Is a diffusion process determined by its intrinsic metric?

J. R. Norris proved that the small time asymptotic liml → 02t•logp(t,x,y) of a symmetric elliptic diffusion on n (or, more general, on a Lipschitz manifold) is determined by the intrinsic metric defined in terms of the associated Dirichlet form. Here we ask the question: Is the Dirichlet form (or the diffusion process) determined uniquely by its intrinsic metric (i.e. by its small time asymptotic)? The answer is NO. For any symmetric elliptic diffusion there exists another one with the same small time asymptotic but with strictly smaller diffusion coefficients. However, the answer is YES if a priori we know that the diffusion coefficients are continuous.