Journal of Applied Mathematics and Stochastic Analysis, 16:1 (2003), 45-67.
Printed in the U.S.A. ©2003 by North Atlantic Science Publishing Company
QUASISTABLE GRADIENT AND HAMILTONIAN
SYSTEMS WITH A PAIRWISE INTERACTION
RANDOMLY PERTURBED BY WIENER PROCESSES
A. SKOROKHOD
Institute of Mathematics
Kiev, Ukraine
and
Michigan State University
Department of Statistics and Probability
East Lansing, MI 48824 USA
(Received February, 2002; Revised May, 2002)
Infinite systems of stochastic differential equations for randomly perturbed particle systems in
with pairwise interacting are considered. For gradient systems these equations are of the form
and for Hamiltonian systems these equations are of the form
Here
is the position of the th particle,
is its velocity,
where the function
is the potential of the system,
is a constant,
is a sequence of independent standard Wiener processes.
Let
be a sequence of different points in
with
, and
be a sequence in
. Let ~
be the trajectories of the
-particles gradient system for which
~
, and let
be the trajectories of the -particles Hamiltonian
system for which
,
,
. A system is called quasistable if for all
integers ! the joint distribution of
! or ~
! has a limit as
. We investigate conditions on the potential function and on the initial conditions under
which a system possesses this property.
Key words: Configuration Space, Gradient and Hamiltonian Systems, Stochastic Differential
Equations, Weak Convergence of Measures.
AMS subject classifications: 60H10, 60G46, 60K35.
1 Introduction
We consider a finite or an infinite sequence of
-valued stochastic processes
45
46
A. SKOROHOD
"
where is a finite or infinite interval of the natural numbers by which the evolution of a set of
pairwise interacting particles in the space
is described. The interaction is determined by a
potential
satisfying the condition
(P)
#
where the function #
is smooth, #
for
$ %, and for some &
the relation
lim $
$'
#
))
$
(
is fulfilled, %
,(
are constants.
The motion of a gradient system is determined by the system of the stochastic differential
equations
(1)
"
"
The number of equations coincides with the number of particles.
is the sequence of
independent standard Wiener processes in
. These equations are solved with some initial
conditions
"
, where
"
, is a sequence of different points in
, for which lim
if is an infinite interval.
The motion of a Hamiltonian system is determined by the system of the stochastic
differential equations
"
Here
initial conditions
"
is the velocity of the
(2)
th particle. Equations (2) are solved with the
,
where
" is another sequence in
.
The existence of the solution to infinite systems of kind (1) was considered for smooth
functions
by J. Fritz (see [2]). Finite gradient and Hamiltonian systems with the
potentials satisfying condition (P) were considered by the author (see [5]).
Quasistability. Let sequences
*
and
*
be given. For any
, there exists the solution to system (1)
, satisfying the initial
conditions
and the solution to system (2) ~
~
This follows from the results of [5].
satisfying the initial conditions
+
~
Quasistable Gradient
47
Gradient system (1) is called quasistable if for any ! the joint distribution of the
!
stochastic processes ~ (t),
! has as a limit
in the space , .
Hamiltonian system (2) is called quasistable if for any ! the joint distributions of the
stochastic processes ~
! has a limit
in the same space. Note that the
quasistability is determined by the potential and initial conditions.
The main goal of this article is the investigation of the conditions on the potential and
initial conditions of gradient and Hamiltonian systems under which they are quasistable.
2 The Spaces
and
It is convenient to consider gradient systems (1) in the configuration space which is the set of
locally finite counting measures on the Borel -algebra
of the space
. So, a
measure " satisfies the property:
(LF) the support . of the measure is a sequence
*
of different points in
for which
&/
for all /
/
if
.
The topology in
&
is generated by the weak convergence of measures:
/
for
0,
"
0
is the set of continuous functions
1
and for a continuous function
1
3
with bounded support. Denote
2
"
0
with bounded support
'
2
'
.
