  In this paper, we give probabilistic interpretations of both, the forward and
the inverse problem of electrical impedance tomography with possibly
anisotropic, merely measurable conductivities: Using the theory of symmetric
Dirichlet spaces, Feynman-Kac type formulae corresponding to different
electrode models on bounded Lipschitz domains are derived. Moreover, we give a
probabilistic interpretation of the Calder\'on inverse conductivity problem in
terms of reflecting diffusion processes and their corresponding boundary trace
processes.
