Periodicity in distribution. I. Discrete systems
We consider the existence of periodic in distribution solutions to the difference equations in a Banach space. A random process is called periodic in distribution if all its finite-dimensional distributions are periodic with respect to shift of time with one period. Only averaged characteristics of...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202011328 |
| Summary: | We consider the existence of periodic in distribution
solutions to the difference equations in a Banach space. A random
process is called periodic in distribution if all its
finite-dimensional distributions are periodic with respect to
shift of time with one period. Only averaged characteristics of a
periodic process are periodic functions. The notion of the
periodic in distribution process gave adequate description for
many dynamic stochastic models in applications, in which dynamics
of a system is obviously nonstationary. For example, the
processes describing seasonal fluctuations, rotation under impact
of daily changes, and so forth belong to this type. By now, a considerable
number of mathematical papers has been devoted to periodic and
almost periodic in distribution stochastic processes. We
give a survey of the theory for certain classes of the linear
difference equations in a Banach space. A feature of our treatment
is the analysis of the solutions on the whole of axis. Such an
analysis gives simple answers to the questions about solution
stability of the Cauchy problem on +∞, solution stability
of analogous problem on −∞, or of existence solution for
boundary value problem and other questions about global behaviour
of solutions. Examples are considered, and references to
applications are given in remarks to appropriate theorems. |
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| ISSN: | 0161-1712 1687-0425 |