1. Introduction
Stochastic Loewner evolution (SLE) is a family of random growth process introduced by Oded Schramm [1] to study the scaling limit of loop-erased random walk (LERW) and uniform spanning tree (UST). The family of random growth process is described by the classic Loewner differential equation driven by
, where
is a positive parameter, and
is a one-dimensional standard Brownian motion. The behavior of SLE trace depends on the real-valued parameter
; usually we write SLE as
to illustrate that the behavior of SLE traces is related to
. When
, the trace of
is a simple curve; when
, the trace is no longer a simple curve; when
, the trace fills the whole space.
SLE is an important and very cutting-edge research topic in today’s mathematics field, which involves random processes, complex analysis and statistical physics. It is closely related to the scale limit of the grid model in statistical physics. Many mathematicians believe that different
describes the scale limits of different discrete models. In statistical physics, the scale limits of many two-dimensional systems are conjectured by theoretical physicists to be conformal invariant under critical conditions, but it has not been not proven by rigorous mathematical methods. Since Oded Schramm introduced
, a lot of conjectures have been proven, see [2] - [8].
The stochastic coupling technique is a useful tool in studying reversibility of stochastic Loewner evolution (SLE). Dapeng Zhan proved the coupling of the chordal SLE in the process of proving the reversibility of the chordal SLE in [9]. He then proved the coupling of the annulus SLE and the whole-plane SLE in [10], and on this basis he proved that the whole-plane SLE is reversible, which is closely related with Julien Dub’s work on SLE couple relationships in [11] [12]. The stochastic couplings of strip SLE has not been studied so far. The research of this paper will lay the foundation for the study of stirp SLE reversibility.
This paper is organized as follows. In Section 2, we give some symbols that will be used frequently in this paper. The definition of strip
is introduced in Section 3. In Section 4, we construct a continuous local martingale M based on (4) (5), and then prove that M is bounded. On this basis, we prove that for
, there is a coupling of two strip
process on the strip domain.
2. Symbols
In this article, we will use the following symbols: Let
,
,
,
. The conformal map in this paper refers to a univalent analytic function. Let f be the conformal in
, and
, f is said to be conformal map from
onto
, denoted as
. Further, if
,
is points or collections in
, and f extension map
onto
, denoted as
.
Many of the functions in this text have two variables, the first of which represent time, and the second is not. In this case, We use
and
to represent the partial derivative of the first variable, and
is used to represent the differentials of
. We’ll use
frequently. For convenience, we will write 2 in the position of subscript, namely
.
3. Strip Loewner Equation
In this section we give a brief description of the definition and some basic concepts of the strip Loewner equation, and more detailed background can be found in [13] [14].
Definition 3.1. Let
,
. Let
be the solution of
(1)
For each
, let
be the set of
at which
is not defined. Then
and
are called the strip Loewner hulls and maps driven by
. For each
,
is a bounded random growth hull in
and
,
,
,
.
Let K is a bounded hull in
, and
. Then there is a constant
, and a map
determined by K, such that
,
,
.
is called the capacity of K with respect to
in
, denote
. Then, for the above strip Loewner hulls, the capacity of
is t.
Let
is a semi-martingale, whose stochastic part is
and drift part is a continuously differentiable function. Then
(2)
a.s. for any
,
exists. It is a continuous curve in
, who starte from
. We call
the strip Loewner trace driven by
. For each
,
is the unbounded branch of
. Particularly, when
,
is a simple curves, for each
,
.
On the other hand, Let
be a simple curves in
, and only intersecting with
when
. Let
be the capacity of
with respect to
in
. Then
is a continuous increase function, which maps
to
(
is a constant in
). there exist some
so that
is a strip Loewner trace driven by
.
Definition 3.2. Let
,
. Let
be the maximal solution to the SDE:
(3)
where
is a strip Loewner maps driven by
. We call the strip Loewner trace driven by
the strip
trace in
started from a with marked point b.
4. Coupling of Two Strip SLE Trace
In this chapter we will discuss the stochastic coupling of the traces of strip SLE. We prove that for
, when certain ODE is satisfied, we can couple two strip SLE trace. That is, we have the following theorem.
Theorem 4.1. Let
,
, Suppose
is a positive function that satisfies
(4)
(5)
Let
,
,
, then
, there is a coupling of two curves:
and
, such that for
,
(i)
is the strip
trace in
started from
with marked point
.
(ii) If
is a stopping time with respect to
, then conditioned on
, After a time-change,
, is the strip
trace in
started from
with marked point
, where
is the first time that
visits
, if such time not exist set to be
.
4.1. Ensemble
Let
,
,
and
are the strip Loewner map and trace driven by
. Define
Fix
,
, let
is the first time that
visits
. Define
Then
is a simple curves start from
, when
,
. Let
, then
is a continuous increase function, which maps
to
, where
.
is a strip Loewner trace driven by some
.
Let
be a strip Loewner trace map by
. For
, let
,
,
(6)
map
to
, map
to
.
