Green function estimates for subordinate Brownian motions : stable and beyond (CROSBI ID 187700)
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Kim, Panki ; Mimica, Ante
engleski
Green function estimates for subordinate Brownian motions : stable and beyond
A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates of the Green functions of these subordinate Brownian motions in any bounded $C^{; ; ; 1, 1}; ; ; $ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{; ; ; 1, 1}; ; ; $ open set with explicit decay rate. Unlike \cite{; ; ; KSV2, KSV4}; ; ; , our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent \[ \phi(\lambda)=\log(1+\lambda^{; ; ; \alpha/2}; ; ; )\ \ \ \ (0<\alpha\leq 2, d > \alpha)\] and \[ \phi(\lambda)=\log(1+(\lambda+m^{; ; ; \alpha/2}; ; ; )^{; ; ; 2/\alpha}; ; ; -m)\ \ \ \ (0<\alpha<2, \, m>0, d >2)\, . \]
geometric stable process; Green function; Harnack inequality; Poisson kernel; harmonic function; potential; subordinator; subordinate Brownian motion
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