section 3 are given in a general frame for easier adaptation to other examples.
2 Two hard core models
2.1 Globules model
We want to construct a model for interacting globules. Each globule is spherical with random radius oscillating between a minimum and a maximum value. Its center is a point in R
d
, d ≥ 2. The
number n of globules is fixed. Globules configurations will be denoted by
x = x
1
, ˘ x
1
, . . . , x
n
, ˘ x
n
with x
1
, . . . , x
n
∈ R
d
and ˘
x
1
, . . . , ˘ x
n
∈ R where x
i
is the center of the i
th
globule and ˘ x
i
is its radius. An allowed globules configuration is a configuration
x satisfying
∀i r
−
≤ ˘x
i
≤ r
+
and ∀i 6= j |x
i
− x
j
| ≥ ˘x
i
+ ˘ x
j
So, in an allowed configuration, spheres do not intersect and their radii are bounded from below by the minimum value r
−
0 and bounded from above by the maximum value r
+
r
−
. In this paper, the symbol
| · | denotes the Euclidean norm on R
d
or R
d+1n
or some other Euclidean space, depending on the context.
Let A
g
be the set of allowed globules configurations : A
g
= ¦
x ∈ R
d+1n
, ∀i r
−
≤ ˘x
i
≤ r
+
and ∀i 6= j |x
i
− x
j
| ≥ ˘x
i
+ ˘ x
j
© The random motion of reflecting spheres with fluctuating radii is represented by the following
stochastic differential equation :
E
g
X
i
t = X
i
0 + Z
t
σ
i
XsdW
i
s + Z
t
b
i
Xsds +
n
X
j=1
Z
t
X
i
s − X
j
s ˘
X
i
s + ˘ X
j
s d L
i j
s ˘
X
i
t = ˘ X
i
0 + Z
t
˘ σ
i
Xsd ˘ W
i
s + Z
t
˘b
i
Xsds −
n
X
j=1
L
i j
t − L
+ i
t + L
− i
t In this equation,
Xs is the vector X
i
s, ˘ X
i
s
1 ≤i≤n
. The initial configuration X0 is an
A
g
-valued random vector. The W
i
’s are independent R
d
-valued Brownian motions and the ˘ W
i
’s are independent one-dimensional Brownian motions, also independent from the W
i
’s. The diffusion coefficients σ
i
and ˘ σ
i
, and the drift coefficients b
i
and ˘b
i
are functions defined on A
g
, with values in the d × d
matrices for σ
i
, values in R
d
for b
i
, and values in R for ˘ σ
i
and ˘b
i
. To make things simpler with the summation indices, we let L
ii
≡ 0. A solution of equation
E
g
is a continuous A
g
-valued process {Xt, t ≥ 0} satisfying equation E
g
for some family of local times L
i j
, L
+ i
, L
− i
such that for each i, j :
E
′ g
L
i j
≡ L
ji
, L
i j
t = Z
t
1I
|X
i
s−X
j
s|= ˘ X
i
s+ ˘ X
j
s
d L
i j
s , L
+ i
t = Z
t
1I
˘ X
i
s=r
+
d L
+ i
s and
L
− i
t = Z
t
1I
˘ X
i
s=r
−
d L
− i
s 144
Remark 2.1 : Condition E
′ g
means that the processes L
i j
, L
+ i
and L
− i
only increase when X is on
the boundary of the sets ¦
x, |x
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