Strong Feller properties for degenerate SDEs with jumps

Dong, Zhao; Peng, Xuhui; Song, Yisheng; Zhang, Xicheng

doi:10.48550/arxiv.1312.7380

Cited by 3 publications

(7 citation statements)

References 3 publications

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“…Remark 1.2. If σ(x) = σ is constant, in [40] and [12], we have already proved this theorem. Moreover, if we define vector fields…”

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confidence: 78%

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Nonlocal Hormander's hypoellipticity theorem

Zhang

2013

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Consider the following nonlocal integro-differential operator: for α ∈ (0, 2),where σ : R d → R d × R d and b : R d → R d are two C ∞ b -functions, and p.v. stands for the Cauchy principal value. Let B 1 (x) := σ(x) and B j+1 (x) := b(x) • ∇B j (x) − ∇b(x) • B j (x) for j ∈ N. Under the following Hörmander's type condition: for any x ∈ R d and some n = n(x) ∈ N,by using the Malliavin calculus, we prove the existence of the heat kernel ρ t (x, y) to the operator L (α) σ,b as well as the continuity of x → ρ t (x, •) in L 1 (R d ) for each t > 0. Moreover, when σ(x) = σ is constant, under the following uniform Hörmander's type condition: for some j 0 ∈ N,we also prove the smoothness of (t, x, y) → ρ t (x, y) with

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“…Remark 1.2. If σ(x) = σ is constant, in [40] and [12], we have already proved this theorem. Moreover, if we define vector fields…”

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confidence: 78%

“…We remark that the generator of stochastic equation (1.6) also takes the form (1.4) with constant σ and nonlinear b, which is a typical Hörmander's type operator if we assume U ′′ > 0 (cf. [12]).…”

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confidence: 99%

Nonlocal Hormander's hypoellipticity theorem

Zhang

2013

Preprint

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“…Proof of Theorem 1.1. Now we can finish the proof of Theorem 1.1 by the same argument as in [7]. We divide the proof into two steps.…”

Section: Lemma 42mentioning

confidence: 93%

“…When A 1 = 0 and κ(z) = c|z| −d−α , Theorems 1.1 and 1.3 have been proved in [22] and [7] by using the Malliavin calculus for subordinated Brownian motions (cf. [11]).…”

Section: Introductionmentioning

confidence: 99%

“…However, their results do not cover the present general case (see also [23,24,21] for some related works). Compared with [22] and [7], in this work we shall use Bismut's approach to prove Theorems 1.1 and 1.3, and need to assume that the Lévy measure is absolutely continuous with respect to the Lebesgue measure. It is noticed that in [7], the Lévy measure can be singular and the drift is allowed to be arbitrarily growth, which cannot be dealt with in the current settings.…”

Section: Introductionmentioning

confidence: 99%

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Regularity of density for SDEs driven by degenerate Lévy noises

Song

Zhang

2015

Electron. J. Probab.

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By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.A 1 A * 1 B n−1 (x).

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Regularity of density for SDEs driven by degenerate Lévy noises

Song¹,

Zhang²

2014

Preprint

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Strong Feller properties for degenerate SDEs with jumps

Cited by 3 publications

References 3 publications

Nonlocal Hormander's hypoellipticity theorem

Nonlocal Hormander's hypoellipticity theorem

Regularity of density for SDEs driven by degenerate Lévy noises

Regularity of density for SDEs driven by degenerate Lévy noises

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