances of the four teams (team 1 to 4) in this block is
$(d_{13}+d_{14}+d_{23}+d_{24}+d_{31}+d_{32}+d_{41}+d_{42})+2\times(d_{12}+d_{34})$,
where edges $\langle 1,2\rangle$ and $\langle 3,4\rangle$ are belonging to the
minimum weight perfect matching $M$. Traveling distances for games for the arc
in Fig. 6 are
$(d_{13}+d_{14}+d_{23}+d_{24}+d_{31}+d_{32}+d_{41}+d_{42})+(d_{31}+d_{32}+d_{41}+d_{42})+2\times(d_{34})$,
where the second term $(d_{31}+d_{32}+d_{41}+d_{42})$ is surplus. This kind of
arc appears at the top right and leftmost in Figs. 2 and 3; we assume that
teams on the gray vertices (teams 25 to 28 if $n=30$) take this surplus.
Traveling distances for games for the arc in Fig. 7 are
$(d_{13}+d_{1x}+d_{24}+d_{2x}+d_{31}+d_{3y}+d_{42}+d_{4y}+d_{x1}+d_{x2}+d_{y3}+d_{y4})+(d_{x3}+d_{x4}+d_{y1}+d_{y2})+(d_{12}+d_{34})$.
Teams on the gray vertex (i.e., teams 29 and 30 for $n=30$) take the surplus
$(d_{x3}+d_{x4}+d_{y1}+d_{y2})$.

At the last block of the first phase, we have different home away patterns as
shown in Fig. 6 and Fig.8. Traveling distances for games for the arc in Fig. 6
are, as stated in the previous paragraph,
$(d_{13}+d_{14}+d_{23}+d_{24}+d_{31}+d_{32}+d_{41}+d_{42})+(d_{31}+d_{32}+d_{41}+d_{42})+2\times(d_{34})$,
where the second term $(d_{31}+d_{32}+d_{41}+d_{42})$ is surplus. Traveling
distances for games for the arc in Fig. 8 are
$(d_{13}+d_{1x}+d_{24}+d_{2x}+d_{31}+d_{3y}+d_{42}+d_{4y}+d_{x1}+d_{x2}+d_{y3}+d_{y4})+(d_{x3}+d_{x4})+(d_{1x}+d_{2x}+d_{3y}+d_{4y}+d_{13}+d_{24})+d_{34}$.
Team $x$ (team 29 if $n=30$) takes the surplus $d_{x3}+d_{x4}$.

We now analyze the total traveling distances of the teams in the first phase.
For each edge belonging to the minimum weight perfect matching $M$, at most
two teams use it for one block, and there are $n/2-4$ blocks. Hence the total
traveling distances related to the matching $M$ is

$\left(\frac{n}{2}-4\right)\times 2\times d(M)=(n-8)\cdot d(M).$ (3)

We then consider the surplus for the last block; surpluses for (normal) edges
with two vertices and that for the edge with an intermediate vertex, namely
$(d_{31}+d_{32}+d_{41}+d_{42})$ for Fig. 6 and
$(d_{1x}+d_{2x}+d_{3y}+d_{4y}+d_{13}+d_{24})$ for Fig. 8. As stated in Section
3, there are $n/2-4$ possibilities for the initial position of teams on the
black vertices. If an edge with weight $d_{ij}$ appears as the above surpluses
for an initial position, the edge cannot appear as surpluses for different
initial positions. Thus the total surpluses for the last slot of the first
phase is at most $\Delta/2(n/2-4)$ on average. Choosing the best initial
position, the surplus for the last slot is at most

$\Delta/2(n/2-4)=\Delta/(n-8).$ (4)

In the second phase, we evaluate traveling distances of teams 1 to $n-8$ (set
$T_{1}$ with $n-8$ teams) and teams $n-7$ to $n$ (set $T_{2}$) independently.
We first consider teams in $T_{1}$ with Fig. 9. For example, team 7 goes to
the home venues of teams 4,5,6,8,9,10 and 12. Except for single trips from the
home venue of team 7 to those of the other teams, team 7 takes the surplus
$(d_{74}+d_{75}+d_{76}+d_{78}+d_{79}+d_{7,10}+d_{7,12})$. With the triangle
inequality, it is at most $(d_{56}+7\times d_{78}+d_{9,10})+(d_{46}+3\times
d_{68}+3\times d_{8,10}+d_{10,12})$. Similarly, team 8 visits the home venues
of teams 3,5,6,7,9,10 and 11. Except for single trips, team 8 takes the
surplus $(d_{83}+d_{85}+d_{86}+d_{87}+d_{89}+d_{8,10}+d_{8,11})$, and it is at
most $(d_{34}+d_{56}+d_{78}+d_{9,10}+d_{11,12})+(d_{46}+3\times d_{68}+3\times
d_{8,10}+d_{10,12})$. By considering all the teams in $T_{1}$, the surplus for
teams in $T_{1}$ in the second phase is at most

$\displaystyle 14\times(d_{12}+d_{34}+\cdots+d_{n-9,n-8})$
$\displaystyle+16\times(d_{24}+d_{46}+\cdots+d_{n-10,n-8}+d_{n-8,2})$
$\displaystyle\leq 14\times d(M)+16\times(d(T)+d(M)).$ (5)

Teams belonging to the set $T_{2}$ play a double round-robin tournament of
eight teams in the second phase. We evaluate surpluses in this schedule except
for the single trips of the teams. Among teams in $T_{2}$, teams
$n-5,n-4,\ldots,n$ have already taken some surpluses in the first phase. We
can evaluate the surpluses of those teams $i$ within both of the phases as
$s(i)$. For the remaining surpluses, team $n-7$ takes $t(n-7)$, team $n-6$
takes $t(n-6)$ and we have $2\times d_{n-7,n-6}$, where edge $\langle
n-7,n-6\rangle$ belongs to $M$. The rules (a) and (b) for the numbering of the
teams give the following inequalities:

$\displaystyle s(n-5)+s(n-4)+\cdots+s(n)\leq 6\Delta/n,$ (6) $\displaystyle
t(n-7)+t(n-6)\leq 12\Delta/\\{n(n-6)\\}.$ (7)

Finally, we evaluate the total traveling distance of the double round-robin
tournament we constructed. We assume that the number of teams $n$ is at least
30; for the case with $n<30$ teams, it is possible to enumerate all the
possible schedules and choose an optimal solution in constant time (since 30
is a constant number). Under this assumption, we can use the following
inequality

$\Delta/(n-8)+12\Delta/\\{n(n-6)\\}\leq 2\Delta/n.$ (8)

With the sum of single trips of $n$ teams ($=\Delta$) and equations (3) to
(8), the total traveling distance is at most

$\displaystyle\left(1+\frac{8}{n}\right)\Delta+(n+6)\cdot
d(M)+16\cdot(d(T)+d(M))$ $\displaystyle\leq$
$\displaystyle\left(1+\frac{8}{n}\right)(\Delta+n\cdot
d(M))+\frac{16}{n}(n\cdot(d(T)+d(M)).$ (9)

By using two kinds of lower bounds (1) and (2), we have derived a 1 + $24/n$
approximation for TTP(2) with $n=4m+2\ (\geq 30)$ teams. With the 1 + $16/n$
approximation algorithm for TTP(2) with $n=4m\ (\geq 12)$ teams by Thielen2012
and a naive enumeration algorithm for the case with a constant number of teams
($n<30$), we attain the first $1+O(1/n)$ approximation algorithm for TTP(2).

##  5 CONCLUSIONS AND FUTURE WORK

This paper studied an approximation algorithm for the traveling tournament
problem with constraints that the maximum length of home/away consecutive
games is two. Thielen2012 proposed a 1 + $16/n$ approximation algorithm for
the problem with $n=4m$ teams. In this paper we proposed a 1 + $24/n$
approximation algorithm for the case with $n=4m+2$ teams, this new result
completes a 1 + $O(1/n)$ approximation algorithm for TTP(2).

A remained open problem is to reveal the complexity of the problem. It was
showed by Thielen2011 that TTP($k$) is NP-hard for $k\geq 3$. However, the
complexity of TTP(2) is still open. Another direction of future research is a
single round-robin tournament version of TTP(2). It is easy to design a
constant factor approximation algorithm (because any feasible solution yields
a 4 approximation), but attaining better results (such as 1 + $O(1/n)$
approximation) are future work.

## References

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# Martingale solution of stochastic hybrid Korteweg - de Vries - Burgers
equation

Anna Karczewska  Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
<EMAIL_ADDRESS>and  Maciej Szczeciński  Faculty of Management,
Computer Science and Finance
Wrocław University of Economics, Komandorska 118/120, 53-345 Wrocław, Poland
<EMAIL_ADDRESS>
###### Abstract.

In the paper, we consider a stochastic hybrid Korteweg - de Vries - Burgers
type equation with multiplicative noise in the form of cylindrical Wiener
process. We prove the existence of a martingale solution to the equation
studied. The proof of the existence of the solution is based on two
approximations of the considered problem and the compactness method. First, we
introduce an auxiliary problem corresponding to the equation studied. Then, we
prove the existence of a martingale solution to this problem. Finally, we show
that the solution of the auxiliary problem converges, in some sense, to the
solution of the equation under consideration.

