%This is file: geilo0f.tex. Outline of six lectures to be given in Geilo, %Norway, July-August, 1996 on Stochastic Differential Systems with Memory, %Sixth Workshop on Stochastic Analysis. %Author: Salah Mohammed %Date: July 24, 1996 (11:10pm) \input amstex \documentstyle{amsppt} %\magnification=\magstep2 %\nopagenumbers \magnification=\magstep1 \NoBlackBoxes \loadbold \loadmsbm \TagsOnRight \font\eightsmc=cmssqi8 \font\b=cmbx10 scaled \magstephalf \font\bb=cmr10 scaled \magstep4 \font\cc=cmbx10 scaled \magstep3 \font\aa=cmbx10 scaled \magstep2 \font\a=cmbx10 scaled \magstep1 \font\d=cmr10 scaled \magstep1 \font\bi=cmbxti10 scaled \magstep1 \font\ii=cmti10 scaled \magstep1 \font\sl=cmsl12 scaled \magstep1 \document \catcode`\@=11 \def\logo@{} \catcode`\@=\active \NoRunningHeads \NoBlackBoxes \TagsAsMath \TagsOnRight \parindent=.5truein \parskip=3pt \hsize=6truein \vsize=7.8truein \raggedbottom \loadmsam \loadbold \loadmsbm % Substitute for missing cmcsc8: \font\eightsmc=cmbx8 %Other math symbols %\define \x{\times} \let \< = \langle \let \> = \rangle \let \| = \Vert %Trecie's AmS-TeX macros for this file %Greek letters %\define \a {\alpha} %\redefine \b {\beta} \redefine \dl {\delta} \redefine \D {\Cal D} \redefine \e {\epsilon} \define \F {\Cal F} \define \G {\Gamma} \redefine \g {\gamma} %\redefine \ll {\lambda} \define \br {\bold R} \define \be {\bold E} \define \R {\bold R} \redefine \R {\bold R} \define \C {\bold C} \define \bp {P} \define \s {\sigma} \define \th {\theta} \redefine \O {\Omega} \redefine \o {\omega} \redefine \t {\tau} \define \ub {\subset} \define \cl {\Cal L} \define \E {\Cal E} \define \X {\Cal X} \redefine \w {\o} \define \bP {\Bbb P} \define \A {\Cal A} %Define the roman "e" used in math equations \define \ee {\hbox{e}} %Other macros \redefine \i {\infty} \define \pt {\partial} \define \bl {{\Cal L}} \define \ube {\subseteq} %\define\qd{{\unskip\nobreak\hfil\penalty50\hskip2em\vadjust{} % \nobreak\hfil$\square$\parfillskip=0pt\finalhyphendemerits=0\par}} \define\qd{{\unskip\nobreak\hfil\penalty50\hskip2em\vadjust{} \nobreak\hfil$\square$\parfillskip=0pt\finalhyphendemerits=0\par}} %\topmatter \document \vbox to 1.0truein{} %\magnification=\magstep1 {\aa \centerline{{ \a STOCHASTIC DIFFERENTIAL SYSTEMS }} \line{} \centerline{{\a WITH MEMORY }} \line{} \line{} \centerline{{ \bf THEORY, EXAMPLES AND APPLICATIONS }} \vbox to 1.0truein{} {\aa \centerline{{ \a Geilo, Norway : July 29-August 4, 1996 }} \vbox to 1.0truein{} %\author {\a \centerline{ Salah-Eldin A. Mohammed }} \line{} {\d \centerline{Southern Illinois University} \line{} \centerline{Carbondale, \ IL \ 62901--4408 \ USA } \line{} \centerline{Web page: \bf{http://salah.math.siu.edu}}} %\vbox to 0.5truein{} %\rightheadtext{I} %\leftheadtext{I} \newpage %\baselineskip=36truept %\baselineskip=24truept %\baselineskip=30truept \baselineskip=18truept %\input geilo0 \centerline{Outline of Lectures} \bigskip \medskip \noindent {\a Lecture I. Existence.} \bigskip {\d \item{1.} Simple examples: The noisy feedback loop: $dx(t)= x(t-r) \, dW(t)$. Solution process $x(t)$ is not a Markov process in $\R$. No closed form solution when $r >0$. Compare with the case $r=0$. The logistic time-lag model with Gaussian noise: $$dx(t)= [\alpha -\beta x(t-r)] x(t) \, dt + \sigma x(t) \, dW(t).$$ The classical ``heat-bath" model of R. Kubo: Motion of a large molecule in a viscous fluid. \item{2.