Using Ito's formula and considering the function ( as a function of two variables: (
we can rewrite system (1) using -valued function for which
1
2
"
0
'),
(3)
in the form
1
2
11
)
(
3
2
1
2
4
where
"
0
+%
))
(4)
48
and
A. SKOROHOD
is the set of those " 0 for which '
and ''
are continuous functions.
A -valued stochastic process
is called a weak solution to equation (4) if for all
" 0 the stochastic process
0
1
2
-1
)
(
is a martingale with respect to the filtration
1
If
3
$2
1
$ 25
$
(5)
with the square characteristic
*
2
)
1
)
$2
(6)
$
is a weak solution to system (4) and
1
for
$
" 0 , then the sequence
Compacts in
For any
2
"
is a weak solution to system (1).
and a continuous decreasing function
"
for which
,
there exists a continuous decreasing function
such that
'
'
' &
.
(8)
For any compact set 6 from and any function satisfying the conditions mentioned before,
there exists a function of the form given by relation (8) for which
sup 1
"6
3
2&
1
3
2* 7
Note that the set
is a compact in for any
of the form (8) and 7
" for which relation (7) is fulfilled. Set
. Denote by
the set of those
Quasistable Gradient
sup 1
2
1
2
" 8 9:
49
1
3
2
where
8 9:
"
sup
0
sup
1
3
2
)
)
(10)
)
with the distance
is a separable locally compact space.
It is convenient to consider Hamiltonian systems in the configuration space
the subspace of those locally finite counting measures ; in
for which
"
4
<
Let
1
4;
be a function from
;
2
18
2
;
1
( ;
which is
.;
where
are sequences in the space
solution to equation (2). Introduce a -valued function
;
(9)
. Let
be a
< "
"
. Ito's formula implies the relations
0
3 ;
2
(11)
where
8
+%
and
(
This means that the stochastic process ;
"
the stochastic process
;
1
)
(
;
2
(
is the solution to a martingale problem: for any
18
; $ 2
1
( ; $ 3 ; $ 2
$
(12)
is a continuous martingale with the square characteristic
1; 2
1
; $ 2 $
3 Some Properties of Solutions to Finite Systems
(13)
50
A. SKOROHOD
Assume that
and the function
satisfies the condition (P). In this section we
consider solutions to systems (1) and (2) for which
- /5
/
Lemma 1:a) Let
differential equations
/ be different points in
~
/
with initial value ~
b) Let
,
Proof: Set
( ~
. Then the system of stochastic
~
(14)
has a strong solution and this solution is unique.
/ be points in
. Then system (2) with the initial conditions
,
/ has a strong solution and this solution is unique.
#)
(
(15)
=
Denote by ~
/ the solution to system (14) in which the function (
is changed
to the function (
, the existence and the uniqueness of this solution is a consequence of the
Lipschitz condition of the function (
. Let
inf
/
.
/
Introduce the stopping times
inf ~
inf
Then for
&
&
~
(16)
/
we have
&
~
Let >
/
')
'
~
/
and set
?
/
> ~
~
. @ ~9
~
/
Then
?
.
~
~
9
9
~
(17)
9
where
S
7
@
(
(18)
Quasistable Gradient
.@
7@
@
7 7@ are positive constants, and
1
51
(
(19)
@
is a martingale with the square characteristic
, ~
2
and
~
$
/
$
$
/
,
A
/
9
9
9
Using formulas (18) and (19) we can prove that for some constant 8
depend on , the inequality holds
.
.@
9
9
9
8
9
/ which does not
/)
(20)
9
for any set of different points
x "
/
Formulas (14) and (17) imply that the stochastic process
?
8
/
+
is a non-negative supermartingale on the interval - + 5. So, for any +
relation
7
lim sup B sup ?
+
we have the
7
This implies the relation
B lim
Statement a) of the lemma is proved.
