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2011

Lower bounded semi-Dirichlet forms associated with L\'evy type operators

Schilling, René L., Wang, Jian

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Let $k:E\times E\to [0,\infty)$ be a non-negative measurable function on some locally compact separable metric space $E$.

We provide some simple conditions such that the quadratic form with jump kernel $k$ becomes a regular lower bounded (non-local, non-symmetric) semi-Dirichlet form.
If $E=\R^n$ we identify the generator of the semi-Dirichlet form and its (formal) adjoint.
In particular, we obtain a closed expression of the adjoint of the stable-like generator $-(-\Delta)^{\alpha(x)}$ in the sense of Bass.

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Bass, R.: Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79
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Ma, Z.M. and Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms , Universitext, Springer-Verlag, Berlin 1992
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Then

Uemura, T.: A remark on non-local operators with variable order, Osaka J. Math. 46
2009
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Wang, J.: Symmetric Lévy type operator, Acta Math. Sin. Eng. Ser. 25
Original
2009
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Fukushima, M. and Uemura, T.: Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab. 40
2012
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