2014/12/03 by Fernando, Pani W., Hausenblas, Erika, Razafimandimby, Paul
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1412.1453
We consider the stochastic differential equations of the form \begincases dX^ x(t) = σ(X(t-)) dL(t)
X^ x(0)=x, x∈ℝ^ d, \endcases where σ:ℝ^ d→ ℝ^ d is Lipschitz continuous and L=\L(t):t≥ 0\ is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let (Pt)t≥0 be the Markovian semigroup associated to X defined by ( Pt f) (x) := 𝔼 [ f(X^ x(t))], t≥ 0, x∈ ℝd, f∈ Bb(ℝd). Let B be a pseudo--differential operator characterized by its symbol q. Fix ρ∈ℝ. In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that | B Pt u |Hρ2 ≤ C t-γ | u |Hρ2, ∀ u ∈ Hρ2(ℝd ), tgt;0.