By Johanna Dettweiler
During this paintings we analzyse the Stochastic Cauchy challenge pushed by means of a cylindrical Wiener procedure. Given the lifestyles of strategies we convey regularity of the trails of the answer. In dependence on houses of the operators within the equation or on geometrical homes of the underlying Banachspace we derive area time regularity effects for the trails of the answer.
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Extra resources for Path Regularity for Stochastic Differential Equations in Banach Spaces
6 Let B be γ-radonifying from H into Xβ for some β > 21 . Then there exists a solution of SCP. 9]. 2] with a different approach using mainly covariance functions and a further assumption on the growth of S. Our approach does not need further assumptions on S and seems to be more directly by considering the paths of the weak solution. 7 Let 0 < α < 12 . Assume that Φ(t) := t−α S(t)B : H → E is stochastically Pettis integrable. Then (SCP) has a weak solution which has a continuous modification. Proof.
Tm ∈ τ . Then for every Tn n = 1, . . , m and every x ∈ X there exist operators Tn,k ∈ τ, k ∈ N and such that (Tn − Tn,k )x Y → 0 as k → ∞ for all n = 1, . . , m. Hence we have for certain constants Cn , n = 1, . . , m, m m ξn Tn xn L2 (Ω;Y ) ≤ ξn (Tn − Tn,k )xn L2 (Ω;Y ) n=1 n=1 m + ξn Tn,k xn L2 (Ω;Y ) Cn (Tn − Tn,k )xn +C n=1 m ≤ m Y n=1 ξn xn L2 (Ω;X) . n=1 Since the first term goes to zero as k tends to infinity, the result follows. 4 Let G be an index set and let Tn (s) ∈ B(X, Y ) for n ∈ N and s ∈ G.
Define S : L2 (0, T ; H) → E by T Sf := g(t)Bf (t) dt, f ∈ L2 (0, T ; H). 0 Then gB represents S and we have T 2 m n 2 fn (t)g(t)Bhm dt γ mn S γ (L2 (0,T ;H),E) = E 0 2 ξm Bhm = E = g 2 B 2γ (H,E) . 7 Observe that SS ∗ = g 2L2 (0,T ) BB ∗ . Since by assumption BB ∗ is a Gaussian covariance operator, the same is true for SS ∗ and the result follows. For H = R the above definitions simplify by canonically identifying L(R, E) with E. Accordingly, a function ϕ : (0, T ) → E which is weakly-L2 is said to represent an operator T ∈ L(L2 (0, T ), E) if T T f, x∗ = ϕ(t), x∗ f (t) dt, f ∈ L2 (0, T ), x∗ ∈ E ∗ , 0 and we write ϕ ∈ γ(0, T ; E) if the operator T = Iϕ is γ-radonifying.