Heat kernels of non-symmetric Lévy-type operators

Grzywny, Tomasz; Szczypkowski, Karol

doi:10.1016/j.jde.2019.06.013

Cited by 32 publications

(120 citation statements)

References 101 publications

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“…Recently, heat kernel estimates for Markov processes with jumps have been extensively studied due to their importance in theory and applications (see [8,10,3,12,14,16,19,20,21,24,22,23,25,26,29,30,32,33,35,36,38,39,40,41,44,45,46,47,49,51,52,55,56] and references therein). In [21], the authors obtained heat kernel estimates for pure jump symmetric Markov processes on metric measure space (M, d, µ), where M is a locally compact separable metric measure space with metric d and a positive Radon measure µ satisfying volume doubling property.…”

Section: Introductionmentioning

confidence: 99%

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

Bae¹,

Kang²,

Kim³

et al. 2019

Ann. Probab.

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In this paper, we consider a symmetric pure jump Markov process X on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density p(t, x, y) of X and their stabilities when the jumping kernel for X has general mixed polynomial growths. Unlike [22], in our setting, the rate function which gives growth of jumps of X may not be comparable to the scale function which provides the borderline for p(t, x, y) to have either near-diagonal estimates or off-diagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than 1, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general sub-Gaussian bounds, we obtain heat kernel estimates which generalize [3, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function F related to walk dimension of underlying space. As an application, we prove that the finite moment condition in terms of F on such symmetric Markov process is equivalent to a generalized version of Khintchine-type law of iterated logarithm at infinity.

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Section: Introductionmentioning

confidence: 99%

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

Bae¹,

Kang²,

Kim³

et al. 2019

Ann. Probab.

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“…Over the last decade, fluctuation theory stimulated the study of potential and spectral theory for symmetric Lévy processes (and in particular those with completely monotone jumps); see, e.g., [7,8,9,10,12,14,24,25,26,33,34,35,36,37,38,39,41,53,54,55,72] and the references therein. There are, however, very few papers where similar problems are studied for asymmetric processes, see [27,28,40,63,75]. The results of the present article may therefore stimulate the development of potential theory and spectral theory for asymmetric Lévy processes.…”

Section: Introductionmentioning

confidence: 69%

Fluctuation theory for Lévy processes with completely monotone jumps

Kwaśnicki¹

2019

Electron. J. Probab.

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We study the Wiener-Hopf factorization for Lévy processes X t with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of X t up to an independent exponential time; (b) the Laplace transform of the supremum of X t up to a fixed time T , as a function of T . The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent f (ξ) of X t , including a peculiar structure of the curve along which f (ξ) takes real values.

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“…where k 1 , k 2 ∈ N 2 . Hence, by Hölder inequality, (21), (10), (15) and (20), for p > 2 α−1 , we get…”

Section: Gradient Estimatesmentioning

confidence: 86%

“…where |k| = k 1 + k 2 . By scaling property and [23, Lemma 3.1] (see also [15,10] for more general setting),…”

Section: Preliminariesmentioning

confidence: 99%

Uniform pointwise asymptotics of solutions to quasi-geostrophic equation

Jakubowski

Serafin

2020

Nonlinearity

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We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in L 2/(α−1) (R 2 ). Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in L p (R 2 ), p > 2/(α − 1), are also discussed.where (P t ) t 0 is the semi-group generated by ∆ α/2 . Although all of the aforementioned results provide precise bounds for L p norms of the solutions, they do not say much about pointwise behaviour of these solutions. In particular, there are not known any lower bounds. In fact, this is rather common problem in the theory of nonlinear 2010 Mathematics Subject Classification. 35B40; 35K55; 35S10.

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Heat kernels of non-symmetric Lévy-type operators

Abstract: We construct the fundamental solution (the heat kernel) p κ to the equation ∂ t = L κ , where under certain assumptions the operator L κ takes one of the following forms,

Cited by 32 publications

References 101 publications

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

Fluctuation theory for Lévy processes with completely monotone jumps

Uniform pointwise asymptotics of solutions to quasi-geostrophic equation

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