For z
∈ B0, ˘
x
i
+ ˘ x
j
: | y
i
+ z
i
− y
j
+ z
j
| − ˘y
i
+ ˘ z
i
+ ˘ y
j
+ ˘ z
j
≤ |z
i
| + |z
j
| − 2˘x
i
+ ˘ x
j
+ |˘z
i
| + |˘z
j
| ≤ 2|z| − 2˘x
i
+ ˘ x
j
thus y + z
∈ D
c i j
. This proves that B
y, ˘ x
i
+ ˘ x
j
⊂ D
c i j
, hence N
D
i j
x
= N
D
i j
x, ˘ x
i
+ ˘ x
j
. The general Uniform Exterior Sphere property does not hold, but the property restricted to
A
g
holds for D
i j
with constant α
i j
= 2r
−
because ˘ x
i
+ ˘ x
j
≥ 2r
−
for x
∈ A
g
. For
x
∈ A
g
such that x
∈ ∂ D
i j
, let us define l
i j x
= n
i j
x. For another configuration y ∈ ∂ D
i j
:
l
i j x
. n
i j
y =
x
i
− x
j
2 ˘ x
i
+ ˘ x
j
. y
i
− y
j
2 ˘ y
i
+ ˘ y
j
+ x
j
− x
i
2 ˘ x
i
+ ˘ x
j
. y
j
− y
i
2 ˘ y
i
+ ˘ y
j
+− 1
2
2
+− 1
2
2
= x
i
− x
j
. y
i
− y
j
2 ˘ x
i
+ ˘ x
j
˘ y
i
+ ˘ y
j
+ 1
2 Since x
i
− x
j
. y
i
− y
j
≥ |x
i
− x
j
|
2
− |x
i
− x
j
|| y
i
− x
i
− y
j
+ x
j
| ≥ |x
i
− x
j
|
2
− p
2 |x
i
− x
j
||x − y|
and ˘ y
i
+ ˘ y
j
≤ ˘x
i
+ ˘ x
j
+ p
2 |x − y|, this leads to :
l
i j x
. n
i j
y ≥
|x
i
− x
j
| − p
2 |x − y|
2 ˘ x
i
+ ˘ x
j
+ p
2
|x − y|
+ 1
2 =
˘ x
i
+ ˘ x
j
˘ x
i
+ ˘ x
j
+ p
2
|x − y|
Consequently l
i j x
. n
i j
y ≥
Æ 1
− β
2 i j
as soon as |x − y| ≤
˘ x
i
+ ˘ x
j
p 2
1 Æ
1 −β
2 i j
− 1
. This proves that D
i j
has the Uniform Normal Cone property restricted to A
g
with any constant β
i j
∈]0, 1[ and with δ
i j
= p
2r
−
1 Æ
1 −β
2 i j
− 1
0. In particular, the property holds for any β
i j
with δ
i j
= r
−
β
2 i j
p 2.
For x
∈ ∂ A
g
, let us construct a unit vector l
x
satisfying assumption iv in proposition 3.4. We first have to define clusters of colliding globules :
C
x
i = ¦
j s.t. |x
i
− x
j
1
| = ˘x
i
+ ˘ x
j
1
, |x
j
1
− x
j
2
| = ˘x
j
1
+ ˘ x
j
2
, . . . , |x
j
k
− x
j
| = ˘x
j
k
+ ˘ x
j
for some j
1
, . . . , j
k
© and their centers of gravity x
′ i
= 1
♯C
x
i X
j ∈C
x
i
x
j
. Let l
x
=
v |v|
with for each i : v
i
= x
i
− x
′ i
and ˘
v
i
= r
−
r
+
− r
−
r
+
+ r
−
2 − ˘x
i
We need an upper bound on the norm of v. Since x
∈ A
g
, each ˘ x
i
is larger than r
−
and smaller than r
+
, thus |
r
+
+r
−
2
− ˘x
i
| ≤
r
+
−r
−
2
. Moreover, if j belongs to the cluster C
x
i around the i
th
globule, the distance between x
i
and x
j
is at most 2n − 1r
+
, thus |v
i
| ≤ 2n − 1r
+
:
|v|
2
=
n
X
i=1
|v
i
|
2
+ r
−
r
+
− r
− 2
n
X
i=1
r
+
+ r
−
2 − ˘x
i 2
≤ 4nn − 1
2
r
2 +
+ n r
2 −
4 4n
3
r
2 +
We also need lower bounds on the scalar products of v with normal vectors on the boundaries of
the D
i+
, D
i −
, D
i j
. If ˘ x
i
= r
+
, then ˘ v
i
=
r
−
r
+
−r
−
r
−
−r
+
2
thus v.n
i+
=
r
−
2
. Similarly, if ˘ x
i
= r
−
, then
v.n
i −
=
r
−
2
. If |x
i
− x
j
| = ˘x
i
+ ˘ x
j
then x
′ i
= x
′ j
thus :
v.n
i j
x = x
i
− x
′ i
.