###### Key words and phrases:

KdV equation, Burgers equation, mild solution

###### 2010 Mathematics Subject Classification:

93B05, 93C25, 45D05, 47H08, 47H10

##  1\. Introduction

The deterministc hybrid Korteweg - de Vries - Burgers (hKdVB for short)
equation has been derived by Misra, Adhikary and Shuka [10] and Elkamash and
Kourakis [5] in the context of shock excitations in multicomponent plasma. The
hKdVB equation, derived in stretched coordinates
$\xi=\epsilon^{\frac{1}{2}}(x-vt),~{}\tau=\epsilon^{\frac{3}{2}}t$ ($v$ is the
phase velocity of the wave) has the form

(1.1) $u_{\tau}+Au\,u_{\xi}+Bu_{3\xi}=Cu_{2\xi}-Du.$

In (1.1), $u(\xi,\tau)$ represents electrostatic potential or electric field
pulse in the reference frame moving with the velocity $v$. Indexes denote
partial derivatives, that is, $u_{\tau}=\partial u/\partial\tau$,
$u_{2\xi}=\partial^{2}u/\partial\xi^{2}$ and so on. Constants $A,B,C,D$ are
related to parameters describing properties of plasma [5, Eq. (27)].

Although the equation (1.1) was derived for dissipative dispersive waves in
multicomponent plasma, it can be applied in several other physical systems,
e.g., surface water waves and the motion of optical impulses in fibers. For
some particular values of constants $A,B,C,D$ the hKdVB equation (1.1) reduces
to the particular cases:

  * •

the Korteweg - de Vries equation, when $C=D=0$;

  * •

the damped (dissipative) KdV equation, when $C=0$;

  * •

the Burgers equation, when $B=D=0$;

  * •

the KdV-Burgers equation, when $D=0$;

  * •

the damped Burgers equation, when $B=0$.

The term with $A\neq 0$ introduces nonlinearity, that with $B\neq 0$ is
responsible for dispersion, $C\neq 0$ supplies diffusive term and $D\neq 0$
introduces damping. All equations of these kinds were widely studied 30-40
years ago, and most of physical ideas have been already understood (see, e.g.,
Lev Ostrovsky’s book [11] and references therein). On the other hand, during
the last few years, one can observe renewal of interest in this fields, mostly
due to extensions to higher order equations.

Studies of the full generalized hybrid KdB-Burgers equation (1.1) have
appeared in the physical literature only in [5, 10]. Some approximate analytic
solutions and several cases of numerical solutions to (1.1) were subjects of
recent studies in [6].

The paper deals with a stochastic hybrid Korteweg - de Vries - Burgers type
equation. The presence of stochastic noise has deep physical grounds. In the
case of waves in plasma, it can be caused by thermal fluctuations, whereas in
the case of water surface waves by air pressure fluctuations due to the wind.
To the best of our knowledge, our paper is the first one which deals with the
stochastic hKdVB equation.

The main result of the paper, Theorem 2.2 supplies the existence of a
martingale solution to the equation (2.1), which is the stochastic version of
the equation (1.1).

The idea of the proof of the existence of a martingale solution to (2.1) is
the following. First, we introduce an auxiliary problem (2.6) which we can
call $\varepsilon$-approximation of the equation (2.1). Then, in Lemma 2.3 we
prove that the problem (2.6) has a martingale solution. Here we use the
Galerkin approximation (4.1) of (2.6) and the tightness of the family of
distributions of the solutions to the approximation (4.1). Next, in Lemma 2.4
we show two estimates used in the proof of Theorem 2.2. Lemma 2.5 guarantees
the tightness of the family of distributions of solutions to the problem (2.6)
in a proper space. Finally, we prove that the solution to(2.6) converges, in
some sense, to the solution of the equation (2.1).

The paper is organized as follows. In section 2 we define the martingale
solution to some kind of stochastic hybrid Korteweg - de Vries - Burgers
equation (2.1) with a multiplicative Wiener noise on the interval $[0,T]$.
Then we formulate and prove Theorem 2.2. In the proof, some methods introduced
in [7] and extended in [4] have been adapted to the problem considered.

In section 3 Lemmas 2.4 and 2.5 used in the proof of Theorem 2.2 are proved.
Lemma 2.4 contains a version of estimates which are analogous to those
presented in [4] and [7].

In section 4 we give the detailed proof of Lemma 2.3. This lemma formulates
sufficient conditions for the existence of martingale solutions for
$m$-dimensional Galerkin approximation of Korteweg - de Vries - Burgers
equation with a multiplicative Wiener noise for arbitrary $m\in\mathrm{N}$.

##  2\. Existence of martingale solution

Denote $X:=[x_{1},x_{2}]\subset\mathbb{R}$, where
$-\infty<x_{1}<0<x_{2}<\infty$. We consider the following initial value
problem for hybrid Korteweg - de Vries - Burgers type equation

(2.1)
$\begin{cases}du(t,x)+\big{(}Au(t,x)u_{x}(t,x)+Bu_{3x}(t,x)-Cu_{2x}(t,x)+Du(t,x)\big{)}dt\\\
\hskip 34.44434pt=\Phi\left(u(t,x)\right)dW(t)\\\ u(0,x)=u_{0}(x),\quad x\in
X,\quad t\geq 0.\\\ \end{cases}$

In (2.1), $W(t)$, $t\geq 0$, is a cylindrical Wiener process adapted to the
filtration $\left\\{\mathscr{F}_{t}\right\\}_{t\geq 0}$, $u_{0}\in L^{2}(X)$
is a deterministic real-valued function. In (2.1)
$u(\omega,\cdot,\cdot):\mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}$
for all $\omega\in\Omega$. We assume that there exists such $\lambda_{X}>0$,
that

(2.2)
$\left|u(t,x)\right|<\lambda_{X}<\infty\quad\mbox{for~{}all~{}}t\in\mathbb{R}_{+}\mbox{~{}and~{}all~{}}x\in
X.$

This assumption reflects the finiteness of the solutions to the deterministic
equation (1.1) on finite interval $X$ (see, e.g., [10, 6]).

By $H^{1}(X),H^{2}(X),H^{s}(X),s<0$ we denote the Sobolev spaces according to
definitions in [1]. We assume that $\Phi:H^{2}(X)\rightarrow
L_{0}^{2}(L^{2}(X))$ is a continuous mapping which for all $u\in H^{2}(X)$
fulfills the following conditions:

(2.3)
$\displaystyle\mathop{\exists}_{\kappa_{1},\kappa_{2}>0}\quad\left\|\Phi(u(x))\right\|_{L_{0}^{2}(L^{2}(X))}\leq\kappa_{1}\min\left\\{\left|u(x)\right|^{2}_{L^{2}(X)},\left|u(x)\right|_{L^{2}(X)}\right\\}+\kappa_{2};$
(2.4) $\displaystyle\mbox{there~{}exist~{}such~{}functions}~{}a,b\in
L^{2}(X)~{}\mbox{with~{}compact support,~{}that~{}the~{}mapping}$
$\displaystyle
u\mapsto\left(\Phi(u)a,\Phi(u)b\right)_{L^{2}(X)}~{}\mbox{is~{}continuous~{}in~{}}L^{2}(X).$

######  Definition 2.1.

We say that the problem (2.1) has a martingale solution on the interval
$[0,T]$, $0<T<\infty$, if there exists a stochastic basis
$(\Omega,\mathscr{F},\left\\{\mathscr{F}_{t}\right\\}_{t\geq
0},\mathbb{P},\left\\{W_{t}\right\\}_{t\geq 0})$, where
$\left\\{W_{t}\right\\}_{t\geq 0}$ is a cylindrical Wiener process, and there
exists the process $\left\\{u(t,x)\right\\}_{t\geq 0}$ adapted to the
filtration $\left\\{\mathscr{F}_{t}\right\\}_{t\geq 0}$ with trajectories in
the space

$L^{\infty}(0,T;L^{2}(X))\cap L^{2}(0,T;L^{2}(X))\cap C(0,T;H^{s}(X)),\quad
s<0,\quad\mathbb{P}-a.s.,$

such that

$\displaystyle\left\langle u(t,x);v(x)\right\rangle+\int_{0}^{t}\left\langle
Au(t,x)u_{x}(t,x)+Bu_{3x}(t,x)-Cu_{2x}(t,x)+Du(t,x);v(x)\right\rangle ds$
$\displaystyle=\left\langle
u_{0}(x);v(x)\right\rangle+\left\langle\int_{0}^{t}\Phi(u(s,x))dW(s);v(x)\right\rangle,\quad\mathbb{P}-a.s.,$

for all $t\in[0,T]$ and $v\in H^{1}(X)$.

In our consideration we shall assume that the coefficients of the equation
(2.1) fulfill the following condition

(2.5) $B,C,D\geq 0\qquad\mbox{with}\qquad 3B\geq A+1.$

The physical sense of the coefficients $A,B,C,D$ and the fact that $A$ can be
positive or negative, see, e.g. [5, 6, 10] confirms that the condition (2.5)
admits for a broad class of physically meaningful equations which contains all
particular cases listed in section 1.

