  In this paper, we use the theory of symmetric Dirichlet forms to derive
Feynman-Kac formulae for the forward problem of electrical impedance tomography
with possibly anisotropic, merely measurable conductivities corresponding to
different electrode models on bounded Lipschitz domains. Subsequently, we
employ these Feynman-Kac formulae to rigorously justify stochastic
homogenization in the case of a stochastic boundary value problem arising from
an inverse anomaly detection problem. Motivated by this theoretical result, we
prove an estimate for the speed of convergence of the projected mean-square
displacement of the underlying process which may serve as the theoretical
foundation for the development of new scalable stochastic numerical
homogenization schemes.
