On drift, diffusion and geometry (Q2496834)

The author reports on relations between drift, diffusion and geometry. Different sources of ``diffusion models'' in physics and mathematics are examined. Diffusion processes may arise from original models either deterministic, or random but where dynamics (``drift'') and noise (``diffusion'') are clearly delineated. At the end, the author presents a diffusion process, where noise and dynamics are generally intimately entangled in a second-order partial differential operator. The importance of recurrent differential geometric structures -- connections and Riemannian metrics -- needed to properly define a ``diffusion term'' and also to separate drift from diffusion is stressed. In particular, the following questions are considered: Are there implicit geometric structures to properly define a diffusion? How are drift / dynamics and diffusion mixed? Are there geometric structures needed to separate drift and diffusion?