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A Zeta Function For Flows With Positive Templates, Michael C. Sullivan

Articles and Preprints

A zeta function for a map f : M → M is a device for counting periodic orbits. For a topological flow however, there is not a clear meaning to the period of a closed orbit. We circumvent this for flows which have positive templates by counting the “twists” in the stable manifolds of the periodic orbits.

An Extension Of Hörmander’S Theorem For Infinitely Degenerate Second-Order Operators, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hörmander type. The hypotheses of our theorem allow Hörmander's general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. It is based on the Malliavin calculus and requires new sharp estimates for diffusion processes in Euclidean space.

Educating Dilbert, Michael C. Sullivan

Articles and Preprints

A commentary on calculus reform efforts.

Degenerate Stochastic Differential Equations, Flows And Hypoellipticity, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

In this article we study stochastic hereditary systems on R^d, their flows and regularity of their solutions with respect to d-dimensional Lebesgue measure. More specifically we will state and outline the proofs of several results on the following issues:

(i)Existence of smooth densities for solutions of stochastic hereditary equations whose covariances degenerate polynomially (anywhere) on hypersurfaces in R^d.

(ii)Existence of smooth densities for diffusions with degeneracies of infinite order on a collection of hypersurfaces in R^d.

(iii)Extension and refinement of Hormander's hypoellipticity theorem for a large class of highly degenerate second order parabolic …

Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

We establish the existence of smooth densities for solutions of R^d-valued stochastic hereditary differential systems of the form

dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).

In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in R^d and g is an n x d matrix-valued function defined on [0, ∞) x R^d, such that gg^* has …