## Non-local Dirichlet forms and symmetric jump processes

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- by Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen and Moritz Kassmann PDF
- Trans. Amer. Math. Soc.
**361**(2009), 1963-1999

## Abstract:

We consider the non-local symmetric Dirichlet form $(\mathcal {E}, \mathcal {F})$ given by \[ \mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $\mathcal {F}$ the closure with respect to $\mathcal {E}_1$ of the set of $C^1$ functions on $\mathbb {R}^d$ with compact support, where $\mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {R}^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ \kappa _1|y-x|^{-d-\alpha } \leq J(x,y) \leq \kappa _2|y-x|^{-d-\beta } \] for $0<\alpha < \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(\mathcal {E}, \mathcal {F})$. We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to $\mathcal {E}$. Finally we construct an example where the corresponding harmonic functions need not be continuous.## References

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## Additional Information

**Martin T. Barlow**- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
- Email: barlow@math.ubc.ca
**Richard F. Bass**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- Email: bass@math.uconn.edu
**Zhen-Qing Chen**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 242576
- ORCID: 0000-0001-7037-4030
- Email: zchen@math.washington.edu
**Moritz Kassmann**- Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Beringstrasse 6, D-53115 Bonn, Germany
- Email: kassmann@iam.uni-bonn.de
- Received by editor(s): September 29, 2006
- Received by editor(s) in revised form: May 4, 2007
- Published electronically: October 23, 2008
- Additional Notes: The research of the first author was partially supported by NSERC (Canada)

The research of the second author was partially supported by NSF grant DMS-0601783.

The research of the third author was partially supported by NSF grant DMS-0600206.

The research of the fourth author was partially supported by DFG (Germany) through Sonderforschungsbereich 611. - © Copyright 2008 Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann
- Journal: Trans. Amer. Math. Soc.
**361**(2009), 1963-1999 - MSC (2000): Primary 60J35; Secondary 60J75, 45K05, 31B05
- DOI: https://doi.org/10.1090/S0002-9947-08-04544-3
- MathSciNet review: 2465826