2,491 research outputs found
Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity
Retarded stochastic differential equations (SDEs) constitute a large
collection of systems arising in various real-life applications. Most of the
existing results make crucial use of dissipative conditions. Dealing with "pure
delay" systems in which both the drift and the diffusion coefficients depend
only on the arguments with delays, the existing results become not applicable.
This work uses a variation-of-constants formula to overcome the difficulties
due to the lack of the information at the current time. This paper establishes
existence and uniqueness of stationary distributions for retarded SDEs that
need not satisfy dissipative conditions. The retarded SDEs considered in this
paper also cover SDEs of neutral type and SDEs driven by L\'{e}vy processes
that might not admit finite second moments.Comment: page 2
Exponential Mixing for Retarded Stochastic Differential Equations
In this paper, we discuss exponential mixing property for Markovian
semigroups generated by segment processes associated with several class of
retarded Stochastic Differential Equations (SDEs) which cover SDEs with
constant/variable/distributed time-lags. In particular, we investigate the
exponential mixing property for (a) non-autonomous retarded SDEs by the
Arzel\`{a}--Ascoli tightness characterization of the space \C equipped with
the uniform topology (b) neutral SDEs with continuous sample paths by a
generalized Razumikhin-type argument and a stability-in-distribution approach
and (c) jump-diffusion retarded SDEs by the Kurtz criterion of tightness for
the space \D endowed with the Skorohod topology.Comment: 20 page
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
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