arXiv subjects
Publications and source records attributed to David P. Herzog.
Motivated by applications to stochastic differential equations, an extension of Hörmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-wise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.
We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $α_{1}=α_{2}$ such that when $α_{2}>α_{1}$ the system is ergodic and when $α_{2}<α_{1}$ solutions are not defined for all times. Hörmander's hypoellipticity theorem and geometric control theory are also utilized.
We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schroedinger equation in a random delta-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory.