The radial part of a \(\Gamma\)-martingale and a non-implosion theorem
A stochastic process \(X\) taking values in a complete Riemannian manifold is a \(\Gamma\)-martingale if its compositions with \(C^2\) maps are submartingales at least when \(X\) belongs to the region of convexity of the map. For time approaching zero such a process has always a limit which is finite or infinite. The aim of the paper is to give conditions for this limit to be finite, the property which implies that the corresponding diffusion is Feller. To this end behaviour of the radial part \(R\) of the process \(X\) is analyzed. It is shown that if the manifold has sectional curvatures bounded from below by \(- K^2\) then \(R\) is a semimartingale with the semimartingale decomposition \(dR = dN + d \Lambda\), where the local martingale \(N\) satisfies \(d[N,N] \leq \text{trace} d [X,X]\) and the finite variation part has the property \[ I_{(R > 0)} d \Lambda \leq {K \over 2}\quad \text{coth} (KR)\quad \text{trace} d [X,X]. \] This result allows to extend the approach of Hsu to obtain the integral curvature condition which insures that \(X(0 + ) \neq \infty\).
scientific article; zbMATH DE number 796507