} General Formulation. Choice of state space. Pathwise existence and uniquenss of solutions to sfde's under local Lipschitz and linear growth hypotheses on the coefficients. Existence theorem allows for stochastic white-noise perturbations of the memory, e.g. % $$ dx(t) = \biggl \{ \int _{[-r,0]} x (t + s) \, dW(s)\biggr\} \, dW(t) \quad t > 0 $$ % Above sfde is {\ii not\/} covered by classical results of Protter, Metivier and Pellaumail, Doleans-Dade. \item{3.} Mean Lipschitz, smooth and sublinear dependence of the trajectory random field. %\bigskip \newpage \noindent {\a Lecture II. Markov Behavior and the Generator.} \bigskip {\d \item{1.} Markov (Feller) property holds for the trajectory random field. Time homogeneity. \item{2.} Construction of the semigroup. Semigroup is not strongly continuous for positive delay. Domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from $0$, but contains ``quasitame" functions. These are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. \item{3.} Derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions. } \newpage \noindent {\a Lecture III. Regularity. Classification of SFDE's.} \bigskip {\d \item{1.} Pathwise regularity of the trajectory random field in the time variable. $\alpha$-H\"older continuity. \item{2.} Almost sure (pathwise) dependence on the initial state. Non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) \, dW(t)$. Breakdown of linearity and local boundedness. Classification of sfde's into regular and singular types. \item{3.} Results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales. \item{4.} Sussman-Doss type nonlinear sfde's. Existence and compactness of semiflow. \item{5.} Path regularity of general non-linear sfde's with ``smooth memory". } \newpage \noindent {\a Lecture IV. Ergodic Theory of Linear SFDE's.} \bigskip {\d \item{1.} Existence of stochastic semiflows for certain classes of linear sfde's with smooth memory terms. The cocycle and its perfection. {\item{2.} Compactness of the semiflow in the finite memory case. {\item{3.} Ruelle-Oseledec multiplicative ergodic theorem in Hilbert space. Existence of a discrete Lyapunov spectrum. The Stable Manifold Theorem (viz. {\ii random saddles}) for hyperbolic linear sfde's driven by white noise. The case of helix noise. } \newpage \noindent {\a Lecture V. Stability. Examples and Case Studies.} \bigskip {\d \item{1.} Estimates on the maximal exponential growth rate for the singular noisy feedback loop: $dx(t)= \s x(t-r) \, dW(t)$. Stability and instability for small $\s$ (or large $r$) using a Lyapunov functional argument. Comparison with the non-delay case for large $\s$. \item{2.} Derivation of estimates on the top Lyapunov exponent $\lambda_1$ for various examples of one-dimensional regular sfde's driven by white noise or a martingale with stationary ergodic increments. \item{3.} Lyapunov spectrum for sdde $dx(t)= x((t-1)-) \, dN(t)$ driven by a Poisson process $N$. Characterization of the Lyapunov spectrum. } \newpage \noindent {\a Lecture VI. Miscellany } \bigskip {\d \item{1.} Malliavin Calculus of SFDE's. Regularity of the solution $x(t, \o)$ in $\o$. Malliavin smoothness and existence of smooth densities. Classical solution of a degenerate parabolic pde as an application. \item{2.} Small delays. Applications to sode's. A proof of the classical existence theorem for solutions to sode's. \item{3.} Affine systems of sfde's. Lyapunov spectrum. The hyperbolic splitting. Existence of stationary solutions in the hyperbolic case. Application to simple population model. \item{4.} Random delays. Induced measure-valued process. Random families of Markov fields and random generators. \item{5.