Estimates on the transition densities of Girsanov transforms of symmetric stable processes

In this paper, we first study a purely discontinuous Girsanov transform which is more general than that studied in Chen and Song [(2003), J. Funct. Anal. 201, 262-281]. Then we show that the transition density of any purely discontinuous Girsanov transform of a symmetric stable process is comparable to the transition density of the symmetric stable process. The same is true for the Girsanov transform introduced in Chen and Zhang [(2002), Ann. Inst. Henri poincaré 38, 475-505]. As an application of these results, we show that the Green function of Feynman-Kac type transforms of symmetric stable processes by continuous additive functionals of zero energy, when exists, is comparable to that of the symmetric stable process.