One-dimensional bi-generalized diffusion processes
The notion of one-dimensional diffusion or gap diffusion is generalised by allowing the scale function s to have flats, i.e. s is (only) non- decreasing. Using the function s and a non-negative speed measure m an analytical construction of a Markov process X - called bi-generalised diffusion process (BGDP) - is given. It is seen that X does not have in general the strong Markov property and right continuous paths. In fact, sample paths are described using a gap diffusion when X is off the flats of s and a stationary process when in the flats. Moreover, conditions for the convergence of finite-dimensional distributions in a sequence of BGDP's are given. These results are used to study metastable behaviour in statistical physics, asymptotic behaviour of a diffusion in random environment and approximations of diffusion processes of gene frequencies. Consequences of the existence of bi-generalised birth and death processes for the Stieltjes' moment problem are also discussed.
asymptotic behaviour of a diffusion in random environment
scientific article; zbMATH DE number 4149307