Harnack inequality for some discontinuous Markov processes with a diffusion part

Song, Renming

doi:10.3336/gm.40.1.15

Cited by 19 publications

(17 citation statements)

References 9 publications

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“…Green function estimates (for the whole space) and the Harnack inequality for a class of processes with both continuous and jump components were established in [37,38]. The parabolic Harnack inequality and heat kernel estimates were studied in [39] for Lévy processes in R d that are independent sums of Brownian motions and symmetric stable processes, and in [21] for much more general symmetric diffusions with jumps.…”

Section: Introductionmentioning

confidence: 99%

Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

Chen

Kim

Song

et al. 2012

Trans. Amer. Math. Soc.

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Abstract. For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo-differential operators {Δ + bΔ α/2 ; b ∈ [0, 1]} on R d that evolves continuously from Δ to Δ + Δ α/2 . In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to Δ + bΔ α/2 (or, equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b 1/α ) in C 1,1 open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for non-negative functions which are harmonic with respect to Δ + bΔ α/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.

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Section: Introductionmentioning

confidence: 99%

Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

Chen

Kim

Song

et al. 2012

Trans. Amer. Math. Soc.

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“…The following Harnack inequality is proved along the same lines as the ones in Theorem 3.1 in [22] and Theorem 4.5 in [17]. We omit the details.…”

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confidence: 98%

“…The methodology was introduced in [2] and refined in [1]. We are going to use the notation and the approach from [20], combined with some results and ideas from [17] and [22].…”

Section: Harnack Inequalitymentioning

confidence: 99%

Potential Theory of Geometric Stable Processes

Šikić

Song

Vondraček

2005

Probab. Theory Relat. Fields

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In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x| d log 2 |x|), while the Lévy density behaves like 1/|x| d . We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.

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“…To the best of our knowledge, except for the very recent paper [25], the Harnack inequality has not been established for discontinuous Markov processes with a nondegenerate diffusion part. In this paper we try to fill this gap by considering subordinate Brownian motion using a subordinator with a positive drift term.…”

Section: Introductionmentioning

confidence: 99%

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

2006

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Abstract. Let X be a Lévy process in R d , dU3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.Mathematics Subject Classifications (2000): Primary 60J45, 60J75, Secondary 60J25.

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Harnack inequality for some discontinuous Markov processes with a diffusion part

Abstract: In this paper we establish a Harnack inequality for nonnegative harmonic functions of some discontinuous Markov processes with a diffusion part.

Cited by 19 publications

References 9 publications

Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

Boundary Harnack principle for $Δ+ Δ^{𝛼/2}$

Potential Theory of Geometric Stable Processes

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

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