######  Theorem 2.2.

If the conditions (2.2)-(2.5) hold then for all real-valued $u_{0}\in
L^{2}(X)$ and $0<T<\infty$ there exists a martingale solution to (2.1).

###### Proof.

Let $\varepsilon>0$. Consider the following auxiliary problem

(2.6) $\begin{cases}du^{\varepsilon}(t,x)+[\varepsilon
u^{\varepsilon}_{4x}(t,x)+Au^{\varepsilon}(t,x)u^{\varepsilon}_{x}(t,x)+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)\\\
\hskip
51.6665pt+Du^{\varepsilon}(t,x)]dt=\Phi\left(u^{\varepsilon}(t,x)\right)dW(t)\\\
u^{\varepsilon}_{0}(x)=u^{\varepsilon}(0,x),\quad\varepsilon>0.\end{cases}$

In the proof of theorem we shall use the following lemmas.

######  Lemma 2.3.

For any $\varepsilon>0$ there exists a martingale solution to the problem
(2.6) if the conditions (2.3), (2.4) and (2.5) hold.

######  Lemma 2.4.

There exists $\varepsilon_{0}>0$, such that

(2.7)
$\displaystyle\exists_{C_{1}>0}\forall_{0<\varepsilon<\varepsilon_{0}}~{}\varepsilon\,\mathbb{E}\left(\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{2}(\mathbb{R}))}\right)$
$\displaystyle\leq\tilde{C}_{1},$ (2.8) $\displaystyle\forall_{k\in
X_{k}}\exists_{C_{2}(k)>0}\forall_{0<\varepsilon<\varepsilon_{0}}~{}\mathbb{E}\left(\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{1}(-k,k))}\right)$
$\displaystyle\leq\tilde{C}_{2}(k),$

where
$X_{k}=\big{\\{}k>0:\left|k\right|\leq\min\left\\{-x_{1},x_{2}\right\\}\big{\\}}$.

######  Lemma 2.5.

Let $\mathscr{L}(u^{\varepsilon})$ denote the family of distributions of the
solutions $u^{\varepsilon}$ to (2.6). Then the family
$\mathscr{L}(u^{\varepsilon})$ is tight in $L^{2}(0,T;L^{2}(X))\cap
C(0,T;H^{-3}(X))$.

Now, substitute in Prohorov’s theorem (e.g., see Theorem 5.1 in [2]),
$S:=L^{2}(0,T;L^{2}(X))\cap C(0,T;H^{-3}(X))$ and
$\mathscr{K}:=\left\\{\mathscr{L}(u^{\varepsilon})\right\\}_{\varepsilon>0}$.
Since $\mathscr{K}$ is tight in $S$, then it is sequentially compact, so there
exists a subsequence of
$\left\\{\mathscr{L}(u^{\varepsilon})\right\\}_{\varepsilon>0}$ converging to
some measure $\mu$ in $\bar{\mathscr{K}}$. Because
$\left\\{\mathscr{L}(u^{\varepsilon})\right\\}_{\varepsilon>0}$ is convergent,
then in Skorohod’s theorem (e.g., see Theorem 6.7 in [2]) one can substitute
$\mu_{\varepsilon}:=\left\\{\mathscr{L}(u^{\varepsilon})\right\\}_{\varepsilon>0}$
and $\mu:=\lim_{\varepsilon\rightarrow 0}\mu_{\varepsilon}$. Then there exists
a space
$(\bar{\Omega},\bar{\mathscr{F}},\left\\{\bar{\mathscr{F}}_{t}\right\\}_{t\geq
0},\bar{\mathbb{P}})$ and random variables with values in
$L^{2}(0,T;L^{2}(X))\cap C(0,T;H^{-3}(X))$, such that
$\bar{u}^{\varepsilon}\rightarrow\\!\bar{u}$ in $L^{2}(0,T;L^{2}(X))$ and
$\bar{u}^{\varepsilon}\rightarrow\\!\bar{u}$ in $C(0,T;H^{-3}(X))$. Moreover,
$\mathscr{L}(\bar{u}^{\varepsilon})\\!=\mathscr{L}(u^{\varepsilon})$.

Then due to Lemma 2.4, for any $p\in\mathbb{N}$ there exist constants
$\tilde{C}_{1}(p)$, $\tilde{C}_{2}$ such that

$\displaystyle\mathbb{E}(\sup_{t\in[0,T]}\left|\bar{u}^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{2p})\leq\tilde{C}_{1}(p),\quad\mathbb{E}(\left|\bar{u}^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{2}(X))})\leq\tilde{C}_{2}$

and $\bar{u}^{\varepsilon}(t,x)\in L^{2}(0,T;H^{1}(-k,k))\cap
L^{\infty}(0,T;L^{2}(X)),~{}\mathbb{P}-a.s.$ Then one can conclude that
$\bar{u}^{\varepsilon}\rightarrow\bar{u}$ weakly in
$L^{2}(\bar{\Omega},L^{2}(0,T;H^{1}(-k,k)))$.

Let $x\in\mathbb{R}$ be fixed. Denote

$\displaystyle M^{\varepsilon}(t):=$ $\displaystyle
u^{\varepsilon}(t,x)-u_{0}^{\varepsilon}(x)+\int_{0}^{t}\bigg{[}\varepsilon
u^{\varepsilon}(t,x)_{4x}(t,x)+Au^{\varepsilon}(t,x)u^{\varepsilon}_{x}(t,x)$
$\displaystyle\hskip
94.72192pt+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\bigg{]}ds,$
$\displaystyle\bar{M}^{\varepsilon}(t):=$
$\displaystyle\bar{u}^{\varepsilon}(t,x)-\bar{u}_{0}^{\varepsilon}(x)+\int_{0}^{t}\bigg{[}A\bar{u}^{\varepsilon}(t,x)\bar{u}^{\varepsilon}_{x}(t,x)+B\bar{u}^{\varepsilon}_{3x}(t,x)-C\bar{u}^{\varepsilon}_{2x}(t,x)+D\bar{u}^{\varepsilon}(t,x)\bigg{]}ds.$

Note, that

$\displaystyle M^{\varepsilon}(t)=$ $\displaystyle\hskip
4.30554ptu_{0}^{\varepsilon}(x)-\int_{0}^{t}\bigg{[}\varepsilon
u^{\varepsilon}(t,x)_{4x}(t,x)+Au^{\varepsilon}(t,x)u^{\varepsilon}_{x}(t,x)+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)$
$\displaystyle\hskip
55.97205pt+Du^{\varepsilon}(t,x)\bigg{]}ds+\int_{0}^{t}\left(\Phi\left(u^{\varepsilon}(s,x)\right)\right)dW(s)$
$\displaystyle-
u_{0}^{\varepsilon}(x)+\int_{0}^{t}\bigg{[}Au^{\varepsilon}(t,x)u^{\varepsilon}_{x}(t,x)+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\bigg{]}ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\left(\Phi\left(u^{\varepsilon}(s,x)\right)\right)dW(s).$

So, $M^{\varepsilon}(t)$, $t\geq 0$, is a square integrable martingale with
values in $L^{2}(X)$, adapted to the filtration
$\sigma\left\\{u^{\varepsilon}(s,x),0\leq s\leq t\right\\}$ with quadratic
variation equal

$\left[M^{\varepsilon}\right](t):=\int_{0}^{t}\Phi(u^{\varepsilon}(s,x))\left[\Phi(u^{\varepsilon}(s,x))\right]^{*}ds.$

Substitute in the Doob inequality (e.g., see Theorem 2.2 in [8])
$M_{t}:=M^{\varepsilon}(t)$ and $p:=2p$. Then

(2.9)
$\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|M^{\varepsilon}(t)\right|_{L^{2}(X)}^{p}\right)\right]\leq\left(\frac{p}{p-1}\right)^{p}\mathbb{E}\left(\left|M^{\varepsilon}(T)\right|_{L^{2}(X)}\right).$

Assume $0\leq s\leq t\leq T$ and let $\varphi$ be a bounded continuous
function on $L^{2}(0,s;L^{2}(X))$ or $C(0,s;H^{-3}(X))$. Let $a,b\in
H^{3}_{0}(-k,k)$, $k\in\mathbb{N}$, be arbitrary and fixed. Since
$M^{\varepsilon}(t)$ is a martingale and
$\mathscr{L}(\bar{u}^{\varepsilon})=\mathscr{L}(u^{\varepsilon})$, then (see
[7], p. 377-378)

$\mathbb{E}\Big{(}\left\langle
M^{\varepsilon}(t)-M^{\varepsilon}(s);a\right\rangle\varphi\left(u^{\varepsilon}(t,x)\right)\Big{)}=0\quad\mbox{and}\quad\mathbb{E}\Big{(}\left\langle\bar{M}^{\varepsilon}(t)-\bar{M}^{\varepsilon}(s);a\right\rangle\varphi\left(\bar{u}^{\varepsilon}(t,x)\right)\Big{)}=0.$

