#### Abstract

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state.

Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise , and the classical ``heat-bath" model of R. Kubo , modeling the motion of a ``large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade.

In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from zero. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. This chapter also contains a derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions.

In Chapter III, we study pathwise regularity of the trajectory random field in the time variable and in the initial path. Of note here is the non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) dW(t)$ and a breakdown of linearity and local boundedness. This phenomenon is peculiar to stochastic delay equations. It leads naturally to a classification of sfde's into regular and singular types. Necessary and sufficient conditions for regularity are not known. The rest of Chapter III is devoted to results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales, and Sussman-Doss type nonlinear sfde's.

Building on the existence of a compacting stochastic flow, we develop a multiplicative ergodic theory for regular linear sfde's driven by white noise, or general helix semimartingales (Chapter IV). In particular, we prove a Stable Manifold Theorem for such systems.

In Chapter V, we seek asymptotic stability for various examples of one-dimensional linear sfde's. Our approach is to obtain upper and lower estimates for the top Lyapunov exponent.

Several topics are discussed in Chapter VI. These include the existence of smooth densities for solutions of sfde's using the Malliavin calculus, an approximation technique for multidimensional diffusions using sdde's with small delays, and affine sfde's.

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## Comments

For Proceedings of the Sixth Oslo-Silivri Workshop, Geilo, Norway, July 29-August 4, 1996 (Invited presentation of six hourly lectures), Published in

Stochastic Analysis and Related Topics VI. The Geilo Workshop, 1996, Progress in Probability, vol. 42; ed. L. Decreusefond, Jon Gjerde, B. Oksendal, A.S. Ustunel; Birkhauser (1998), 1-77. The original publication is available at www.springerlink.com. Presentation slides are posted at http://opensiuc.lib.siu.edu/math_misc/28/.