Functional Calculus for the Series of Semigroup Generators Via Transference

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

ALMOST PERIODIC SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS ON BANACH SPACES

We prove the existence of almost periodic solutions to a class of abstract stochastic evolution equations on a Banach space E, [Formula: see text] Both autonomous (A is a C0-semigroup generator) and non-autonomous (A(t) satisfies conditions of Acquistapace–Terreni and generates a strongly continuous evolution family) cases are studied. Results are based on the theory of stochastic integration on Banach spaces of van Neerven and Weis and R-boundedness estimates for semigroups and evolution families due to Hytönen and Veraar. An example is given for a non-autonomous second order boundary value problem on a domain in ℝd.

FLOWS AND INVARIANCE FOR DEGENERATE ELLIPTIC OPERATORS

LetSbe a sub-Markovian semigroup onL2(ℝd) generated by a self-adjoint, second-order, divergence-form, elliptic operatorHwithW1,∞(ℝd) coefficientsckl, and let Ω be an open subset of ℝd. We prove that ifeither C∞c(ℝd) is a core of the semigroup generator of the consistent semigroup onLp(ℝd) for somep∈[1,∞] or Ω has a locally Lipschitz boundary, thenSleavesL2(Ω) invariant if and only if it is invariant under the flows generated by the vector fields ∑dl=1ckl∂lfor allk. Further, for allp∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.