Moreover

$\displaystyle\mathbb{E}$ $\displaystyle\bigg{\\{}\bigg{[}\left\langle
M^{\varepsilon}(t);a\right\rangle\left\langle
M^{\varepsilon}(t);b\right\rangle-\left\langle
M^{\varepsilon}(s);a\right\rangle\left\langle
M^{\varepsilon}(s);b\right\rangle$
$\displaystyle-\int_{s}^{t}\left\langle\left[\Phi\left(u^{\varepsilon}(\xi,x)\right)\right]^{*}a;\left[\Phi\left(u^{\varepsilon}(\xi,x)\right)\right]^{*}b\right\rangle
d\xi\bigg{]}\varphi(u^{\varepsilon}(t,x))\bigg{\\}}=0$
$\displaystyle\mbox{and}\quad\mathbb{E}$
$\displaystyle\bigg{\\{}\bigg{[}\left\langle\bar{M}^{\varepsilon}(t);a\right\rangle\left\langle\bar{M}^{\varepsilon}(t);b\right\rangle-\left\langle\bar{M}^{\varepsilon}(s);a\right\rangle\left\langle\bar{M}^{\varepsilon}(s);b\right\rangle$
$\displaystyle-\int_{s}^{t}\left\langle\left[\Phi\left(\bar{u}^{\varepsilon}(\xi,x)\right)\right]^{*}a;\left[\Phi\left(\bar{u}^{\varepsilon}(\xi,x)\right)\right]^{*}b\right\rangle
d\xi\bigg{]}\varphi(\bar{u}^{\varepsilon}(t,x))\bigg{\\}}=0.$

Denote
$\bar{M}(t):=\bar{u}(t,x)-u_{0}(x)+\int_{0}^{t}\bigg{[}A\bar{u}(t,x)\bar{u}_{x}(t,x)+B\bar{u}_{3x}(t,x)-C\bar{u}_{2x}(t,x)+D\bar{u}(t,x)\bigg{]}ds$.
If $\varepsilon\rightarrow 0$, then
$\bar{M}^{\varepsilon}(t)\rightarrow\bar{M}(t)$ and
$\bar{M}^{\varepsilon}(s)\rightarrow\bar{M}(s)$, $\bar{\mathbb{P}}$ \- a.s. in
$H^{-3}(X)$. Moreover, since $\varphi$ is continuous, then
$\varphi(\bar{u}^{\varepsilon}(s,x))\rightarrow\varphi(\bar{u}(s,x))$,
$\bar{\mathbb{P}}$ \- a.s. So, if $\varepsilon\rightarrow 0$, then

$\displaystyle\mathbb{E}$
$\displaystyle\Big{(}\left\langle\bar{M}^{\varepsilon}(t)-\bar{M}^{\varepsilon}(s);a\right\rangle\varphi(\bar{u}^{\varepsilon}(t,x))\Big{)}\rightarrow\mathbb{E}\Big{(}\left\langle\bar{M}(t)-\bar{M}(s);a\right\rangle\varphi(\bar{u}(t,x))\Big{)}.$

Moreover, since $\Phi$ is a continuous operator in topology $L^{2}(X)$ and
(2.9) holds, so if $\varepsilon\rightarrow 0$, then

$\displaystyle\left\langle\left(\Phi(\bar{u}^{\varepsilon}(s,x))\right)^{*}a;\left(\Phi(\bar{u}^{\varepsilon}(s,x))\right)^{*}b\right\rangle\rightarrow\left\langle\left(\Phi(\bar{u}(s,x))\right)^{*}a;\left(\Phi(\bar{u}(s,x))\right)^{*}b\right\rangle$

and

$\displaystyle\mathbb{E}$
$\displaystyle\bigg{\\{}\bigg{[}\left\langle\bar{M}^{\varepsilon}(t);a\right\rangle\left\langle\bar{M}^{\varepsilon}(t);b\right\rangle-\left\langle\bar{M}^{\varepsilon}(s);a\right\rangle\left\langle\bar{M}^{\varepsilon}(s);b\right\rangle$
$\displaystyle-\int_{s}^{t}\left\langle\left[\Phi\left(\bar{u}^{\varepsilon}(s,\xi)\right)\right]^{*}a;\left[\Phi\left(\bar{u}^{\varepsilon}(s,\xi)\right)\right]^{*}b\right\rangle
d\xi\bigg{]}\varphi(\bar{u}^{\varepsilon}(t,x))\bigg{\\}}$
$\displaystyle\rightarrow\mathbb{E}$
$\displaystyle\bigg{\\{}\bigg{[}\left\langle\bar{M}(t);a\right\rangle\left\langle\bar{M}(t);b\right\rangle-\left\langle\bar{M}(s);a\right\rangle\left\langle\bar{M}(s);b\right\rangle$
$\displaystyle-\int_{s}^{t}\left\langle\left[\Phi\left(\bar{u}(s,\xi)\right)\right]^{*}a;\left[\Phi\left(\bar{u}(s,\xi)\right)\right]^{*}b\right\rangle
d\xi\bigg{]}\varphi(\bar{u}(t,x))\bigg{\\}}.$

Then $\bar{M}(t),t\geq 0,$ is also a square integrable martingale adapted to
the filtration
$\sigma\left\\{\bar{u}(s),0\leq s\leq t\right\\}$ with quadratic variation
$\int_{0}^{t}\Phi(\bar{u}(s,x))\left(\Phi(\bar{u}(s,x))\right)^{*}ds$.

Substitute in the representation theorem (e.g., see Theorem 8.2 in [3]),
$M_{t}:=\bar{M}(t)$,
$[M_{t}]:=\int_{0}^{t}\Phi(\bar{u}(s,x))\times\left(\Phi(\bar{u}(s,x))\right)^{*}ds$
and $\Phi(s):=\Phi(\bar{u}(s,x))$.

Then there exists a process
$\tilde{M}(t)=\int_{0}^{t}\Phi(\bar{u}(s,x))dW(s)$, that
$\tilde{M}(t)=\bar{M}(t)$, $\mathbb{\bar{P}}$ \- a.s., and

$\displaystyle\bar{u}(t,x)-u_{0}(x)+\int_{0}^{t}\bigg{[}A\bar{u}(t,x)\bar{u}_{x}(t,x)+B\bar{u}_{3x}(t,x)-C\bar{u}_{2x}(t,x)+D\bar{u}(t,x)\bigg{]}ds$
$\displaystyle=\int_{0}^{t}\Phi(\bar{u}(s,x))dW(s).$

This implies that

$\displaystyle\bar{u}(t,x)=u_{0}(x)-\int_{0}^{t}\bigg{[}A\bar{u}(t,x)\bar{u}_{x}(t,x)+B\bar{u}_{3x}(t,x)-C\bar{u}_{2x}(t,x)+D\bar{u}(t,x)\bigg{]}ds$
$\displaystyle+\int_{0}^{t}\Phi(\bar{u}(s,x))dW(s),$

so $\bar{u}(t,x)$ is a solution to (2.1), what finishes the proof of Theorem
2.2 . ∎

##  3\. Proofs of Lemma 2.4 and Lemma 2.5

###### Proof of Lemma 2.4.

Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth function fulfilling the
following conditions

  * (i)

$p$ is increasing on $X$;

  * (ii)

$p(x_{1})=\delta>0$;

  * (iii)

$p^{\prime}(x)>\alpha_{X}$ for all $x\in X$;

  * (iv)

$Bp^{\prime\prime\prime}(x)+Cp^{\prime\prime}(x)\leq\gamma<-1$ for all $x\in
X$.

Additionally, let
$F(u^{\varepsilon}):=\int_{X}p(x)(u^{\varepsilon}(x))^{2}dx$. Application of
the Itô formula for $F(u^{\varepsilon})$ yields the formula

$\displaystyle dF(u^{\varepsilon}(t,x))=$ $\displaystyle\left\langle
F^{\prime}(u^{\varepsilon}(t,x));\Phi(u^{\varepsilon}(t,x))\right\rangle
dW(t)-\left\langle F^{\prime}(u^{\varepsilon}(t,x));\varepsilon
u^{\varepsilon}_{4x}+Au^{\varepsilon}(t,x)u_{x}^{\varepsilon}(t,x)\right.$
(3.1)
$\displaystyle\left.+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\right\rangle
dt$
$\displaystyle+\frac{1}{2}\text{Tr}\left\\{F^{\prime\prime}(u^{\varepsilon}(t,x))\Phi(u^{\varepsilon}(t,x))\left[\Phi(u^{\varepsilon}(t,x))\right]^{*}\right\\}dt,$

where $\left\langle
F^{\prime}(u^{\varepsilon}(t,x));v(t,x)\right\rangle=2\int_{X}p(x)u^{\varepsilon}(t,x)v(t,x)dx$
and $F^{\prime}(u^{\varepsilon}(t,x))v(t,x)=2p(x)v(t,x)$.

We will use the following auxiliary result.