Let
/ be the solution to the system of the stochastic differential equations
/ (
/
with the initial conditions
/
52
A. SKOROHOD
The existence and the uniqueness of the strong solution follows from the Lipschitz property of
the function ( . Let below the stopping time
be determined by formula (16) with
~
instead of ~
. Set
#) $ =
$
C
/
/
/
It is easy to calculate that
C
/
/
So, the stochastic process
C
/
+
/#
where #
inf$ # $ , is a non-negative martingale on the interval - + 5 for all +
. Further proof of statement b) is the same as for statement a). The lemma is proved.
,
Free Particles Processes
We consider now the solution to equations (1) and (2) for
particles processes. The stochastic process
~E
D /
in the space (
. They are called free
E
/
/
is called an /-particles free gradient process, and the stochastic process
D
E
/
E
$
$
/
in the same space is called an /-particles free Hamiltonian process. We use the notations
~
D /
~
/
D
/
/
~ ! !
~ E be probability
for the stochastic processes introduced in Lemma 1. Let !
/
/
/
/
measures on the space - 5
which are the distributions of the stochastic processes
~
D /
D
/
~E
D /
D
E
/
on the interval - 5.
~ is absolutely continuous with respect to the measure
Lemma 2: a) The measure !
/
E
~
!/ , and the measure !/ is absolutely continuous with respect to the measure !E/ for all
.
b) Denote
Z/
and
?
4
4?/
?
"
5
/
Quasistable Gradient
~
!
/
~E
!
C/
!/
!E/
C/
~
/
C/
/
C/
53
.
Then
exp
~
d/
~E
D /
~
F / $ ~E $
$
~
~
F / $ ~E $
/
$
/
where
~
F/ $
~E $
(
~E $
(22)
/
and
d/
exp
F
/
E
$
D
$
3
E
/
$
F
/
/
$
E
$
$ ,
/
where
F
/
$
E
E
(
$
$ .
(24)
/
Proof: Denote by !
;/ the distribution of the stochastic process
~
D /
~
/
/
in the space - 5
. This stochastic process was introduced in the proof of Lemma 1.
~ G !
~ E and
It follows from the results of [5] that !
/
/
~
~
F / $ ~E $
exp
~E
D
~
!
H
~E
!
/
/
~
F / $ ~E $
$
/
~
where F / $
$
(25)
/
is calculated by formula (22) with ( instead of ( . It is easy to see that
~
D /
~
D /
~
/
~
/
for
(see formula (16)). This implies formula (21). Formula (23) is proved in the same
way. The lemma is proved.
4 Infinite Free Particles System
54
A. SKOROHOD
Let
be a sequence in
for which
" , and let
, *
arbitrary sequence in
We consider these two sequences of stochastic processes:
~E
E
*
$
$
be an
.
Lemma 3: a) The relation
B
~E
&7
(26)
&
is fulfilled for all
,7
if satisfies condition
(EGFPS) (existence of gradient free particle system) for all
exp
I
&
b) The relation
B
E
&7
(28)
&
is fulfilled for all
,7
if the sequences
satisfy the condition
(EHFPS) (existence of Hamiltonian free particle system) for all
,$
exp
$
(29)
&
Proof: It follows from Kolmogorov's theorem on the convergence of random series that
relation (26) is equivalent to the relation
B
E
.
& 7 &
(30)
Using the inequality
A
we can prove the existence of positive continuous functions
F
7 , <
7 ,
, 7
,
for which the inequalities below are held
F 7
exp
B
E
& 7
< 7
exp
A
This implies statement a). Statement b) can be proved in the same way. The lemma is proved.
Corollary 1: If condition (EGFPS) is fulfilled, then the relations
Quasistable Gradient
~E
1
55
E
2
"
determine a -valued continuous stochastic process ~ E
particles process.
If condition (EHFPS) is fulfilled, then the relations
E
1
which is called a gradient free
E
2
0
"
0
where ;
"
determine a -valued stochastic process
4
called a Hamiltonian free particle process.
Remark 1: Let
satisfy condition (EGFPS), and
which is
E
J
Then condition (EHFPS) is fulfilled.
Introduce for 1 ! & / the stopping times with respect to the filtration
generated by the sequence
~
!/
inf
min ~ E
!/
inf
min
~E
E
E
9
!
