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

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Abstract

We consider the system of stochastic differential equations

$$ dX_t=A(X_{t-})\, dZ_t, $$

where Z 1 t , . . . , Z d t 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.

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References

  1. Bass R.F.: Diffusions and Elliptic Operators. Springer, NewYork (1997)

    Google Scholar 

  2. Bass R.F., Chen Z.-Q.: Systems of equations driven by stable processes. Probab. Theory Rel. Fields 134, 175–214 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bass R.F., Levin D.A.: Harnack inequalities for jump processes. Potential Anal. 17, 375–388 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bass R.F., Tang H.: The martingale problem for a class ofstable-like processes. Stoch. Process. Appl. 119, 1144–1167 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen Z.-Q., Kumagai T.: Heat kernel estimates forstable-like processes on d-sets. Stoch. Process. Appl. 108, 27–62 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Heidelberg (1983)

    MATH  Google Scholar 

  7. Schwab, C.: Private communication

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Richard F. Bass

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Zhen-Qing Chen

Authors
  1. Richard F. Bass
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  2. Zhen-Qing Chen
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Corresponding author

Correspondence to Zhen-Qing Chen.

Additional information

R. F. Bass’s research was partially supported by NSF grant DMS-0601783 and Z.-Q. Chen’s research was partially supported by NSF grant DMS-0600206.

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Bass, R.F., Chen, ZQ. Regularity of Harmonic functions for a class of singular stable-like processes. Math. Z. 266, 489–503 (2010). https://doi.org/10.1007/s00209-009-0581-0

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  • DOI: https://doi.org/10.1007/s00209-009-0581-0

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