######  Lemma 3.1.

[[4], p.242] There exist positive constants $C_{1},C_{2},C_{3}$ such that

$\displaystyle\int_{X}p(x)u^{\varepsilon}(t,x)u^{\varepsilon}_{4x}(t,x)dx\geq$
$\displaystyle\frac{1}{2}\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx-
C_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(X)}$ $\displaystyle-
C_{2}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx;$
$\displaystyle\int_{X}p(x)u^{\varepsilon}(t,x)u^{\varepsilon}_{3x}(t,x)dx=$
$\displaystyle\frac{3}{2}\int_{X}p^{\prime}(x)\left[u^{\varepsilon}_{x}(t,x)\right]^{2}dx-\frac{1}{2}\int_{X}p^{\prime\prime\prime}(x)\left[u(t,x)\right]^{2}dx;$
$\displaystyle\int_{X}p(x)\left[u^{\varepsilon}(t,x)\right]^{2}u^{\varepsilon}_{x}(t,x)dx\geq$
$\displaystyle-
C_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)-\frac{1}{2}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx.$

Similarly as in Lemma 3.1, one has

$\displaystyle\int_{X}p(x)u^{\varepsilon}(t,x)u^{\varepsilon}_{2x}(t,x)dx=$
$\displaystyle\frac{1}{2}\int_{X}p^{\prime\prime}(x)\left[u^{\varepsilon}(t,x)\right]^{2}dx-\int_{X}p(x)\left[u_{x}(t,x)\right]^{2}dx.$

These estimations imply

$\displaystyle\left\langle F^{\prime}(u^{\varepsilon}(t,x));\varepsilon
u_{4x}^{\varepsilon}(t,x)+Au^{\varepsilon}(t,x)u_{x}^{\varepsilon}(t,x)+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\right\rangle$
$\displaystyle\geq$
$\displaystyle\varepsilon\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx-2\varepsilon
C_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(X)}-2\varepsilon
C_{2}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx$
$\displaystyle+3B\int_{X}p^{\prime}(x)\left[u^{\varepsilon}_{x}(t,x)\right]^{2}dx-B\int_{X}p^{\prime\prime\prime}(x)\left[u^{\varepsilon}(t,x)\right]^{2}dx-2AC_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)$
$\displaystyle-A\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx-C\int_{X}p^{\prime\prime}(x)\left[u^{\varepsilon}(t,x)\right]^{2}dx+2C\int_{X}p(x)\left[u_{x}(t,x)\right]^{2}dx$
$\displaystyle+2D\int_{X}p(x)\left[u(t,x)\right]^{2}dx$ $\displaystyle\geq$
$\displaystyle\varepsilon\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx-2\varepsilon
C_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(X)}+\left(3B-A-2\varepsilon
C_{2}\right)C_{2}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx$
$\displaystyle-\int_{X}\left[Bp^{\prime\prime\prime}(x)+Cp^{\prime\prime}(x)\right]\left[u^{\varepsilon}(t,x)\right]^{2}dx-2AC_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)$
$\displaystyle\geq$
$\displaystyle\varepsilon\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx+\left(3B-A-2\varepsilon
C_{2}\right)\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx$
$\displaystyle-\left(\gamma+2\varepsilon
C_{1}\right)\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(X)}-2AC_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right).$

Let
$\varepsilon\leq\min\left\\{\frac{3B-A-1}{2C_{2}},-\frac{1+\gamma}{2C_{1}}\right\\}$.
Then

(3.2) $\displaystyle\left\langle F^{\prime}(u^{\varepsilon}(t,x));\varepsilon
u_{4x}^{\varepsilon}(t,x)+Au^{\varepsilon}(t,x)u_{x}^{\varepsilon}(t,x)+Bu^{\varepsilon}_{3x}(t,x)-Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\right\rangle$
$\displaystyle\geq$
$\displaystyle\varepsilon\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx+\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx+\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(X)}$
$\displaystyle-2AC_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)$
$\displaystyle\geq$
$\displaystyle\varepsilon\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dx+\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dx-2AC_{3}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right).$

Let $\left\\{e_{i}\right\\}_{i\in\mathbb{N}}$ be an orthonormal basis in
$L^{2}(X)$. Then, there exists a constant $C_{4}>0$ such that

(3.3)
$\displaystyle\text{Tr}\left(F^{\prime\prime}(u)\Phi(u)\left[\Phi(u)\right]^{*}\right)=$
$\displaystyle
2\sum_{i\in\mathbb{N}}\int_{X}p(x)\left|\Phi\left(u^{\varepsilon}(t,x)\right)e_{i}(x)\right|^{2}dx\leq
C_{4}\left|\Phi\left(u^{\varepsilon}(t,x)\right)\right|^{2}_{L_{0}^{2}\left(L^{2}(X)\right)}$
$\displaystyle\leq$ $\displaystyle
C_{4}\left(\kappa_{1}\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{2}+\kappa_{2}\right)^{2}.$

From (3.2) and (3.3) we have

$\displaystyle\mathbb{E}F(u^{\varepsilon}(t,x))\leq$ $\displaystyle
F\left(u^{\varepsilon}_{0}\right)-\varepsilon\mathbb{E}\int_{0}^{t}\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dxdt-\mathbb{E}\int_{0}^{t}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dxdt$
$\displaystyle+2AC_{3}\mathbb{E}\int_{0}^{t}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)dt+C_{4}\mathbb{E}\left(\kappa_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(\mathbb{R})}+\kappa_{2}\right)^{2},$

so

$\displaystyle\mathbb{E}F(u^{\varepsilon}(t,x))+\varepsilon\mathbb{E}\int_{0}^{t}\int_{X}p(x)\left[u^{\varepsilon}_{2x}(t,x)\right]^{2}dxdt+\mathbb{E}\int_{0}^{t}\int_{X}p^{\prime}(x)\left[u_{x}(t,x)\right]^{2}dxdt$
$\displaystyle\leq$ $\displaystyle
F\left(u^{\varepsilon}_{0}\right)+2AC_{3}\mathbb{E}\int_{0}^{t}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)dt+C_{4}\mathbb{E}\left(\kappa_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(\mathbb{R})}+\kappa_{2}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
F\left(u^{\varepsilon}_{0}\right)+2AC_{3}\mathbb{E}\int_{0}^{T}\left(1+\left|u^{\varepsilon}(t,x)\right|_{L^{2}(X)}^{6}\right)dt+C_{4}\mathbb{E}\left(\kappa_{1}\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(\mathbb{R})}+\kappa_{2}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
F\left(u^{\varepsilon}_{0}\right)+2AC_{3}\mathbb{E}\int_{0}^{T}\left(1+C_{5}\right)dt+C_{6}=F\left(u^{\varepsilon}_{0}\right)+2AC_{3}T(1+C_{5})+C_{6}\leq
C_{7}.$

Let $\varepsilon_{0}>0$ be fixed. Then for all $0<\varepsilon<\varepsilon_{0}$
one has

$\displaystyle\varepsilon$
$\displaystyle\mathbb{E}\left(\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{2}(X))}\right)=\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}\left[u^{\varepsilon}(t,x)\right]^{2}dxdt+\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}\left[u_{2x}^{\varepsilon}(t,x)\right]^{2}dxdt$
$\displaystyle\leq\varepsilon
C_{8}+\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}\left[u_{2x}^{\varepsilon}(t,x)\right]^{2}dxdt=\varepsilon
C_{8}+\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}\frac{1}{p(x)}p(x)\left[u_{2x}^{\varepsilon}(t,x)\right]^{2}dxdt$
$\displaystyle\leq\varepsilon
C_{8}+\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}\frac{1}{\delta}p(x)\left[u_{2x}^{\varepsilon}(t,x)\right]^{2}dx\leq\varepsilon
C_{8}+\frac{1}{\delta}\varepsilon\mathbb{E}\int_{0}^{T}\int_{X}p(x)\left[u_{2x}^{\varepsilon}(t,x)\right]^{2}dx$
$\displaystyle\leq\varepsilon C_{8}+\frac{1}{\delta}C_{7}\leq
C_{9}(\varepsilon+\frac{1}{\delta})\leq
C_{9}(\varepsilon_{0}+\frac{1}{\delta}),$

which proves the formula (2.7). Moreover, we have

$\displaystyle\mathbb{E}$
$\displaystyle\left(\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{1}(-k,k))}\right)=\mathbb{E}\int_{0}^{T}\int_{-k}^{k}\left[u^{\varepsilon}(t,x)\right]^{2}dxdt+\mathbb{E}\int_{0}^{T}\int_{-k}^{k}\left[u_{x}^{\varepsilon}(t,x)\right]^{2}dxdt$
$\displaystyle\leq
C_{10}+\mathbb{E}\int_{0}^{T}\int_{-k}^{k}\left[u_{x}^{\varepsilon}(t,x)\right]^{2}dx\leq
C_{10}+\mathbb{E}\int_{0}^{T}\int_{X}\left[u_{x}^{\varepsilon}(t,x)\right]^{2}dx$
$\displaystyle\leq
C_{10}+\mathbb{E}\int_{0}^{T}\int_{X}\frac{1}{p^{\prime}(x)}p^{\prime}(x)\left[u_{x}^{\varepsilon}(t,x)\right]^{2}dx.$

Since $p^{\prime}(x)$ is bounded from below on every compact set $X$ by
positive number $\alpha_{X}>0$, then

