Lévy Matters III

Stochastic representation of solution to nonlocal-in-time diffusion

“…If a set C is invariant under T and a core for (L, Dom(L)), then we say that C is an invariant core for (L, Dom(L)). Recall that if C is a dense subspace of Dom(L) and C is invariant under T , then C is an invariant core for (L, Dom(L)) (see [10,Lemma 1.34]). For a given λ ≥ 0 we define the resolvent of T by (λ − L) −1 := ∞ 0 e −λs T s ds, and recall that for λ > 0, (λ − L) −1 : B → Dom(L) is a bijection and it solves the abstract resolvent equation…”

“…. This operator is also known as the generator form of fractional derivatives [30,38], or a Lévy-type generator [10]. (iii) Other particular cases include the fractional derivatives of variable order, which are obtained by taking ρ as the function ρ(t, r) = −r −1−α(t) /Γ(−α(t)) with a suitable function α(t) : R → (0, 1) [27], and tempered Lévy kernels ρ(t, r) = −e −λr r −1−α /Γ(−α), α ∈ (0, 1), λ > 0, [11,51].…”

On association and other forms of positive dependence for Feller processes

“…Throughout this chapter, we often assume that p s (x, ξ) is s-continuous and bounded, i.e. satisfying (5) and (6), respectively.…”

“…It is known that (rich) Feller processes X = (X t ) t≥0 on R d can be characterized by their state-space dependent characteristic triplet (b(x), Σ(x), ν(x, dy)), where b : R d → R d describes the non-random behavior, Σ : R d → R d×d describes the continuous diffusion-like behavior, and ν(·, dy) is a measurable kernel describing the jump behavior of the process. Feller processes are often called Lévy-type processes, given that they behave locally like a Lévy process (for more on Feller processes, see [5]).…”

Existence and Uniqueness Results for Time-Inhomogeneous Time-Change Equations and Fokker–Planck Equations

“…Furthermore, classical results on multiplicative perturbations of Feller generators and time-changed Lévy processes allow for weak regularity assumptions on σ, see e.g. [BSW13,Thm. 4.1] and the original reference [Lum73], [ES85] and the references therein.…”