x
i
−x
j
2 ˘ x
i
+ ˘ x
j
+ x
j
− x
′ j
.
x
j
−x
i
2 ˘ x
i
+ ˘ x
j
−
1 2
r
−
r
+
−r
−
r
+
+r
−
2
− ˘x
i
−
1 2
r
−
r
+
−r
−
r
+
+r
−
2
− ˘x
j
= x
i
− x
j
.
x
i
−x
j
2 ˘ x
i
+ ˘ x
j
−
1 2
r
−
r
+
−r
−
r
+
+ r
−
− ˘x
i
− ˘x
j
=
r
+
2r
+
−r
−
˘ x
i
+ ˘ x
j
−
r
−
r
+
+r
−
2r
+
−r
−
≥
r
−
2
155
because ˘ x
i
+ ˘ x
j
≥ 2r
−
. This proves that assumption iv in proposition 3.4 is satisfied with β
= r
−
2 p
4n
3
r
2 +
= r
−
4r
+
n p
n . The inequality
β p
2 maxmax
i
β
i+
, max
i
β
i −
, max
i, j
β
i j
holds as soon as β
i j
β
2
2. As seen above, the Uniform Normal Cone property restricted to A
g
holds for D
i j
with constants
β
i j
= β
2
4 and δ
i j
=
r
−
β
4
16 p
2
. So, thanks to proposition 3.4, we obtain that A
g
has the Uniform Exterior Sphere property with constant
α
A
g
= 2r
−
β =
r
2 −
2r
+
n p
n
and the Uniform Normal Cone property with constants
δ
A
g
=
r
−
β
4
32 p
2
=
r
5 −
2
13
p 2r
4 +
n
6
and β
A
g
= p
1 − β
2
2
= Ç
1 −
r
2 −
64r
2 +
n
3
Proof of theorems 2.2 and 2.4
As seen in the proof of proposition 4.1, the set A
g
of allowed globules configurations satisfies the assumptions of corollary 3.6. If the diffusion coefficients
σ
i
and ˘ σ
i
and the drift coefficients b
i
and ˘b
i
are bounded and Lipschitz continuous on A
g
for 1 ≤ i ≤ n, then the functions
σ x =
σ
1
x · · ·
· · · ˘
σ
1
x
... ..
. ..
. ...
... ...
.. .
.. .
... σ
n
x
· · · · · ·
˘ σ
n
x
and
bx =
b
1
x
˘b
1
x ..
. b
n
x
˘b
n
x
are bounded and Lipschitz continuous as well. Thus, for m = nd + 1 and
D = A
g
, equation 1 has a unique strong solution
X, and there exists a decomposition of the reflection term
R
·
nsdLs
so that it is equal to
n
X
j=1
Z
·
X
i
− X
j
2 ˘ X
i
+ ˘ X
j
sd L
i j
s , −
n
X
j=1
Z
·
1 2
d L
i j
s − Z
·
d L
i+
s + Z
·
d L
i −
s
1 ≤i≤n
with L
i j
= Z
·
1I
|X
i
s−X
j
s|= ˘ X
i
s+ ˘ X
j
s
c
i j
s dLs, L
+ i
= Z
·
1I
˘ X
i
s=r
−
c
i+
s dLs and L
− i
= Z
·
1I
˘ X
i
s=r
+
c
i −
s dLs for some measurable choice of the c
i j
, c
i+
, c
i −
. For convenience, the local time
1 2
L
i j
has been used in equation E
g
, and denoted by L
i j
again. For an even
C
2
function ϕ on R
d+1
with bounded derivatives, the function Φ x =
P
1 ≤i j≤n
ϕx
i
−
x
j
is C
2
with bounded derivatives and the drift function in equation E
ϕ g
is equal to −
1 2
∇Φ. Thus E
ϕ g
has a unique strong solution. Moreover, if Z = R
A
g
e
−Φx
d x
+∞, theorem 3.3 implies the time-reversibility of the solution with initial distribution d
µx =
1 Z
1I
A
g
xe
−Φx
d x.
