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

Abstract

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 L 2 ( 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 .