} Infinite memory and stationary solutions. } \newpage \noindent {\a References} \bigskip \baselineskip=18truept \parskip=3truept \medskip {\d \item{[AKO]} Arnold, L\., Kliemann, W\. and Oeljeklaus, E\. Lyapunov exponents of linear stochastic systems, in {\it Lyapunov Exponents}, Springer Lecture Notes in Mathematics { \a 1186} (1989), 85--125. \medskip \item{[AOP]} Arnold, L\. Oeljeklaus, E\. and Pardoux, E\., Almost sure and moment stability for linear It\^o equations, in {\it Lyapunov Exponents}, Springer Lecture Notes in Mathematics { \a 1186} (ed\. L\. Arnold and V\. Wihstutz) (1986), 129--159. \medskip \item{[B]} Baxendale, P\.H\., {\it Moment stability and large deviations for linear stochastic differential equations}, in Ikeda, N\. (ed\.) {\it Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics}, Katata and Kyoto (1985), 31--54, Tokyo: Kinokuniya (1987). \medskip \item{[CJPS]} Cinlar, E\., Jacod, J\., Protter, P\. and Sharpe, M\. {\it Semimartingales and Markov processes}, Z\. Wahrsch\. Verw\. Gebiete { \a 54} (1980), 161--219. \medskip \item{[FS]} Flandoli, F\. and Schauml\"offel, K\.-U\. {\it Stochastic parabolic equations in bounded domains: Random evolution operator and Lyapunov exponents}, Stochastics and Stochastic Reports { \a 29}, 4 (1990), 461--485. \medskip \medskip \item{[M1]} Mohammed, S\.-E\.A\. {\it Stochastic Functional Differential Equations}, Research Notes in Mathematics { \a 99}, Pitman Advanced Publishing Program, Boston-London-Melbourne (1984). \medskip \item{[M2]} Mohammed, S\.-E\.A\. {\it Non-linear flows for linear stochastic delay equations}, Stochastics { \a 17}, 3 (1986), 207--212. \medskip \item{[M3]} Mohammed, S\.-E\.A\. {\it The Lyapunov spectrum and stable manifolds for stochastic linear delay equations}, Stochastics and Stochastic Reports { \a 29} (1990), 89--131. \medskip \item{[M4]} Mohammed, S\.-E\.A\. {\it Lyapunov exponents and stochastic flows of linear and affine hereditary systems}, (1992) (Survey article), Birkh\"auser (1992), 141--169. \medskip \item{[MS]} Mohammed, S\.-E\.A\. and Scheutzow, M\.K\.R\. {\it Lyapunov exponents of linear stochastic functional differential equation driven by semimartingales. Part I: The multiplicative ergodic theory}, (preprint, 1990), AIHP (to appear). \medskip \item{[P]} Protter, Ph\.E\. Semimartingales and measure-preserving flows, {\it Ann. Inst. Henri Poincar\'e, Probabilit\'es et Statistiques, \/} vol. 22, (1986), 127-147. \medskip \item{[R]} Ruelle, D\. Characteristic exponents and invariant manifolds in Hilbert space, {\it Annals of Mathematics \/} { \a 115} (1982), 243--290. \medskip \medskip \item{[S]} Scheutzow, M\.K\.R\. {\it Stationary and Periodic Stochastic Differential Systems: A study of Qualitative Changes with Respect to the Noise Level and Asymptotics}, Habiltationsschrift, University of Kaiserslautern, W. Germany (1988). \medskip \item{[Sc]} Schwartz, L\., {\it Radon Measures on Arbitrary Topological Spaces and Cylindrical measures \/}, Tata Institute of Fundamental Research, Oxford University Press, (1973). \medskip \item{[Sk]} Skorohod, A\. V\., {\it Random Linear Operators \/}, D. Reidel Publishing Company (1984). \medskip \item{[SM]} de Sam Lazaro, J\. and Meyer, P\.A\. Questions de th\'eorie des flots, {\it Seminaire de Probab. IX,\/} Springer Lecture Notes in Mathematics { \a 465}, (1975), 1--96. \medskip \item{[E]} Elworthy, K. D.,Stochastic differential equations on manifolds, Cambridge (Cambridgeshire) ; New York : Cambridge University Press, (18), 326 p. ; 23 cm. 1982, London Mathematical Society lecture note series ; 70. \medskip \item{[D]} Dudley, R\.M\., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, {\it J. Functional Analysis \/},1, (1967), 290-330. \medskip \item{[FS]} Flandoli, F\. and Schauml\"offel, K\.