1 Introduction
1.1 Notation
1. Probability: The symbol E∁ will denote the complement of the event E, and X
⊕ Y will denote the independent sum of the random variables X and Y .
2. Topology and measures: The boundary of the set B will be denoted by
∂ B and the closure of B will be denoted by clB, that is clB := B
∪∂ B; the interior of B will be denoted by ˙B and B
ε
will denote the ε-neighborhood of B. We will also use the notation ˙
B
ε
:= { y ∈ B : B
ε
+ y ⊂ B}, where B + b :=
{ y : y − b ∈ B} and B
t
:= {x ∈ R
d
: |x| t}. By a bounded rational rectangle
we will mean a set B ⊂ R
d
of the form B = I
1
× I
2
× · · · × I
d
, where I
i
is a bounded interval with rational endpoints for each 1
≤ i ≤ d. The family of all bounded rational rectangles will be denoted by
R. M
f
R
d
and M
1
R
d
will denote the space of finite measures and the space of probability measures, respectively, on R
d
. For µ ∈ M
f
R
d
, we define kµk := µR
d
. |B| will denote the Lebesgue measure of B. The symbols “
w
⇒” and“
v
⇒” will denote convergence in the weak topology and in the vague topology, respectively.
3. Functions: For f , g
0, the notation f x = O gx will mean that f x ≤ C gx if x x with some x
∈ R, C 0; f ≈ g will mean that f g tends to 1 given that the argument tends to an appropriate limit. For N
→ R functions the notation f n = Θgn will mean that c
≤ f ngn ≤ C ∀n, with some c, C 0. 4.
Matrices: The symbol I
d
will denote the d-dimensional unit matrix, and r A will denote the
rank of a matrix A.
5. Labeling: In this paper we will often talk about the ‘i
th
particle’ of a branching particle system. By this we will mean that we label the particles randomly, but in a way that does not depend
on their spatial position.
1.2 A model with self-interaction
Consider a dyadic i.e. precisely two offspring replaces the parent branching Brownian motion BBM in R
d
with unit time branching and with the following interaction between particles: if Z denotes the process and Z
i t
is the i
th
particle, then Z
i t
‘feels’ the drift 1
n
t
X
1 ≤ j≤n
t
γ · Z
j t
− · ,
where γ 6= 0 , that is the particle’s infinitesimal generator is
1 2
∆ + 1
n
t
X
1 ≤ j≤n
t
γ · Z
j t
− x · ∇.
1.1 Here and in the sequel, n
t
is a shorthand for 2
⌊t⌋
, where ⌊t⌋ is the integer part of t. If γ 0, then
this means attraction, if γ 0, then it means repulsion.
1940
To be a bit more precise, we can define the process by induction as follows. Z is a single particle at
the origin. In the time interval [m, m + 1 we define a system of 2
m
interacting diffusions, starting at the position of their parents at the end of the previous step at time m
− 0 by the following system of SDE’s:
dZ
i t
= dW
m,i t
+ γ
2
m
X
1 ≤ j≤2
m
Z
j t
− Z
i t
dt; i = 1, 2, . . . , 2
m
, 1.2
where W
m,i
, i = 1, 2, . . . , 2
m
; m = 0, 1, ... are independent Brownian motions.
Remark 1 Attractive interaction. If there were no branching and the interval [m, m + 1 were extended to [0,
∞, then for γ 0 the interaction 1.2 would describe the ferromagnetic Curie- Weiss model, a model appearing in the microscopic statistical description of a spatially homogeneous
gas in a granular medium. It is known that as m → ∞, a Law of Large Numbers, the McKean-Vlasov
limit holds and the normalized empirical measure ρ
m
t := 2
−m 2
m
X
i=1
δ
Z
i t
tends to a probability measure-valued solution of ∂
∂ t ρ =
1 2
∆ρ + γ
2 ∇ · ρ∇ f
ρ
, where f
ρ
x := R
R
d
|x − y|
2
ρd y. See p. 24 in [8] and the references therein. ⋄
Remark 2 More general interaction. It seems natural to replace the linearity of the interaction by a more general rule. That is, to define and analyze the system where 1.2 is replaced by
dZ
i t
= dW
m,i t
+ 2
−m
X
1 ≤ j≤2
m
g |Z
j t
− Z
i t
| Z
j t
− Z
i t
|Z
j t
− Z
i t
| dt; i = 1, 2, . . . , 2
m
, where the function g : R
+
→ R has some nice properties. In this paper we treat the gx = γx case. This is part of a future project with J. Feng.
⋄
1.3 Existence and uniqueness
Notice that the 2
m
interacting diffusions on [m, m+1 can be considered as a single 2
m
d-dimensional Brownian motion with linear and therefore Lipschitz drift
b : R
2
m
d
→ R
2
m
d
:
b x
1
, x
2
, ..., x
d
, x
1+d
, x
2+d
, ..., x
2d
, ..., x
1+2
m
−1d
, x
2+2
m
−1d
, ..., x
2
m
d
=: γβ
1
, β
2
, ..., β
2
m
d T
, where
β
k
= 2
−m
x
bk
+ x
bk+d
+ ... + x
bk+2
m
−1d
− x
k
, 1 ≤ k ≤ 2
m
d, and b
k ≡ k mod d, 1 ≤ bk ≤ d. This yields existence and uniqueness for our model.
1941
1.4 Results on the self-interacting model
Kategori