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 × ℝ^d, i.e. if E_t := {x ∈ ℝ^d (t, x) ∈ E}, t ∈ ℝ, then either E_s ⊂ E_t, ∀_s ≤ t, or E_s ⊃ E_t, ∀_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 ℝ^d, 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.