About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains

We construct and analyze in a very general way time inhomogeneous (possibly also degenerate or reflected) diffusions in monotonely moving domains E⊂R×Rd, i.e. if Et≔{x∈Rd|(t,x)∈E}, t∈R, then either Es⊂Et, ∀s⩽t, or Es⊃Et, ∀s⩽t, s,t∈R. Our major tool is a further developed L2(E,m)-analysis with well chosen reference measure m. Among few examples of completely different kinds, such as e.g. singular diffusions with reflection on moving Lipschitz domains in Rd, non-conservative and exponential time scale diffusions, degenerate time inhomogeneous diffusions, we present an application to what we name skew Bessel process on γ. Here γ is either a monotonic function or a continuous Sobolev function. These diffusions form a natural generalization of the classical Bessel processes and skew Brownian motions, where the local time refers to the constant function γ≡0.