$\displaystyle\mathbb{E}\left(\left|u^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{1}(-k,k))}\right)$
$\displaystyle\leq
C_{10}+\frac{1}{\alpha_{X}}\mathbb{E}\int_{0}^{T}\int_{X}p^{\prime}(x)\left[u_{x}^{\varepsilon}(t,x)\right]^{2}dx\leq
C_{10}+\frac{1}{\alpha_{X}}C_{7}\leq C_{11}.$

This proves inequality (2.8). ∎

###### Proof of Lemma 2.5.

Let $k\in X_{k}$ be arbitrary and fixed and let
$0<\varepsilon<\varepsilon_{0}\leq 1$. Then

(3.4) $\displaystyle u^{\varepsilon}(t,x)$
$\displaystyle=u_{0}^{\varepsilon}(x)-\\!\int_{0}^{t}\\!\bigg{[}\varepsilon
u^{\varepsilon}_{4x}(t,x)+Au^{\varepsilon}(t,x)u^{\varepsilon}_{x}(t,x)+Bu^{\varepsilon}_{3x}(t,x)$
$\displaystyle\hskip 60.27759pt-
Cu^{\varepsilon}_{2x}(t,x)+Du^{\varepsilon}(t,x)\bigg{]}ds+\\!\int_{0}^{t}\\!\left(\Phi(u^{\varepsilon}(s,x))\right)dW(s).$

Denote

$\displaystyle J_{1}$ $\displaystyle:=u_{0}^{\varepsilon}(x);$ $\displaystyle
J_{2}:=-\varepsilon\int_{0}^{t}u^{\varepsilon}_{4x}(t,x)ds;$ $\displaystyle
J_{3}$
$\displaystyle:=-A\int_{0}^{t}u^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)ds;\quad$
$\displaystyle J_{4}:=-B\int_{0}^{t}u^{\varepsilon}_{3x}(t,x)ds;$
$\displaystyle J_{5}$
$\displaystyle:=C\int_{0}^{t}u^{\varepsilon}_{2x}(t,x)ds;$ $\displaystyle
J_{6}:=-D\int_{0}^{t}u^{\varepsilon}(t,x)ds;$ $\displaystyle J_{7}$
$\displaystyle:=\int_{0}^{t}\left(\Phi(u^{\varepsilon}(s,x))\right)dW(s).$

Now, we start estimating the above terms.

From the assumption,
$\mathbb{E}\left|J_{1}\right|^{2}_{W^{1,2}(0,T;H^{-2}(-k,k))}=C_{1}$, where
$C_{1}>0$.

Next, there exists a constant $C_{2}>0$, that

$\left|-\varepsilon
u^{\varepsilon}_{4x}(t,x)\right|_{H^{-2}(-k,k)}=\varepsilon\left|u^{\varepsilon}_{4x}(t,x)\right|_{H^{-2}(-k,k)}\leq
C_{2}\varepsilon\left|u^{\varepsilon}(s,x)\right|_{H^{2}(-k,k)}.$

So, due to Lemma 2.4, there exists a constant $C_{3}(k)>0$ such that

$\displaystyle\mathbb{E}$ $\displaystyle\left|-\varepsilon
u^{\varepsilon}_{4x}(t,x)\right|^{2}_{L^{2}(0,T;H^{-2}(-k,k))}=\mathbb{E}\int_{0}^{T}\left|-\varepsilon
u^{\varepsilon}_{4x}(t,x)\right|^{2}_{H^{-2}(-k,k)}ds$ $\displaystyle\leq
C_{2}^{2}\varepsilon^{2}\mathbb{E}\int_{0}^{T}\left|u^{\varepsilon}(s,x)\right|^{2}_{H^{2}(-k,k)}ds\leq
C_{3}(k).$

Therefore we can write $\hskip
4.30554pt\mathbb{E}\left|J_{2}\right|^{2}_{W^{1,2}(0,T,H^{-2}(-k,k))}\leq
C_{4}(k)$, where $C_{4}(k)>0$.

Now, we will use the following result from [4].

######  Lemma 3.2.

([4], p. 243) There exists a constant $C_{5}(k)$ such that the following
inequality holds
$\quad\left|u^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)\right|_{H^{-1}(-k,k)}\leq
C_{5}(k)\left|u^{\varepsilon}(s,x)\right|^{\frac{3}{2}}_{L^{2}(-k,k)}\left|u^{\varepsilon}(s,x)\right|^{\frac{1}{2}}_{H^{1}(-k,k)}.$

Due to Lemma 3.2 there exist positive constants $C_{6},C_{7}(k),C_{8}(k)$ such
that

$\displaystyle|-Au^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)|_{H^{-2}(-k,k)}=A\left|u^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)\right|_{H^{-2}(-k,k)}\leq
C_{6}A\left|u^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)\right|_{H^{-1}(-k,k)}$
$\displaystyle\leq
AC_{7}(k)\left|u^{\varepsilon}(s,x)\right|^{\frac{3}{2}}_{L^{2}(-k,k)}\left|u^{\varepsilon}(s,x)\right|^{\frac{1}{2}}_{H^{1}(-k,k)}$
$\displaystyle\leq
AC_{7}(k)\left|u^{\varepsilon}(s,x)\right|_{L^{2}(-k,k)}\left|u^{\varepsilon}(s,x)\right|^{\frac{1}{2}}_{L^{2}(-k,k)}\left|u^{\varepsilon}(s,x)\right|^{\frac{1}{2}}_{H^{1}(-k,k)}$
$\displaystyle\leq
AC_{7}(k)\left[\left(2k\lambda_{X}^{2}\right)^{\frac{1}{2}}\right]\left|u^{\varepsilon}(s,x)\right|^{\frac{1}{2}}_{H^{1}(-k,k)}\leq
AC_{8}(k)\lambda_{X}\left|u^{\varepsilon}(s,x)\right|_{H^{1}(-k,k)}.$

Due to Lemma 2.4 there exists a constant $C_{9}(k)>0$, that we can write

$\displaystyle\mathbb{E}$
$\displaystyle\left|-Au^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)\right|^{2}_{L^{2}(0,T;H^{-2}(-k,k))}=\mathbb{E}\int_{0}^{T}\\!\\!\\!\left|-Au^{\varepsilon}(s,x)u^{\varepsilon}_{x}(s,x)\right|^{2}_{H^{-2}(-k,k)}ds$
$\displaystyle\leq
A^{2}C_{8}^{2}(k)\lambda_{X}^{2}\mathbb{E}\int_{0}^{T}\\!\\!\\!\left|u^{\varepsilon}(s,x)\right|^{2}_{H^{1}(-k,k)}ds=A^{2}C_{8}^{2}(k)\lambda_{X}^{2}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{1}(-k,k))}\leq
A^{2}C_{9}(k)\lambda_{X}^{2}.$

Threfore, we obtain $\hskip
4.30554pt\mathbb{E}\left|J_{3}\right|^{2}_{W^{1,2}(0,T,H^{-2}(-k,k))}\leq
C_{10}(k)$, where $C_{10}(k)>0$.

Next, there exist constants $C_{11},C_{12}>0$, that

$\displaystyle\left|-Bu^{\varepsilon}_{3x}(t,x)\right|_{H^{-2}(-k,k)}$
$\displaystyle=B\left|u^{\varepsilon}_{3x}(t,x)\right|_{H^{-2}(-k,k)}\leq
BC_{11}\left|u^{\varepsilon}(s,x)\right|_{H^{1}(-k,k)}$ $\displaystyle\leq
BC_{12}\left|u^{\varepsilon}(s,x)\right|_{H^{2}(-k,k)}.$

Lemma 2.4 implies the existence of a constant $C_{13}(k)>0$, that we can write
the following estimates

$\displaystyle\mathbb{E}$
$\displaystyle\left|-Bu^{\varepsilon}_{3x}(t,x)\right|^{2}_{L^{2}(0,T;H^{-2}(-k,k))}=\mathbb{E}\int_{0}^{T}\left|-Bu^{\varepsilon}_{3x}(t,x)\right|^{2}_{H^{-2}(-k,k)}ds$
$\displaystyle\leq
B^{2}C^{2}_{12}\mathbb{E}\int_{0}^{T}\left|u^{\varepsilon}(s,x)\right|^{2}_{H^{2}(-k,k)}ds=B^{2}C^{2}_{12}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(-k,k))}$
$\displaystyle\leq
B^{2}C^{2}_{12}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(\mathbb{R}))}\leq
B^{2}C_{13}.$

So, we obtain $\hskip
4.30554pt\mathbb{E}\left|J_{4}\right|^{2}_{W^{1,2}(0,T,H^{-2}(-k,k))}\leq
C_{14}$, where $C_{14}>0$.

