Uniqueness and Non Uniqueness of Semigroups Generated by Singular Diffusion Operators This book addresses both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator i.e. a second order linear differential operator without zeroth order term which is a priori defined on test functions over some (finite or infinite dimensional) state space only uniquely determines a strongly continuous semigroup on a corresponding weighted Lp space. Particular emphasis is placed on phenomena causing non uniqueness as well as on the relation between different notions of uniqueness appearing in analytic and probabilistic contexts