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 . Here 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 ≡ 0. © 2004 Elsevier Inc. All rights reserved. MSC: 60J60; 31C25; 47D06; 47D07; 35K65; 35K20; 60J55