Aspects of Micro-Local Analysis and Geometry in the Study of Lévy-Type Generators

On Adjoint Additive Processes

“…Results of this type had been extended to Feller processes generated by pseudo-differential operators with state space dependent symbols, see R. Schilling [15]. In [8] it was pointed out that with the help of the metric d ψ (ξ, η) = ψ interpretation. For additive processes we are not aware of explicit results of this type, however by a standard procedure we can consider additive processes with state space R n as time-homogeneous Markov processes with state space R n+1 , see for example in the context of pseudo-differential operators the work [2].…”

“…As indicated in [7], in particular Theorem 7.1, subordination in the sense of Bochner, see [16] for the general theory, shall lead to further examples. Readers with an interest in state of the art results of the theory of Markov processes related to pseudo-differential operators are referred to Schilling et al [3] as well as to F. Kühn [10] and the forthcoming survey [8]. Whether it is possible to extend our considerations to the classes of processes constructed in [2] using the symbolic calculus of Hoh [5] and in [18] using the ideas of [6] with the help of x and t dependent negative definite symbols remains an open question.…”

Higher order eigenvalues for non-local Schrödinger operators

“…to consider the symbol of a generators as Hamiltonian function. A good starting point is to look at H(q, p) = ψ(p)+V (q) where ψ is a certain convex, coercive negative definite function of class C 1 and V is a suitable potential, see [20,31]. As substitute for the harmonic oscillator it was proposed to consider H(q, p) = ψ(p)+ψ * (q) where ψ * is the conjugate convex function (Legendre transform) of ψ.…”

Topics on Transient Dirichlet Forms: Fundamental Solutions, Potential Kernels and Metric Measure Spaces

“…For the following example we will discuss a continuous negative definite function introduced in [24].…”