I will present a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. It turns out that these transformations are very useful in Euler schemes for killed diffusions, simplifying the solutions of optimal stopping problems with discounting, and characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions. I will give a description of these transformations and discuss their connections with h-transforms and Schroedinger semigroups, and how one can use them to solve the above problems.