Exotic Brownian motions are diffusion processes given by Dirichlet forms ℇ on L2 (S,μ), where the state space S is the support of a Radon measure μ in ℝd and the energy form ℇ is given by the integral of the canonical square field on ℝd with respect to μ. In most cases we take μ in such a way that μ is singular to the d-dimensional Lebesgue measure and, in particular, μ does not satisfy the doubling condition. The purpose of this paper is to prove the parabolic Harnack inequalities for these diffusions. We will do this with a refinement such that the dependence of μ, Poincaré and Sobolev constants is clarified. Because of the singularity of μ, diffusions are expected to behave in rather an unusual fashion. These Harnack inequalities will be used to show exotic properties of our diffusion processes in a forthcoming paper.
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