Liouville theorems for non-local operators

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Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise (2008, background): “1), and to deduce from it the strong Feller property, adapting an argument of [27] to the infinite dimensional case.”
Estimates of the derivatives for parabolic operators with unbounded coefficients (2005, background): “Counterexamples are also given in [18] in the one-dimensional case.”
Structural properties of semilinear SPDEs driven by cylindrical stable processes (2008, background): “ the Bismut-Elworthy-Li formula is not available in the non-Gaussian case. A related formula, but requiring a non trivial Gaussian component in the L´evy noise, was established in ﬁnite dimensions in [20]. 2 Preliminaries H will denote a real separable Hilbert with inner product h·,·i and norm | · |”
Flow of diffeomorphisms for SDEs with unbounded H (2009, background): “C (Rd). However, to be precise, in [4], it is assumed that (x) is an invertible d d matrix and so the expression of J1 in [4, Section 1.5] contains 1(Xx s ) instead of our (Xx s )a 1(Xx s ). We briey explain why (12) holds following the proof of [ 25, Theorem 5.1 ].”