Abstract
In this paper we consider the cylindrical càdlàg property of a solution to a linear equation in a Hilbert space H, driven by a Levy process taking values in a possibly larger Hilbert space U. In particular, we are interested in diagonal type processes, where processes on coordinates are functionals of independent α-stable symmetric processes. We give the equivalent characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a càdlàg version of stable processes described as integrals of deterministic functions with respect to symmetric α-stable random measures with α∈[1,2).
Highlights
We apply the same techniques to obtain a sufficient condition for existence of a càdlàg version of stable processes described as integrals of deterministic functions with respect to symmetric α-stable random measures with α ∈ [1, 2)
We first consider a linear equation in a Hilbert space H given by dXt = AXtdt, X0 = x, where A is a generator of a C0 semigroup (S(t))t 0 on H
We may perturb the linear equation by a Lévy process Z = (Zt)t∈T which takes values in U, where U is a Hilbert space H ⊂ U
Summary
By the Liu Zhai result [6] we know that in the α-stable case, there exists a càdlàg version of X = (Xt)t∈T in H if and only if Z takes values in H, which is equivalent to σnα < ∞. It should be mentioned that the case of α ∈ As it will be proved, our approach works in the much more general setting of diagonal type evolution equations implying a nice sufficient condition for the cylindrical càdlàg property for all diagonal type equations. We give sufficient conditions for existence of càdlàg modifications of stable processes of the form.
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