-U\. Stochastic parabolic equations in bounded domains: Random evolution operator and Lyapunov exponents, {\it Stochastics and Stochastic Reports\/} {\bf 29}, 4 (1990), 461--485. \medskip \item{[H]} Hale, J\.K\., {\it Theory of Functional Differential Equations\/}, Springer-Verlag, New York, Heidelberg, Berlin, (1977). \medskip \item{[Ha]} Has'minski\v i, R\. Z\., {\it Stochastic Stability of Differential Equations\/}, Sijthoff \& Noordhoff (1980). \medskip \item{[KN]} Kolmanovskii, V\.B\. and Nosov, V\.R\., {\it Stability of Functional Differential Equations \/}, Academic Press, London, Orlando (1986) . \medskip \item{[K]} Kushner, H\.J\., On the stability of processes defined by stochastic differential-difference equations, {\it J. Differential equations\/}, 4, (1968), 424-443. %\medskip \medskip \item{[Ma]} Mao, X\.R\., {\it Exponential Stability of Stochastic Differential Equations\/}, Pure and Applied Mathematics, Marcel Dekker, New York-Basel-Hong Kong (1994). \medskip \item{[MT]} Mizel, V\.J\. and Trutzer, V\., Stochastic hereditary equations: existence and asymptotic stability, {\it J. Integral Equations\/}, (1984), 1-72 \medskip \item{[M1]} Mohammed, S\.-E\.A\., {\it Stochastic Functional Differential Equations\/}, Research Notes in Mathematics {\bf 99}, Pitman Advanced Publishing Program, Boston-London-Melbourne (1984). \medskip \item{[M2]} Mohammed, S\.-E\.A\., Non-linear flows for linear stochastic delay equations, {\it Stochastics\/} {\bf 17}, 3 (1986), 207--212. \medskip \item{[M3]} Mohammed, S\.-E\.A\., The Lyapunov spectrum and stable manifolds for stochastic linear delay equations, {\it Stochastics and Stochastic Reports \/} {\bf 29} (1990), 89--131. \medskip \item{[M4]} Mohammed, S\.-E\.A\., Lyapunov exponents and stochastic flows of linear and affine hereditary systems, (1992) (Survey article), in {\it Diffusion Processes and Related Problems in Analysis, Volume II\/}, edited by Pinsky, M\., and Wihstutz, V\. Birkh\"auser (1992), 141--169. \medskip \item{[MS]} Mohammed, S\.-E\.A\. and Scheutzow, M\.K\.R\., Lyapunov exponents of linear stochastic functional differential equation driven by semimartingales. Part I: The multiplicative ergodic theory, {\it Ann. Inst. Henri Poincar\'e, Probabilit\'es et Statistiques, \/}Vol. 32, no. 1 (1996), 69-105. \medskip \item{[PW1]} Pardoux, E\. and Wihstutz, V\., Lyapunov exponent and rotation number of two-dimensional stochastic systems with small diffusion, {\it SIAM J. Applied Math.\/}, 48, (1988), 442-457. \medskip \item{[PW2]} Pinsky, M\. and Wihstutz, V\., Lyapunov exponents of nilpotent It\^o systems, {\it Stochastics\/}, 25, (1988), 43-57. \medskip \item{[R]} Ruelle, D\. Characteristic exponents and invariant manifolds in Hilbert space, {\it Annals of Mathematics \/} {\bf 115} (1982), 243--290. \medskip %\medskip \item{[S]} Scheutzow, M\.K\.R\. {\it Stationary and Periodic Stochastic Differential Systems: A study of Qualitative Changes with Respect to the Noise Level and Asymptotics}, Habiltationsschrift, University of Kaiserslautern, W. Germany (1988). \medskip \item{[Sc]} Schwartz, L\., {\it Radon Measures on Arbitrary Topological Vector Spaces and Cylindrical Measures \/}, Tata Institute of Fundamental Research, Oxford University, Academic Press, London, Orlando (1986) . \medskip } \end