For some constant $C_{15}>0$, we have

$\left|Cu^{\varepsilon}_{2x}(t,x)\right|_{H^{-2}(-k,k)}=C\left|u^{\varepsilon}_{2x}(t,x)\right|_{H^{-2}(-k,k)}\leq
CC_{15}\left|u^{\varepsilon}(s,x)\right|_{L^{2}(-k,k)}\leq
CC_{16}\left|u^{\varepsilon}(s,x)\right|_{H^{2}(-k,k)}.$

Lemma 2.4 implies the existence of a constant $C_{17}(k)>0$ such that

$\displaystyle\mathbb{E}$
$\displaystyle\left|Cu^{\varepsilon}_{2x}(t,x)\right|^{2}_{L^{2}(0,T;H^{-2}(-k,k))}=\mathbb{E}\int_{0}^{T}\left|Cu^{\varepsilon}_{2x}(t,x)\right|^{2}_{H^{-2}(-k,k)}ds$
$\displaystyle\leq
C^{2}C^{2}_{16}\mathbb{E}\int_{0}^{T}\left|u^{\varepsilon}(s,x)\right|^{2}_{H^{2}(-k,k)}ds=C^{2}C^{2}_{16}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(-k,k))}$
$\displaystyle\leq
C^{2}C^{2}_{16}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(\mathbb{R}))}\leq
C^{2}C_{17}.$

Hence, we received $\hskip
4.30554pt\mathbb{E}\left|J_{5}\right|^{2}_{W^{1,2}(0,T,H^{-2}(-k,k))}\leq
C_{18}$, where $C_{18}>0$.

There exists a constant $C_{19}>0$ such that

$\left|-Du^{\varepsilon}(t,x)\right|_{H^{-2}(-k,k)}=D\left|u^{\varepsilon}(t,x)\right|_{H^{-2}(-k,k)}\leq
DC_{19}\left|u^{\varepsilon}(s,x)\right|_{H^{2}(-k,k)}.$

Due to Lemma 2.4 for some constant $C_{20}(k)>0$, we obtain

$\displaystyle\mathbb{E}$
$\displaystyle\left|-Du^{\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{-2}(-k,k))}=\mathbb{E}\int_{0}^{T}\left|-Du^{\varepsilon}(t,x)\right|^{2}_{H^{-2}(-k,k)}ds$
$\displaystyle\leq
D^{2}C^{2}_{19}\mathbb{E}\int_{0}^{T}\left|u^{\varepsilon}(s,x)\right|^{2}_{H^{2}(-k,k)}ds=D^{2}C^{2}_{19}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(-k,k))}$
$\displaystyle\leq
D^{2}C^{2}_{19}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T;H^{2}(\mathbb{R}))}\leq
D^{2}C_{20}.$

This implies $\hskip
4.30554pt\mathbb{E}\left|J_{6}\right|^{2}_{W^{1,2}(0,T,H^{-2}(-k,k))}\leq
C_{21}$, where $C_{21}>0$.

Insert in Lemma 2.1 from [7] $f(s):=\Phi(u(s,x))$, $K\\!=\\!H\\!=\\!L^{2}(X)$.
Then $\mathscr{I}(f)(t)\\!=\\!\int_{0}^{t}\Phi(u(s,x))dW(s)$ and for all
$p\geq 1$ and $\alpha<\frac{1}{2}$ there exists a constant
$C_{22}(p,\alpha)>0$, that

$\mathbb{E}\left|\int_{0}^{t}\Phi(u^{m}(s,x))dW(s)\right|^{2p}_{W^{\alpha(p),2p}(0,T;L^{2}(X))}\leq
C_{22}(2p,\alpha)\mathbb{E}\left(\int_{0}^{T}\left|\Phi(u^{m}(s,x))\right|^{2p}_{L_{2}^{0}(L^{2}(X))}ds\right).$

Therefore, due to the condition (2.3), we can write

$\mathbb{E}\left|\int_{0}^{t}\Phi(u^{m}(s,x))dW(s)\right|^{2p}_{W^{\alpha,2p}(0,T;L^{2}(X))}\leq
C_{23}(p,\alpha),\mbox{~{}where~{}}C_{23}>0.$

Substitution $p:=1$ in the above inequality yields

(3.5)
$\mathbb{E}\left|J_{7}\right|^{2}_{W^{\alpha,2}(0,T;L^{2}(X))}=\mathbb{E}\left|\int_{0}^{t}\Phi(u(s,x))dW(s)\right|^{2}_{W^{\alpha,2}(0,T;L^{2}(X))}\leq
C_{23}(2,\alpha)=C_{24}(\alpha).$

Let $\beta\in\left(0,\frac{1}{2}\right)$ and
$\alpha\in\left(\beta+\frac{1}{2},\infty\right)$ be arbitrary fixed. Note,
that the following relations hold

  * $W^{\alpha,2}(0,T;L^{2}(\mathbb{R}))\subset W^{\alpha,2}(0,T;H^{-2}([-k,k))\quad$ and

  * $W^{1,2}(0,T,H^{-2}(-k,k))\subset W^{\alpha,2}(0,T,H^{-2}(-k,k))$.

Therefore, there exists a constant $C_{25}(\alpha)>0$, such that

$\displaystyle\mathbb{E}$
$\displaystyle\left|u^{m}(s,x)\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}^{2}=\mathbb{E}\left|\sum_{i=1}^{7}J_{i}\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}^{2}\leq\mathbb{E}\left(\sum_{i=1}^{7}\left|J_{i}\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\right)^{2}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{i=1}^{7}\left|J_{i}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}+2\sum_{i=1}^{6}\sum_{j=i+1}^{7}\left|J_{i}\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\left|J_{j}\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\sum_{i=1}^{7}\left|J_{i}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}+2\sum_{i=1}^{6}\sum_{j=i+1}^{7}\left(\left|J_{i}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}+\left|J_{j}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[8\sum_{i=1}^{7}\left|J_{i}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\right]=8\sum_{i=1}^{7}\left[\mathbb{E}\left|J_{i}\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\right]\leq
C_{25}(\alpha).$

Moreover, one has

  * $W^{\alpha,2}(0,T,H^{-2}(-k,k))\subset C^{\beta}(0,T;H^{-3}_{loc}(-k,k)\quad$ and

  * $W^{\alpha,2}(0,T,H^{-2}(\mathbb{R}))\subset W^{\alpha,2}(0,T,H^{-2}(-k,k))$.

So, there exist constants $C_{27}(k),C_{28}(k,\alpha)>0$, that

(3.6)
$\displaystyle\mathbb{E}\left|u^{\varepsilon}(s,x)\right|_{C^{\beta}(0,T;H^{-3}(-k,k)}^{2}\leq$
$\displaystyle
C_{26}\mathbb{E}\left|u^{\varepsilon}(s,x)\right|_{W^{\alpha,2}(0,T,H^{-3}(-k,k))}^{2}\leq
C_{27}(k,\alpha)$
$\displaystyle\mathbb{E}\left|u^{\varepsilon}(s,x)\right|_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}\leq$
$\displaystyle C_{28}(k,\alpha).$

Let $\eta>0$ be arbitrary fixed. Lemma 2.4 implies the existence of a constant
$C_{30}(k)>0$, that

(3.7)
$\displaystyle\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T,H^{-1}(-k,k))}\leq$
$\displaystyle
C_{29}(k)\mathbb{E}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T,H^{-1}(\mathbb{R}))}\tilde{C}_{2}=C_{30}(k).$

Substituting in [4, Lemma 2.1]
$\alpha_{k}:=\eta^{-1}2^{k}\left(C_{30}(k)+C_{27}(k,\alpha)+C_{28}(k,\alpha)\right)$
and using Markov inequality [12, p. 114] for
$X:=\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T,H^{-1}(-k,k))}+\left|u^{\varepsilon}(s,x)\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}+\left|u^{\varepsilon}(s,x)\right|_{C^{\beta}(0,T;H^{-3}_{loc}(-k,k)}^{2}$
and
$\varepsilon:=\eta^{-1}2^{k}\left(C_{30}(k)+C_{27}(k,\alpha)+C_{28}(k,\alpha)\right)$,
one obtains

$\displaystyle\mathbb{P}\Big{(}u^{\varepsilon}\in
A\left(\left\\{\alpha_{k}\right\\}\right)\Big{)}$
$\displaystyle=1-\mathbb{P}\Big{(}\left|u^{\varepsilon}(s,x)\right|^{2}_{L^{2}(0,T,H^{-1}(-k,k))}+\left|u^{\varepsilon}(s,x)\right|^{2}_{W^{\alpha,2}(0,T,H^{-2}(-k,k))}$
$\displaystyle+\left|u^{\varepsilon}(s,x)\right|_{C^{\beta}(0,T;H^{-3}_{loc}(-k,k))}^{2}\geq\eta^{-1}2^{k}\left(C_{30}(k)+C_{27}(k,\alpha)+C_{28}(k,\alpha)\right)\Big{)}$
$\displaystyle=1-\frac{C_{30}(k)+C_{27}(k,\alpha)+C_{28}(k,\alpha)}{\eta^{-1}2^{k}\left(C_{30}(k)+C_{27}(k,\alpha)+C_{28}(k,\alpha)\right)}=1-\frac{\eta}{2^{k}}>1-\eta.$

