Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises
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چکیده
and Applied Analysis 3 if and only if L(γ K ) is σ-additive, where γ K is the canonical cylindrical finitely additive set-valued function (also called a Gaussian distribution) on K. The following is our standing assumption: Assumption 1. Space E ⊂ H ∩ L is a Hilbert space such that for some δ ∈ (0, 1/2), A −δ : E → H ∩ L 4 is γ-radonifying. (12) Remark 2. Under the above assumption, we have the facts that E ⊂ H and the Banach space U is taken as H ∩ L (see [11, 14, 15] for more details and related results). In fact, space E is the reproducing kernel Hilbert space of noise W(t) on H ∩ L. It is well known that subordinators form the subclass of increasing Lévy processes, which can be thought of as a random model of time evolution (see [16]). We will present some properties of the subordinator Lévy process Y(t), t ≥ 0, then review briefly the stochastic integral with respect to Lévy process Y(t). Definition 3 (see [1, 2, 11]). Let E be a Banach space, and let Y = (Y(t), t ≥ 0) be an E-valued stochastic process defined on a probability space (Ω,F,P). Stochastic processY is called a Lévy process if (L1) Y(0) = 0, a.s.; (L2) processYhas independent and stationary increments; and (L3) process Y is stochastically continuous, that is, for all δ > 0 and for all s ≥ 0, lim t→ s P (|Y (t) − Y (s)| > δ) = 0. (13) A subordinator Lévy process is an increasing onedimensional Lévy process. For p > 0, Sub(p) denotes the set of all subordinator Lévy processesZ, whose intensitymeasure ρ satisfies the condition
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تاریخ انتشار 2014