* /
%
9
!
* /
% .
Lemma 4: a) Let condition (EGFPS) be fulfilled. Then for any
relation
!/
is held.
b) Let condition (EHFPS) be fulfilled. Then for any
lim
B
/
,!*
the relation
!/
is held.
Proof: Let us prove statement a). It suffices to prove the relation
$
for any
,7
inf
9
/
~E $
9
7
. This follows from the inequality
B inf
$
The lemma is proved.
inf
9
/
~E $
9
, !*
~
lim
B
/
B inf
lim
/
*
B inf
7
9 /
$
~E $
7
the
56
A. SKOROHOD
5 Quasistability
First, we calculate the conditional expectation K ~ /
!
*
is generated by the Wiener processes
~
9=
/
/
( ~ E9 $
L
!
where the filtration
! . We use the notation
~E $
$
9
9
To simplify writing, we assume that
~
and %
. It follows from formula (21) that
exp ~ /
/
1~ / 2
where 1~ / 2 is the square characteristic of the martingale ~ / This implies the representation
~
~
/
&$ &
&$ &
~
$
/
(31)
$
/
Remark 2: Let
/
=
/
E
9
(
E
$
$
$
9
9
Then the density /
(see formula (23)) is represented by relation (31) with
Lemma 5: Let ! & /. Then
~
exp
K ~/
!/
H! / $
L
/
instead ~ /
!
(32)
$
H! / $
$
!
and for
!, the functions H! /
are determined by the formulas
H! /
&
*
&
<
&
(33)
!/
where
<
K
9
( ~ E9
MN
O
inf O'
(34)
O
~E
9
O
O 9O " 9O) 4
O)
~E
L
9
O
O
O&
( ~E
and
!/
!
O
O
Quasistable Gradient
MN
9O " -!
9 4
/
/5P
49/ 4
O
4
" - /5 3 - /5
/
O& / 9
/
57
/ O&
O
" -!
/5
Proof: First note that the right-hand side of formula (32) gives the general representation
!
for exponential martingales with respect to the filtration
*
with the expectation 1
If the functions H! /
! satisfy formula (32), then we can write the relation
&
*
&
&
H! /
&
&
*
&
*
!
~
K
K
H! /
!
&
&
&
9 =
&
&
&
( ~ E9O
/
~E
O
!
L
/
O
O
9O
O
L
!
O
9O
O
~
/
To determine the function H! / , we have to collect all the terms in the right-hand side of the
last equality representing the integral
H! / $
$
It is easy to see these terms correspond to the sequences 9O O
satisfy the conditions
9 4
49/ 4
4
4
" MN
Let
be a standard Wiener process in
conditional expectations is valid
K
#
and
$
$
which
O
/
, and $ & # &
$ L
O
#
. Then the formula for
$
Here K L
is the conditional expectation with respect to give
. This relation and
description of H! / imply formulas (33), (34). The lemma is proved.
Let the sequences 9O O
and O O
satisfying the condition
/
9 4 49 4 4
" MN
be fixed. Introduce notations
O
!
O
O
!
O
9
9
Denote
<
K
!/
( ~ E9O
E
O&
O
49
9 4
/4
(35)
~E
O
O
9O
9O
O
O
O
O
( ~E
~E
L
!
.
58
A. SKOROHOD
Here K E is the expectation with respect to the joint distribution of the Wiener processes
9"
We will use some graphs in the set D
9
O "
O
Denote by Q
the set of all connected graphs with the set of vertices
R Q
,
D
and the set of edges
K Q S
A graph Q) " Q
9O
9O OO "
O
is called minimal if
7( % K Q)
7( %
The set of all minimal graphs is denoted by T Q
Lemma 6: There exist some constants 7
which the inequality is valid
<
49
!/
exp
9 4
9O
O
,
.
