H.R. Hughes | Mathematics | SIU

H.R. Hughes

Associate Professor

Associate Professor; Ph.D., Northwestern University, 1988. Stochastic processes and stochastic differential geometry.

Research Interests

1. Riemannian geometry and stochastic analysis are applied to address the question of whether one can feel the shape of space by looking at properties of Brownian motion.
2. A variational approach is applied to stochastic optimal control and related problems. Necessary conditions yield stochastic differential equations and efficient numerical methods for solving these problems.

Selected Publications

1. "New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates," with John Gregory, to appear J. Math. Anal. Appl.
2. "Efficient constrained optimization: from the deterministic past to the stochastic future," with John Gregory, Nonlinear Anal. 63 (2005), 763-774.
3. "A new theory and the efficient methods of solution of strong, pathwise, stochastic variational problems," with John Gregory, Methods Appl. Anal. 11 (2004), 303-316.
4. "Products of constant curvature spaces with a Brownian independence property," Proc. Amer. Math. Soc. 126 (1998), 3417-3425.
5. "Curvature conditions on Riemannian manifolds with Brownian harmonicity properties," Trans. Amer. Math. Soc. 347 (1995), 339-361.
6. "Brownian exit distributions from normal balls in S³ \times H³," Ann. Probab. 20 (1992), 655-659.