Hence,
(7)
For
, let
(8)
(9)
By [15], Section 8.1
(10)
So for
,
(11)
From (6) we get
(12)
Differentiate (12) with respect to
, we get
(13)
Let
, then
Hence,
(14)
The Taylor expansion of
near
is:
(15)
Let
, from (7), (15) and L’Hopital’s Rule,
(16)
Differentiate (14) with respect to
. Let
, from (7), (15)and L’Hopital’s Rule,
(17)
and
map
conformal onto
, map
conformal onto
. So exist
, such that
(18)
Similarly exist
, such that
(19)
From (6),
(20)
Similarly,
(21)
Comparing (20) with (21), we get
(22)
Define
,
,
:
(23)
From (18), (19), (23),
(24)
Since
is an odd function,
is an even. Define
(25)
Differentiate (11) with respect to z, we have
(26)
so
(27)
(28)
From (26) and (28) we have
(29)
Differentiate (29) with respect to z, we have
(30)
Let
in (11), (27), (29), (30) we have
(31)
(32)
(33)
(34)
From (34) we have
(35)
Define
(36)
As
, when
,
,
,
, so
. Hence from (35), (36) we have
(37)
4.2. Transformations of ODE
Lemma 4.2. If positive function
satisfy (4),
, then
(38)
Proof.
so
Integral on both sides, we have
i.e.
so
Thus
From (4), we get
Hence,
Differentiate with respect to x, we have
4.3. Martingales in Two Time Variables
Let
be as Theoerm 4.1. Let
be two independent Brownian motion,
. For
, Let
, be the solution of
(39)
then
are independent. When
,
is a.s. a simple curves, denoted by
.
are driven by
, Thus, They are
-adapted.
is
-adapted,
is
-adapted, so
are
-adapted.
is determined by
, hence,
is
-adapted. From (6) we get,
is
-adapted. From (7) we have,
is
-adapted. From (8), (9), (23) we get,
and
are
-adapted.
Fix
and a
-stopping time
. Let
, then
is a filtration.
is independent of
, so it is a
-Brownian motion. Hence, (39) is
-adapted SDE.
From (23), (16), (8), (11), we get
(40)
Let
,
,
. Suppose
satisfy (4). From (24), we define
:
(41)
From Itô formula and (4), (40) we have
(42)
From Itô formula and (17) we have
(43)
Let
From Itô formula and (43), we get
(44)
From (19) and (32) we get
(45)
Thus
(46)
Define
:
(47)
Lemma 4.3. Let
be a Itô process in
, let
, then
Proof. Process from Itô formula we have
Then
Thus
and
From (47), Lemma (3.1), (44), (46), (37), (42),
(48)
Define
:
(49)
Obviously, M is a positive, and
.
Proposition 4.4. (i) Fix any
-stopping time
,
is a
-adapted continuous local martingale, and
(50)
(ii) Fix any
-stopping time
,
is a
-adapted continuous local martingale, and
Proof. (i) From Lemma 4.1, we have
then
(51)
From (48) and (51) we get
(ii) Similarly,
Let
be the set of simple curves between
and
with only two endpoint in
, for
, let
be the first time that
visit J. Let
then
, when
,
. thus,
.
Proposition 4.5. (Boundedness) Fix
, then
is bounded on
by a constant depend on
and
only.
Proof. We say a function is uniformly bounded if the absolute value of function is bounded on
by a constant depend on
and
only.
Define
From [10], Lemma 4.4,
are uniformly bounded, when
, exist
, such that
(52)
is a decreasing function, so
is uniformly bounded.
and
are uniformly bounded, so
is uniformly bounded.
From (36) we get,
is uniformly bounded on
.
From (5) we have
so
is a continuous function with period
. Then
is uniformly bounded on
. Thus,
is uniformly bounded on
.
Since
It is suffices to proof that
is uniformly bounded on
.
From (49), (47) we have,
(53)
Let
,
. From (23) we see that
. It is suffices to proof that
and
are uniformly bounded. From (31) we have
and
are uniformly bounded, so
is uniformly bounded.
So
is uniformly bounded.
Let
, then
It is suffices to proof that
is uniformly bounded. In fact,
Similarly, we can prove that the other parts of the formula above are uniformly bounded. Thus,
is uniformly bounded is proved.
4.4. Coupling Measure
Let
denote the distribution of
,
. Let
.
and
be independent, so
is the joint distribution of
and
. Fix
, from the properties of local martingale and proposition 4.1,
.
Define
by
, then
is a probability measure. Let
and
are marginal measure of
.
So,
. Suppose
and
are the joint distribution of
. For each
, we have the joint distribution of
is
.
The proof of Theorem 4.1: Fix an
-stopping time
. From (39), (50) and Girsnov theorem. Under the measure of
, exist an
-Brownian motion
, such that
satisfy
-adapted SDE:
From the formula above and (6), (14) and Itô formula,
Since
,
, from (8), there is a Brownian motion
such that for
,
Thus, after a time-change,
, is a strip a
trace in
started from
with marked point
. This shows that, conditioning on
, after a time-change,
is a strip
trace in
started from
with marked point
.
5. Conclusion
In this paper, A bounded continuous local martingale M based on ordinary differential Equation ((4), (5)) is constructed. On this basis, we prove that for
, there is a coupling of two strip
traces on the strip domain. The method in this article can provide reference for the study of stochastic coupling of SLE on disk and other regions. The conclusion of this paper can be used to study the reversibility of SLE on the strip domain.