, and a graph Q) " T Q
4
=
O
7
9O
O
~
K Q)
U
"K Q)
where
O
O
max O) & O , ( %
O
~
K Q)
~E
)
Q)
)
9O
Q)
O&
9O
9O 4
O)
O
O
~E
O
L
O&
O
O
9O) 4
KE
O"
O
O
V K Q)
Proof: Let
:"
W
W
:
W
W
We can write the inequality
<
KE
(
49
!/
9O
O
9 4
O
4
:
?O
O
L:
KE
O
9O
O&
KE
O
9O
O
O !
O
O
for
?O
W
LW
9O
O
O
O
O
W
LW
(36)
Quasistable Gradient
59
where
L
O
O
O&
It is easy to see that for some 7
, we can write the inequality
K(
?
:
: L
& 7
?"
So
K ( ?O
9O
:
O
L
7
O
O
O
: L
and
KE
(
9O
O
:
?O
O
O
L:
7
O
O
(37)
O
where
L
O
In addition, for some constant 7
we can write the inequality
KE
9O
9O
O
O
W
O
(38)
7
O
O
Now we proceed to estimation of the expectation
~
K
1
KE
9O
O
O
O
O
?O
Let a minimal graph Q) be given. We call the vertex
, the origin of the graph and denote it
by . For any " R Q) denote by O
the minimal number of edges connecting with
4 so O
. We call O
the Irvel of the vertex . Note that if , ) " K Q) , then
)
O
O )|
.
Now we construct the graph Q) for which the statement of the lemma is valid. Let
9
Denote by Q)
Q 9
#
% where
O#
max O &
9
" MN
, (%
%
the minimal graph with the edges
9 4
, (%
!
O
It is easy to see that the inequality is valid
-!
/5
%
#
x9O#
O#
60
A. SKOROHOD
~
K
1
KE
~E
9O
Q')
O"
~E
O
~
K Q)
O
O
(39)
where
Q)
O&
9O
"
O
9O
" K Q)
O
)
Q)
-
5X
Q)
Denote
Q)
O&
9O
"
O
9O
" K Q)
O
=
9O
O
and
Q)
Q)
Q)
Q)
*
*
Introduce the r.v.
O
~E
9O
O
~E
O
O&
O
?
It is easy to see that the sets of r.v. O O "
Q) are independent for different
)
as the sets O O "
Q . Using the Cauchy inequality we can write
~
K
KE
O
L
KE
Q)
O"
KE
O
KE
%*
O"
L
Q)
O"
L
O
%
as well
.
Q)
Note that
Q)
where the sets
Q)
Q) are determined by the sets
D
O"
%
Q)
9O
O
,
which are determined by their properties;
a)
D
D
)
)
)
b) D
D )
Y if
,
c) 7( % D
O
i0 " D
d) the set D contains only one element ? for which the relation O ?
fulfilled.
Property b) implies the relation
KE
O"
O
%
Q)
KE
O"
O
%
Q)
is
Quasistable Gradient
Let
be the Wiener process in
, and ? "
a constant 7
for which the inequality is held
K
. Then for any
7 exp
?
61
?
"
L
, there exists
=
So
KE
/
$9
/9
9
9
9
/
9
$9
L
9
)
7 exp
where
$9
$9
/
=
This implies the inequality
/9
6 /
/ 4$
)
7@ exp
$
and
?/
imply the relation
~
K
=
This inequality and the following one
)
9
*
9
7@7( % exp
9O
O"
9
$9
9
with some 7 @
/
9
9
=
O
9
~
K Q)
L
Q)
(40)
with some
. This formula and formulas (37),(38) imply the proof of the lemma.
The Condition of the Quasistability of Gradient Systems
Consider the graph Q) 9
9 4
introduced in the proof of Lemma 6. Let
O
X!5
and
%
7(
%
O
Set ?
% Denote by T Z Q ?
?% the set of minimal ordered
O
graphs Q with R Q
C and the origin ?% . Let B ?% be the set of invertible mapping
C
C for which
?%
?% For Q) , Q)) " T Z Q C 4?% we write Q) [ Q)) if
))
)
the relation Q
Q is fulfilled for some " ?% . For Q) " T Z Q C 4?% , we set
Q)
7( %
Q)
"
It is evident that Q)
Q)) if Q) [ Q)) .
