We establish the basic results concerning the problem of constructing operator realizations of the formal differential expression ∇ · a · ∇ − b · ∇ with measurable matrix a and vector field b having critical-order singularities as the generators of Markov semigroups in L p and C∞. 2000 Mathematics Subject Classification. 31C25, 47B44 (primary), 35D70 (secondary). Key words and phrases. Markov semigroups, form-bounded vector fields, regularity of solutions, Feller semigroups. The research of D.K. is supported by the Natural Sciences and Engineering Research Council of Canada and the Fonds de recherche du Québec -Nature et technologies. 1 2 2− d−2 d √ δ , ∞[. 1 1 The maximal interval of quasi bounded solvability for A − V , with 0 ≤ V ≤ δA + c(δ), 0 < δ < 1, isÎm := ]r(δ), r ′ (δ)[, r ′ (δ) := 2 1− √ 1−δ d d−2 . This was proved in 1995 by Yu.A. Semënov, based on ideas set forth in [31]. The fact that the semigroup associated with the Schrödinger operator −∆ − V , V ∈ L d 2 ,∞ , can be extended to a C0 semigroup on L r (R d ) for every r ∈Îm was first observed in [17]. * → reads: if b ∈ Ws (s > 1), then b ∈ F δ 2 with δ = δ( b 2 Ws ) < ∞.

The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a L p variant of De Giorgi's method.

We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the L d class, the weak L d class, as well as some vector fields that are not even in L 2+ε loc for any ε > 0.

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