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Non -- homogeneous zero -- range interaction processes
Dmitri Khmelev \Lambda
Eugene Spodarev y
Abstract
In this r'esum'e we formulate the results concerning the limiting behaviour (as time t !1)
of a non--homogeneous zero--range interaction particle system. Similar (homogeneous) processes
were introduced by Spitzer (1970) and investigated by E. Waymire (1980), E. D. Andjel (1982),
A. Galves and H. Guiol (1997). As this system is not ergodic we describe the class of its
stationary distributions and explore the character of non--stability in particular cases.
Consider the following particle system: a number of indistinguishable particles is located
in a countable set of sites J . Transitions of particles at a site i are made after random periods
of time Ü --- i. i. exponentially d. r. v. with parameter fl i 8 i 2 J : if site i is not empty then
one (and only one) particle chosen randomly at this site instantly moves to any site j w. p.
p ij . Transitions are made independently for any i 2 J . Probabilities of jumps from i to j
form together matrix P = (p ij ) i;j2J ; 8 i 2 J
P
j2J
p ij = 1. One can describe the state of the
system by process j(t) = (j i (t); i 2 J) ; t – 0; where j i (t) is a number of particles in cell i
at time t. Denote fl = P
i2J
fl i Ÿ 1. Let Z+ = f0; 1; 2; : : :g [f1g: Introduce B --- a oe--algebra
generated by open sets in the product topology on the state space W =Z J
+ of our system.
Theorem 1 (Existence) If sup
i2J
fl i ! 1; sup
i2J
P
j2J
fl j p ji ! 1 then there exists unique Feller
process j(t) :
(\Omega ; =; P) ! (W;B) that describes our
system,(\Omega ; =; P) --- some probability
space.
Suppose the Markov chain with state space J and transition matrix P to be homoge­
neous, aperiodic, and irreducible. Let us state
Conjecture A: There exists a non--trivial invariant measure ú = (ú i ) i2J for P (not neces­
sarily finite).
Conjecture B: A countable Markov chain with transition matrix P is transient.
For theorems 2 and 3 suppose sup
i2J
ú i =fl i ! 1 and denote ae max = 1=(sup
i2J
ú i =fl i ). For any
ae 2 [0; ae max ] and ae = 1 introduce product measures L aea (\Delta) on (W;B) with marginal factors
l i
aea (\Delta); i 2 J defined in the following two cases: 1) ae 2 [0; ae max ]; 8i or ae = ae max ; ae(ú i =fl i ) ! 1;
2) ae = ae max , ae(ú i =fl i ) = 1 or ae = 1; 8i:
1) l i
aea (k) =
(
(1 \Gamma ae(ú i =fl i )) (ae(ú i =fl i )) k ; k 2 Z+ ;
0; k = 1; 2) l i
aea (k) =
(
0; k 2 Z+ ;
1; k = 1:
\Lambda Moscow State University, Russia. E­mail: dima@vvv.srcc.msu.su
y Friedrich­Schiller Universit¨at Jena, Deutschland. E­mail: seu@minet.uni­jena.de
1

Theorem 2 (Invariant measures) Suppose that conjecture A holds. Then the closed con­
vex hull of the set of measures fL aea (\Delta) : ae 2 [0; ae max ]; ae = 1g belongs to the class M of all
invariant measures for Markov process fj(t)g t–0 .
Theorem 3 (Clustering) Suppose that conjecture A holds and P
i2J
ú i =fl i ! 1.
Let j 0 2 J satisfies ú j 0
=fl j 0
= max
i2J
ú i =fl i . Then 8 k 2 Z+ 8j 0 : P
i2J
j 0
i = 1
lim
t!1
P
n
j j 0
(t) ? k j j(0) = j 0
o
= 1:
Theorem 4 (Devastation) Suppose that one of the following conditions holds:
1. fl ! 1 and conjecture B.
2. fl ! 1 and P
i2J
p ij Ÿ 1.
3. P
i2J
p ij Ÿ 1 and there exists a sequence of finite sets fJ n g, J 1 ae J 2 ae : : :,
S
n
J n = J such
that sup
j2JnJn
fl j \Gamma! 0 as n !1 and matrices P (n) = (p ij ) i;j2Jn are irreducible.
4. Let P
i2J
p ij Ÿ 1. Introduce the terminating Markov chain Ym with state space J and
transition matrix P T = (p ji ) i;j2J . Denote by ¯
P j (J) the probability of terminating for
Ym provided that Y 0 = j. Let ¯
P j (J) = 1 for all j 2 J and sup
j2J
fl j ! 1.
Then for any j(0) 2 Z J
+ the process j(t) ! 0 weakly as t !1.
Theorem 5 (Stochastical boundedness) If 9i 0 2 J : fl i 0
?
P
j2J
fl j p ji 0
and j i 0
(0) 2 Z+
then j i 0
(t) is stochastically bounded.
References
[1] E. Waymire ''Zero -- range interaction at Bose -- Einstein speeds under a positive recurrent
single particle law'' Ann. Prob. 8, 3, 441­450 (1980)
[2] E. D. Andjel ''Invariant measures for the zero -- range process'' Ann. Prob., 10, 525­547
(1982)
[3] A. Galves, H. Guiol ''Relaxation time of the one -- dimensional symmetric zero range
process with constant rate'' Markov Processes Relat. Fields 3, 323­332 (1997)
[4] Kelbert M.Ya., Kontsevich M.L., Rybko A.N. ''Jackson networks on countable grafs'' (in
Russian: ''O setyakh Dzheksona na schetnykh grafakh'') Th. Prob. and Appl., 2, 379­382,
(1988)
[5] D.V. Khmelev, V.I. Oseledets ''Mean­field approximation for stochastic transportation
network and stability of dynamical system'' Preprint N 445 of University of Bremen.
Bremen, June 1999
[6] E. Spodarev ''Transport networks with an infinite number of nodes'' to appear in the
conference issue of Zeitschrift f¨ur angewandte Mathematik und Mechanik, 1999
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