  We study Girsanov's theorem in the context of symmetric Markov processes,
extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov
transformations of ``gradient type.'' We investigate the most general Girsanov
transformation leading to another symmetric Markov process. This investigation
requires an extension of the forward-backward martingale method of Lyons-Zheng,
to cover the case of processes with jumps.