Introduce the functions
U
%
Q) 4
?
?%
?%
62
A. SKOROHOD
exp
\
?% \
\ ?9
? \
?9 ?O "K Q)
?9 ?
9&
Lemma 7: Let a sequence of ordered graphs Q9) 9 8 from T Z Q ?
satisfy the conditions
a)
Q9) [
L Q) if 9 & ,
H
for any graph Q) " T Z Q ?
?% , the relation is true
?%
Q) [Q9)
9 8
Let
held
/
be the configuration with the support
<
9O
O
49
9 4
7%
4
Q)
O"
sup
Q ) " T Z Q C ?%
D
9 8
/5 . Then the inequality is
" -!
U
D
%
~
K
exp
L
Q9)
;
L
L
]
]%
=
/
/
]
]%
The proof follows from Lemma 6, and the formula
<
9O
<
O
!/
49
9 4
4
Q)
O"
49
!/
9 4
4
Q 9
[Q9)
9 4
9 8
We use the notation
F
?
%
sup
?%
U
%
Q9)
?
?%
"
?%
(41)
9 8
F
%
C
C
C
F
?
%
?%
?
?%
(42)
where C S .
, (% C
%
Theorem 1: Assume the value in condition (P) satisfies the inequality & , and the
initial configuration
"
for a gradient system satisfy condition (EGFPS and the
cJnditions:
1) for any
there exists constant (
for which the inequalities are
fulfilled
exp
]
log
\] \
]
Quasistable Gradient
&
2)
log
(
63
log
\] \
Denote
sup
F
F
C
%
C S
4
4
%
"C
then for all
the relation is valid
F
lim loglog
Under these conditions the gradient system is quasistable.
Proof: It suffices to prove the existence of a limit in probability
lim / !
/
and the relation
K lim
/
for all
.
Note that for any sequences
9
9
/!
4
" -!
3 -
there exists a limit in probability
lim<
/
49
!/
9 4
which we denote by
<
49
!
9 4
4
represented by the right part of equality (35). Introduce the stochastic process in
[E
D
[E
4
!
4
!
4[ E!
and functions
%
! /
4
%
by the relations
%
[E
4D !
! /
<
7( %
!
!/
49
[E
D !
9 4
X- !5 %
%
!
[E
4D !
[E
D !
4
43
64
A. SKOROHOD
<
7( %
49
!
9 4
.
4
44
X- !5 %
It follows from Lemma 7 that
\
%
[E
4D !
! /
7%
[E
D !
\= \
sup
~
K
Q ) " T Z Q C ?%
U
L ,
Q9)
%
exp
L
[E
4D !
!
%
[E
D !
L
\
45
=
]
]%
]
! /
[E
4D !
]%
9 8
It follows from Lemma 5 that
H! /
%
% / !
*
&
&
&
[E
D !
&
(46)
Set
H!
%
% / !
*
&
&
&
[E
4D !
!
[E
D !
&
(47)
Investigate the convergence of the series in formula (47). First we note that for some
and ,
, the inequality is valid
&
&
&
,
&
~
Now we obtain a proper estimation for K . Note that it depends on
so we will write
~
K Q)
~
K Q) 49
9O 4
O
O"
)
Q)
,
9 4
the function in the right-hand side of the equality depends only of those 9O 4 O for which
O " ) Q) Introduce the subsets of the set ) Q)
$ satisfying the conditions
9 9
)
9
9
and min
Q)
$
9
Y , (%
9
max
9
Set
E
9O
9
O"
For some ,
9
9&
,
O
9
we can write the inequality
E
O
O
% 9 & $ , (%
$
%
Quasistable Gradient
~
K Q)
KE
,
65
9
9 $
Let Q) be a minimal ordered graph with the vertices ]
It follows from condition 1) of the theorem that
U
%
Q)
]
]%
] % and the origin ] %
]
]%
%
(
%
%
(48)
where
log
%
Using Lemmas 6 and 7 for the estimation of K E
~
K Q9) 49
log
%
9
\
\
we can obtain the inequality
9 4
9
9
" MN
!
