process X
t
cannot reach the boundary ∂ D
1
before the first ǫ-decoupling time τ
1
, and therefore we can consider that X
t
is a free Brownian motion in R
d
, that is, we can reduce the proof of Theorem 3.1 to the case when D
1
= R
d
. We will give the proof of the Theorem 3.1 first in the 1-dimensional case, then we will extend it to
the case of polygonal domains in R
d
, and we will conclude with the proof in the general case.
3.1 The 1-dimensional case
From Remark 3.4 it follows that in order to construct the mirror coupling in the 1-dimensional case, it suffices to consider D
1
= R and D
2
= 0, a, and to show that for an arbitrary choice x ∈ [0, a] of the starting point of the mirror coupling,
ǫ ∈ 0, a sufficiently small and W
t t
≥0
a 1-dimensional Brownian motion starting at W
= 0, we can construct a strong solution on [0, τ
1
] of the following system
X
t
= x + W
t
3.7 Y
t
= x + Z
t
+ L
Y t
3.8 Z
t
= Z
t
G Y
s
− X
s
dW
s
3.9 where
τ
1
= inf s
0 : |X
s
− Y
s
| ǫ is the first
ǫ-decoupling time and the function G : R
→ M
1 ×1
≡ R is given in this case by G x =
¨ −1,
if x 6= 0
+1, if x = 0
. 3.10
Remark 3.5. Before proceeding with the proof, it is worth mentioning that the heart of the con-
struction is Tanaka’s formula. To see this, consider for the moment a = ∞, and note that Tanaka
formula x + W
t
= x + Z
t
sgn x + W
s
dW
s
+ L
t
x + W gives a representation of the reflecting Brownian motion
x + W
t
in which the increments of the martingale part of
x + W
t
are the increments of W
t
when x +W
t
∈ [0, ∞, respectively the opposite minus of the increments of W
t
in the other case L
t
x + W denotes here the local time at 0 of x + W
t
. Since x + W
t
∈ [0, ∞ is the same as x + W
t
= x + W
t
, from the definition of the function G it follows that the above can be written in the form
x + W
t
= x + Z
t
G
x + W
s
− x + W
s
dW
s
+ L
x+W t
, which shows that a strong solution to 3.7 – 3.9 above in the case a =
∞ is given explicitly by X
t
= x + W
t
, Y
t
= x + W
t
and Z
t
= R
t
sgn x + W
s
dW
s
. We have the following:
511
Proposition 3.6. Given a 1-dimensional Brownian motion W
t t
≥0
starting at W = 0, a strong
solution on [ 0,
τ
1
] of the system 3.7 – 3.9 is given by
X
t
= x + W
t
Y
t
= a −
x + W
t
− a Z
t
= R
t
sgn W
s
sgn a − W
s
dW
s
,
where τ
1
= inf ¦
s 0 :
X
s
− Y
s
ǫ ©
and sgn x =
¨ +1,
if x ≥ 0
−1, if x
. Proof.
Since ǫ a, it follows that for t
≤ τ
1
we have X
t
= x + W
t
∈ −a, 2a, and therefore Y
t
= a −
x + W
t
− a =
− x + W
t
, x + W
t
∈ −a, 0 x + W
t
, x + W
t
∈ [0, a] 2a
− x − W
t
, x + W
t
∈ a, 2a .
3.11
Applying the Tanaka-Itô formula to the function f z = |a − |z − a|| and to the Brownian motion
X
t
= x + W
t
, for t ≤ τ
1
we obtain Y
t
= x +
Z
t
sgn x + W
s
sgn a − x − W
s
d x + W
s
+ L
t
− L
a t
= x +
Z
t
sgn x + W
s
sgn a − x − W
s
dW
s
+ Z
t
ν
D
2
Y
s
d
L
s
+ L
a s
,
where L
t
= sup
s ≤t
x + W
s −
and L
a t
= sup
s ≤t
x + W
s
− a
+
are the local times of x + W
t
at 0, respectively at a, and
ν
D
2
0 = +1, ν
D
2
a = −1. From 3.11 and the definition 3.10 of the function G we obtain
sgn x + W
s
sgn a − x − W
s
=
−1, x + W
s
∈ −a, 0 +1,
x + W
s
∈ [0, a] −1,
x + W
s
∈ a, 2a =
¨ +1,
X
s
= Y
s
−1, X
s
6= Y
s
= G Y
s
− X
s
, and therefore the previous formula can be written equivalently
Y
t
= x + Z
t
+ Z
t
ν
D
2
Y
s
d L
Y s
, where
Z
t
= Z
t
G Y
s
− X
s
dW
s
and L
Y t
= L
t
+ L
a t
is a continuous nondecreasing process which increases only when x + W
t
∈ {0, a}, that is only when Y
t
∈ ∂ D
2
. 512
3.2 The case of polygonal domains
