In this paper we prove well-posedness and stability of a class of stochastic delay differential equations with singular drift. Moreover, we show local wellposedness under localized assumptions.

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations, see [10]. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.Keywords: Stochastic delay differential equations, stochastic functional differential equation, strong Feller property, singular drift, Zvonkin's transformation. MSC2010: Primary 34K50; secondary 60B10, 60B12, 60H10. Notation 1.1. We denote by · OP and · HS the operator norm and respectively the Hilbert-Schmidt norm for matrices A ∈ R d×d , i.e.Additionally, we write for a, b ∈ [−∞, +∞] a ∧ b := min{a, b}, a ∨ b := max{a, b}.Notation 1.2. In the sequel, let r > 0 be an arbitrary but fixed number and define

We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms of simple hitting probabilities of the random walk on the graph.