9 8
, (%
$
F
These inequalities and formula (45) imply the inequality
\
&
&
&
7% F
%
[E
D !
!
[E
D !
\
8
/
%
sup
$
~E $
%
%
sup exp
$
L$
$
=
(49)
where 7 is a constant, and 8 was introduced in Lemma 7. It can be shown that
%
lim
log 8 L% log %
.
This implies the local uniform convergence in of the series in formula (47).
This implies the continuity of the functions ;H ! %
for all % and continuity of the
function H! . Introduce the function
;
exp
H! $
$
\ H! $ \
!
Inequality (50) implies the relation
!/
in probability as /
;
. Introduce the stopping time
+
inf
\ H! $ \
!
$
+
$
(50)
66
A. SKOROHOD
Then the stochastic process ;
^
is a martingale, and
+
K;
^
+
The proof of the theorem is a consequence of the relation
B
+
lim
+
5.2 The Condition of Quasistability of Hamiltonian Systems
Let
/ be the solution to system (2) with the initial condition
/
/
,
"
/
"
where
/ is the set of different points from
.
Denote by /
the density of the distribution of the stochastic process
D
/
,D
$ $
/
/
$
$
/
with respect to the distribution of the stochastic process
$
D
E
$ $
D
E
$
$
#
$
/
#
$
.
This density is determined by formula (23).
Let ! & /, set
K
/!
/
L
!
Lemma 8: a) The formula is fulfilled
/!
~
H/ ! $
exp
~
\ H/ ! $ \
$
$
52
!
~
where the function H / !
~
is determined by formula @@ with the functions < / !
~
instead of the functions < / ! , and the function < / !
are determined by formula 34
with the functions E9
9 / instead of the functions ~ E9
9 / .
b) Let
9
where the set MN
9
4
" MN
is determined by Lemma _ , and the functions
~
< /!
49
9 4
4
Quasistable Gradient
67
are determined by formula 35 with ~ E9
9 / instead of
exist some constants 7
,
, and a graph Q) " T Q
fulfilled
~
\< / !
49
9 4
4
\
9
9 / . Then there
for which the inequality is
E
9
~
9 4 K Q)
L
U % Q) L @ E
?
?%
~
where
and K where introduced in Lemma 6, ?
?% are the vertices of the graph
Q) , and the function U % were introduced before Lemma I .
The proof can be performed in the same way as for Lemmas 5 and 6.
Introduce the function
inf
$
9
9
% 9
~
Theorem 2: Let &
and the initial condition
"
of a Hamiltonian system
satisfy condition
Z log
for all
Let the configuration
" with
support .
*
satisfies conditions 1), 2) of Theorem 1. Then the Hamiltonian
system is quasistable.
The proof is very similar to the proof of Theorem 1.
9
$
9
Acknowledgment
I would like to recognize Professor Yu. Kondratiev with whom I had many fruitful discussions
on the quasistability theme. I also had useful discussions with Professors R. Khasminskii, N.
Krylov, and M. Portenko. I am very grateful to all of them.
References
[1] Fritz, J., Gradient dynamics of infinite point systems, Annals of Prob. 15:4 (1987), 78514.
[2] Girsanov, I.V., On a transformation of some class of stochastic processes by absolute
continuous changing of measures, Theory of Prob. and Applic. 5 (1960), 314-330.
[3] Skorokhod, A.V., On differentiability of measures corresponding to stochastic processes
II: Markov processes, Theory of Prob. and Applic. 5 (1960), 49-53.
[4] Skorokhod, A.V., Stochastic Equations for Complex Systems, Riedel 1988.
[5] Skorokhod, A.V., On the regularity of many particle dynamical systems perturbed by
white noise, J. Appl. Math. and Stoch. Anal. 9 (1996), 427-437.
[6] Skorokhod, A.V., On infinite systems of stochastic differential equations, JH. Methods of
Funct. Anal. and Topol. 4 (1999), 54-61.