Tightness and Boundedness of Stochastic Flows

In Basak and Kannan, (Basak, G.K.; Kannan, D. Volume nullification and asymptotic flatness of degenerate diffusions. Stoch. Anal. Appl. 1996, 14, 131–148) the authors made a conjectural remark about the relationship between the boundedness and tightness of a degenerate diffusion process { Xtx}, (restated in the introductory section below). In the present article we investigate these relationships for the Brownian and Lévy flows in particular; some of these results hold for more general forward stochastic flows. Particularly, we show that (a) when the flow Xtx is an Lp-random field and is also tight for each x, then | Xtx- Xty| as t , then the flow cannot be tight for all x. We establish some relationship between tightness and asymptotic behavior of the infinitesimal generator of Xtx, and also discuss the almost sure boundedness of the flow.