156
4.2 Linear molecule model
The set of allowed chains configurations is the intersection of 2n + 1 sets : A
c
=
n −1
\
i=1
D
i −
\
n −1
\
i=1
D
i+
\ D
−
\ D
+
\ D
=
where D
i −
= ¦
x ∈ R
d n+2
, |x
i
− x
i+1
| ≥ ˘x
−
© D
i+
= ¦
x ∈ R
d n+2
, |x
i
− x
i+1
| ≤ ˘x
+
© D
−
= ¦
x ∈ R
d n+2
, ˘ x
−
≥ r
−
© D
+
= ¦
x ∈ R
d n+2
, ˘ x
+
≤ r
+
© D
=
= ¦
x ∈ R
d n+2
, ˘ x
−
≤ ˘x
+
© The boundaries of these sets are smooth, and simple derivation computations give the unique unit
vector normal to each boundary at each point of ∂ A
c
we are not interested in other points. Actually,
D
−
, D
+
and D
=
are half-spaces, which makes the computations and checking of UES and UNC properties very simple.
• The vector n
−
= 0, . . . , 0, 1, 0 is normal to the boundary ∂ D
−
at each point x
∈ ∂ D
−
• Similarly, n
+
= 0, . . . , 0, 0, −1 is normal to the boundary ∂ D
+
at each point x
∈ ∂ D
+
• The vector normal to the boundary ∂ D
=
at each point x
∈ ∂ D
=
is n
=
= 0, . . . , 0,
−1 p
2
,
1 p
2
• At every x ∈ A
c
∩ ∂ D
i −
for 1 ≤ i ≤ n − 1, the vector normal to ∂ D
i −
is n = n
i −
x defined
by : n
i
= x
i
− x
i+1
˘ x
−
p 3
n
i+1
= x
i+1
− x
i
˘ x
−
p 3
˘ n
−
= − 1
p 3
other components equal zero
• At every x ∈ A
c
∩ ∂ D
i+
, the vector normal to ∂ D
i+
is n = n
i+
x defined by :
n
i
= x
i+1
− x
i
˘ x
+
p 3
n
i+1
= x
i
− x
i+1
˘ x
+
p 3
˘ n
+
= 1
p 3
other components equal zero The proof of theorem 2.5 is similar to the proof of theorems 2.2 and 2.4, and will be omitted. It
relies on the following proposition and uses theorem 3.3 and corollary 3.6 to obtain the existence, uniqueness, and reversibility of the solution of
E
c
. As in the proof of theorem 2.2, the local times are multiplied by a suitable constant to provide a more convenient expression.
So, in order to prove theorem 2.5, we only have to check that A
c
satisfies the assumptions of proposition 3.4, i.e. that the following proposition holds :
Proposition 4.2. A
c
satisfies properties U ES
r
2 −
2r
+
n p
2n
and U N C
s 1
− r
2 −
24r
2 +
n
3
, r
5 −
p 3
2
12
r
4 +
n
6
.
Moreover, for each x in
∂ A
c
, the set N
A
c
x
of normal vectors is equal to :
n ∈ S
d n+2
, n =
X
∂ D
i −
∋x
c
i −
n
i −
x +
X
∂ D
i+
∋x
c
i+
n
i+
x + 1I
∂ D
−
xc
−
n
−
+ 1I
∂ D
+
xc
+
n
+
+ 1I
∂ D
=
xc
=
n
=
with c
i −
, c
i+
, c
−
, c
+
, c
=
≥ 0
157
Proof of proposition 4.2
