Some Transformations of Diffusions by Time Reversal

Time Reversal of Diffusions' by U. G. Haussmann And

It is shown that if a diffusion process, {Xt: 0 < t < 1}, on Rd satisfies dXt = b(t, Xt) dt + a(t, Xt) dwt then the reversed process, {Xt: 0 < t < 1} where Xt = Xl t , is again a diffusion with drift b and diffusion coefficient a, provided some mild conditions on b, a, and p(, the density of the law of X(, hold. Moreover b and a are identified. 1. Introduction. It is well known that a Markov pr...

Continuous Time Perpetuities and the Time Reversal of Diffusions

Drift Transformations of Symmetric Diffusions, and Duality

Starting with a symmetric Markov diffusion process X (with symmetry measure m and L(m) infinitesimal generator A) and a suitable core C for the Dirichlet form of X, we describe a class of derivations defined on C. Associated with each such derivation B is a drift transformation of X, obtained through Girsanov’s theorem. The transformed process X is typically non-symmetric, but we are able to sh...

Some Anisotropic Diffusions

Since the seminal works of Mumford-Shah [16] and Perona-Malik [18], variational and PDE-based methods have found a wide range of applications in Image Processing and have generated steady interest in the mathematical community. The latter is at least partly due to the difficulties one encounters in developing a mathematical theory encopassing equations which are often ill-posed albeit practical...

Reverse Time Diffusions

This paper is a continuation of work commenced in [l] and considers a diffusion in R" given as the solution of a family of stochastic differential equations. The problem discussed is to suppose the direction of the time parameter is reversed, that is, time evolves in anegative direction and the same diffusion process is observed, with the filtration generated by the reversed process. There is a...