Let $K$ be a mapping such that for $\eta>0$,
$K\left(\eta\right):=A\left(\left\\{a_{k}^{(\eta)}\right\\}\right)$, where
$\left\\{a_{k}^{(\eta)}\right\\}$ is an increasing sequence of positive
numbers, which can, but does not have to, depend on $\eta$. Note, that due to
[4, Lemma 2.1], the set $K(\eta)$ is compact for all $\eta>0$. Moreover,
$\mathbb{P}\left\\{K\left(\eta\right)\right\\}>1-\eta$, then the family
$\mathscr{L}\left(u^{\varepsilon}\right)$ is tight. ∎

##  4\. Proof of Lemma 2.3

###### Proof.

Let $\left\\{e_{i}\right\\}_{i\in\mathbb{N}}$ be an orthonormal basis in space
$L^{2}(X)$. Denote by $P_{m}$, for all $m\in\mathbb{N}$, the orthogonal
projection on $Sp(e_{1},...,e_{m})$. Consider finite dimensional Galerkin
approximation of the problem (2.6) in space $P_{m}L^{2}(X)$ in the form

(4.1)
$\begin{cases}du^{m,\varepsilon}(t,x)+\left(\varepsilon\theta\left(\frac{\left|u^{m,\varepsilon}_{4x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{4x}(t,x)+A\theta\left(\frac{\left|u^{m,\varepsilon}_{x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{x}(t,x)\right.\\\
\left.+B\theta\left(\frac{\left|u^{m,\varepsilon}_{3x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{3x}(t,x)-C\theta\left(\frac{\left|u^{m,\varepsilon}_{2x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{2x}(t,x)+Du^{m,\varepsilon}(t,x)\right)dt\\\
=P_{m}\Phi\left(u^{m,\varepsilon}(t,x)\right)dW^{m}(t)\\\
u^{m,\varepsilon}_{0}(x)=P_{m}u^{\varepsilon}(0,x),\end{cases}$

where $\theta\in C^{\infty}(\mathbb{R})$ fulfills conditions

(4.2) $\begin{cases}\theta(\xi)=1,\quad&\textrm{when}\quad\xi\in[0,1]\\\
\theta(\xi)\in[0,1],\quad&\textrm{when}\quad\xi\in(1,2)\\\
\theta(\xi)=0,\quad&\textrm{when}\quad\xi\in\left.[2,\infty)\right..\end{cases}$

Let $m\in\mathbb{N}$ be arbitrary fixed and

$\displaystyle b(u(t,x)):=$
$\displaystyle\varepsilon\theta\left(\frac{\left|u^{m,\varepsilon}_{4x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{4x}(t,x)+A\theta\left(\frac{\left|u^{m,\varepsilon}_{x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{x}(t,x)$
$\displaystyle+B\theta\left(\frac{\left|u^{m,\varepsilon}_{3x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{3x}(t,x)-C\theta\left(\frac{\left|u^{m,\varepsilon}_{2x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{2x}(t,x)+Du^{m,\varepsilon}(t,x),$
$\displaystyle\sigma(u(t,x)):=$ $\displaystyle
P_{m}\Phi(u^{m,\varepsilon}(t,x)).$

Then

$\displaystyle\left|b(u(t,x))\right|_{L^{2}(X)}\leq\varepsilon\left|\theta\left(\frac{\left|u^{m,\varepsilon}_{4x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{4x}(t,x)\right|_{L^{2}(X)}$
$\displaystyle+A\left|\theta\left(\frac{\left|u^{m,\varepsilon}_{x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{x}(t,x)\right|_{L^{2}(X)}+B\left|\theta\left(\frac{\left|u^{m,\varepsilon}_{3x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{3x}(t,x)\right|_{L^{2}(X)}$
$\displaystyle+C\left|\theta\left(\frac{\left|u^{m,\varepsilon}_{2x}(t,x)\right|^{2}}{m}\right)u^{m,\varepsilon}_{2x}(t,x)\right|_{L^{2}(X)}+D\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}$
$\displaystyle=:\varepsilon J_{1}+AJ_{2}+BJ_{3}+CJ_{4}+DJ_{5}.$

Note, that

$J_{2}=\begin{cases}0,\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{x}(t,x)\right|\geq\sqrt{2m}\\\
\lambda\left|u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{x}(t,x)\right|_{L^{2}(X)},\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{x}(t,x)\right|\leq\sqrt{2m},\end{cases}$

where $\lambda\in[0,1]$, so

$\displaystyle
J_{2}\leq\left|u^{m,\varepsilon}(t,x)u^{m,\varepsilon}_{x}(t,x)\right|_{L^{2}(X)}\leq\sqrt{2m}\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}.$

Similarly,

$J_{1}=\begin{cases}0,\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{4x}(t,x)\right|\geq\sqrt{2m}\\\
\lambda\left|u^{m,\varepsilon}_{4x}(t,x)\right|_{L^{2}(X)},\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{4x}(t,x)\right|\leq\sqrt{2m},\end{cases}$

$J_{3}=\begin{cases}0,\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{3x}(t,x)\right|\geq\sqrt{2m}\\\
\lambda\left|u^{m,\varepsilon}_{3x}(t,x)\right|_{L^{2}(X)},\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{3x}(t,x)\right|\leq\sqrt{2m},\end{cases}$

and

$J_{4}=\begin{cases}0,\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{2x}(t,x)\right|\geq\sqrt{2m}\\\
\lambda\left|u^{m,\varepsilon}_{2x}(t,x)\right|_{L^{2}(X)},\quad\textrm{when}\quad\left|u^{m,\varepsilon}_{2x}(t,x)\right|\leq\sqrt{2m},\end{cases}$

where $\lambda\in[0,1]$, so

$J_{1},J_{3},J_{4}\leq\sqrt{2m}.$

Therefore one gets

$\displaystyle\left|b(u^{m,\varepsilon}(t,x))\right|_{L^{2}(X)}=\varepsilon
J_{1}+AJ_{2}+BJ_{3}+CJ_{4}+DJ_{5}$ $\displaystyle\leq$
$\displaystyle\,\varepsilon\sqrt{2m}+A\sqrt{2m}\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+B\sqrt{2m}+C\sqrt{2m}+D\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}$
$\displaystyle=$
$\displaystyle\left(A\sqrt{2m}+D\right)\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+\sqrt{2m}\left(\varepsilon+B+C\right).$

Moreover, due to the condition (2.3), there exist constants
$\kappa_{1},\kappa_{2}>0$, such that

$\left\|\Phi(u^{m}(t,x))\right\|_{L_{0}^{2}(L^{2}(X))}\leq\kappa_{1}\left|u^{m}(t,x)\right|_{L^{2}(X)}+\kappa_{2},$

so

$\displaystyle\left|b(u^{m,\varepsilon}(t,x))\right|_{L^{2}(X)}+\left\|\sigma(u^{m}(t,x))\right\|_{L_{0}^{2}(L^{2}(X))}$
$\displaystyle\leq$
$\displaystyle\left(A\sqrt{2m}+D\right)\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+\sqrt{2m}\left(\varepsilon+B+C\right)+\kappa_{1}\left|u^{m}(t,x)\right|_{L^{2}(X)}+\kappa_{2}$
$\displaystyle=$
$\displaystyle\left(A\sqrt{2m}+D+\kappa_{1}\right)\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+\sqrt{2m}\left(\varepsilon+B+C\right)+\kappa_{2}.$

Let $\kappa:=\max\left\\{\kappa_{1},\kappa_{2}\right\\}$ and
$\Lambda=\max\left\\{A,\varepsilon+B+C\right\\}$. Then

$\displaystyle\left|b(u^{m,\varepsilon}(t,x))\right|_{L^{2}(X)}+\left\|\sigma(u^{m}(t,x))\right\|_{L_{0}^{2}(L^{2}(X))}$
$\displaystyle\leq$
$\displaystyle\left(\Lambda\sqrt{2m}+\kappa+D\right)\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+\Lambda\sqrt{2m}+\kappa+D$
$\displaystyle=$
$\displaystyle\left(\Lambda\sqrt{2m}+\kappa+D\right)\left(\left|u^{m,\varepsilon}(t,x)\right|_{L^{2}(X)}+1\right).$

Therefore, by [9, Proposition 3.6], and [9, Proposition 4.6], for all
$m\in\mathbb{N}$ there exists a martingale solution to (4.1). Moreover,
applying the same methods as in section 3 one can show that for all $m$ the
following inequalities hold

$\displaystyle\exists_{C_{1}(\varepsilon)>0}\mathbb{E}\left(\left|u^{m,\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{2}(X))}\right)$
$\displaystyle\leq$ $\displaystyle\tilde{C}_{1}(\varepsilon),$
$\displaystyle\forall_{k\in
X_{k}}\exists_{C_{2}(k,\varepsilon)>0}\mathbb{E}\left(\left|u^{m,\varepsilon}(t,x)\right|^{2}_{L^{2}(0,T;H^{1}(-k,k))}\right)$
$\displaystyle\leq$ $\displaystyle\tilde{C}_{2}(k,\varepsilon)$

and the family of distributions $\mathscr{L}(u^{m,\varepsilon})$ is tight in
$L^{2}(0,T;L^{2}(X))\cap C(0,T;H^{-3}(X))$. Then application of the same
methods as on pages 2.2–3 leads to proof of the existence of martingale
solution to (2.6) with the conditions (2.3), (2.4) and (2.5). ∎

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