In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an absorbing set for a
random dynamical system
In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space ''S'', a set of ma ...
is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
. A dynamical system is a system in which a
function describes the
time dependence of a
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
in a
geometrical space.
The absorbing set eventually contains the image of any
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
under the cocycle ("flow") of the random dynamical system. As with many concepts related to random dynamical systems, it is defined in the
pullback sense.
Definition
Consider a random dynamical system ''φ'' on a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
separable metric space (''X'', ''d''), where the noise is chosen from a
probability space (Ω, Σ, P) with
base flow ''θ'' : R × Ω → Ω. A
random compact set ''K'' : Ω → 2
''X'' is said to be absorbing if, for all ''d''-bounded
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
sets ''B'' ⊆ ''X'', there exists a (
finite)
random time ''τ''
''B'' : Ω → 0, +∞) such that
:
This is a definition in the pullback sense, as indicated by the use of the negative time shift ''θ''
−''t''.
See also
*
Glossary of areas of mathematics
*
Lists of mathematics topics
*
Mathematics Subject Classification
The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. ...
*
Outline of mathematics
Mathematics is a field of study that investigates topics such as number, space, structure, and change.
Philosophy Nature
* Definitions of mathematics – Mathematics has no generally accepted definition. Different schools of thought, particul ...
References
* (See footnote (e) on p. 104)
Random dynamical systems
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