Regularity of Harmonic functions for a class of singular stable-like processes

Richard F. Bass; Zhen Qing Chen

doi:10.1007/s00209-009-0581-0

Regularity of Harmonic functions for a class of singular stable-like processes

Richard F. Bass, Zhen Qing Chen^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

Abstract

We consider the system of stochastic differential equations dX_t = A(X_t-)dZ_t, where Z_t¹, . . . , Z_t^d are independent one-dimensional symmetric stable processes of order α, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are Hölder continuous, but a Harnack inequality need not hold. The Lévy measure associated with the vector-valued process Z is highly singular.

Original language	English
Pages (from-to)	489-503
Number of pages	15
Journal	Mathematische Zeitschrift
Volume	266
Issue number	3
DOIs	https://doi.org/10.1007/s00209-009-0581-0
Publication status	Published - 2010
Externally published	Yes

Keywords

Harmonic function
Harnack inequality
Hölder continuity
Krylov-Safonov technique
Pseudo-differential operator
Stable-like process
Support theorem

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abstract = "We consider the system of stochastic differential equations dXt = A(Xt-)dZt, where Zt1, . . . , Ztd are independent one-dimensional symmetric stable processes of order α, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are H{\"o}lder continuous, but a Harnack inequality need not hold. The L{\'e}vy measure associated with the vector-valued process Z is highly singular.",

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AU - Chen, Zhen Qing

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AB - We consider the system of stochastic differential equations dXt = A(Xt-)dZt, where Zt1, . . . , Ztd are independent one-dimensional symmetric stable processes of order α, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are Hölder continuous, but a Harnack inequality need not hold. The Lévy measure associated with the vector-valued process Z is highly singular.

KW - Harmonic function

KW - Harnack inequality

KW - Hölder continuity

KW - Krylov-Safonov technique

KW - Pseudo-differential operator

KW - Stable-like process

KW - Support theorem

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EP - 503

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3

ER -

Regularity of Harmonic functions for a class of singular stable-like processes

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Keywords

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