Abstract
We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Lévy process.
Highlights
Introduction and preliminariesLet α ∈ (0, 2) and d = 1, 2, . . . . For the rotation invariant α-stable Lévy process on Rd with the Lévy measure ν(dy) =c | y |α+d dy, y ∈ Rd \{0}, (1)the asymptotic behavior of its transition densities p(t, x, y) is well known, i.e., p(t, x, y) ≈ min t −d /α, |y − t x |α+d, t > 0, x, y ∈ Rd .Estimates of densities for more general classes of stable and other jump Lévy processes gradually extended
Estimates of heat kernels on metric measure spaces having the volume doubling property were obtained by Barlow et al [1], Chen and Kumagai [7,8], and Grigor’yan et al [10]
Upper estimates for heat kernels of symmetric jump processes with small jumps of high intensity were obtained by Mimica in [25]
Summary
We extend the results obtained in [29] and give estimates from above for a wider class of semigroups with the intensity of jumps lighter than stable processes. We note that the assumption (A.1)(c) is satisfied for every nonincreasing function φ : (0, ∞) → (0, 1] such that φ(a)φ(b) ≤ c φ(a + b), a, b > 1, for some positive constant c. If (A.1)–(A.4) are satisfied, there exists the constants C1 and C2 such that for every nonnegative φ ∈ Bb(Rd ) and ε ∈ (0, ε0 ∧ 1), we have etAε φ(x ) ≤ C1eC2t φ(y) min t −d/α, tφ(|y − x|) |y − x|α+d dy + e−tbε(x)φ(x), for every x ∈ Rd. The proof of Theorem 1 is given in Sect. Under the assumption that the corresponding jump kernels are comparable with certain rotation invariant functions, they prove the existence and obtain estimates of the densities Note that the new condition (A.1)(c), which is pivotal for our further investigations, is necessary for the two-sided sharp bounds similar to the right-hand side of (3)
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