zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quasistable gradient and Hamiltonian systems with a pairwise interaction randomly perturbed by Wiener processes. (English) Zbl 1097.60051
Ukrainian mathematical congress – 2001, Kiev, Ukraine, August 21–23, 2001. Proceedings. Section 9. Probability theory and mathematical statistics. Kyïv: Instytut Matematyky NAN Ukraïny (ISBN 966-02-2616-0). 149-159 (2002).

Summary: Infinite systems of stochastic differential equations for randomly perturbed particle systems in d with pairwise interaction are considered. For gradient systems these equations are of the form

dx k (t)=F k (t)dt+σdw k (t)

and for Hamiltonian systems these equations are of the form

dx ˙ k (t)=F k (t)dt+σdw k (t)·

Here x k (t) is the position of the kth particle, x ˙ k (t) is its velocity,

F k =- jk U x (x k (t)-x j (t)),

where the function U: d is the potential of the system, σ>0 is a constant, {w k (t),k=1,2,} is a sequence of independent standard Wiener processes.

Let {x k } be a sequence of different points in d with |x k |, and {v k } be a sequence in d . Let {x ˜ k N (t),kN} be the trajectories of the N-particles gradient system for which x ˜ k N (0)=x k ,kN, and let {x k (t),kN} be the trajectories of the N-particles Hamiltonian system for which x k N (0)=x k ,x ˙ k (0)=v k ,kN. A system is called quasistable if for all integers m the joint distribution of {x k N (t),km} or {x ˜ k N (t),km} has a limit as N. We investigate conditions on the potential function and on the initial conditions under which a system possesses this property.

MSC:
60H10Stochastic ordinary differential equations
60G46Martingales and classical analysis
60K35Interacting random processes; statistical mechanics type models; percolation theory