Semimartingales and geometric inequalities on manifolds

The purpose of this paper is twofold. First, we use resultson Jacobi fields to study the stochastic differential equations(SDEs) for exp_Xt( exp_Xt^-1(Y_t)) with specially constructed coupledsemimartingales X and Y on a complete, simply connected Riemannianmanifold M with constant sectional curvature. Secondly, we applythese SDEs to obtain an analogue for M of a result of Borellconcerning an inequality relating the solutions of the parabolicequation / t = 1/2 – h, with Dirichlet boundary condition,on three convex sets in Euclidean space. From the latter, therefollows an inequality involving the first eigenvalues of theLaplacian on those convex sets with the Dirichlet boundary condition,analogous to an inequality in Euclidean space which is equivalentto the Brunn–Minkowski inequality of these eigenvaluesobtained by Brascamp and Lieb.