# Articles and Preprints on Line

## -Salah Mohammed

Lyapunov Exponents of Linear and Stochastic Functional Differential Equations Driven by Semimartingales, Part I: The Multiplicative Ergodic Theory (with M. Scheutzow), Annals of Institute of Henri Poincare, Probabilites  e'tet Statistiques Vol. 32, 1, (1996), 69-105.

#### Abstract:

We consider a class of stochastic linear functional differential systems driven by semimartingales with stationary ergodic increments. We allow smooth convolution-type dependence of the noise terms on the history of the state. Using a stochastic variational technique we construct a compactifying stochastic semiflow on the state space. As a necessary ingredient of this construction we prove a general perfection theorem for cocycles with values in a topological group (Theorem 3.1). This theorem is an extension of a previous result of de Sam Lazaro and Meyer. A mutlipicative Ruelle-Oseledec ergodic theorem then gives the existence of a discrete Lyapunov spectrum and a saddle-point property in the hyperbolic case.

Degenerate Stochastic Differential Equations, Flows and Hypoellipticity," (with Denis Bell) (Invited Paper), AMS Summer Research Institute on Stochastic Analysis, Ithaca, Cornell 1993, Proceedings of Symposia in Pure Mathematics, American Mathematical Society , Vol. 57, Stochastic Analysis, American Mathematical Society, Providence, Rhode Island (1995), 553-564.

#### Abstract:

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 operators: Hormander's Lie algebra condition is allowed to fail exponentially fast on the degeneracy hypersurfaces, which are imbedded in submanifolds of dimension less than $d$. The exponential decay rate near the degeneracy surface is found to be optimal.

Our proofs are based on the Malliavin calculus and require new sharp estimates for Ito processes in Euclidean space.

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An Extension of Hormander's Theorem for Infinitely Degenerate Second-Order Operators," (with Denis Bell), Duke Mathematical Journal, Vol.78, No. 3, (1995), 453-475.

#### Abstract:

We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hormander type. The hypotheses of our theorem allow Hormander'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. An analytic proof of this result is not known.

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• Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift" (with Denis Bell), The Annals of Probability, Vol. 23, No. 4, (1995), 1875-1894.

#### Abstract:

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 $R^n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a non-anticipating functional defined on the space of paths in $R^d$ and $g$ is an $R^{nxd}$-matrix-valued function defined on $[0,\infty)xR^d$ such that $gg^*$ has degeneracies of polynomial order on a hypersurface in $R^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

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• Lyapunov Exponents of Linear Stochastic Functional Differential Equations, Part II: Examples and Case Studies" (With M. Scheutzow), (preprint 1995) The Annals of Probability , Vol. 25, No. 3, (1997), 1210-1240. (Reprint available upon request).

#### Abstract:

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semiflow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\bar lambda_1(\sigma)$ of the trajectories expressed in terms of the noise variance $\sigma$. Roughly speaking we show that for small $\sigma$, $\bar lambda_1(\sigma)$ behaves like $-\sigma^2/2$, while for large $\sigma$, it grows like $\log \sigma$. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent $\lambda_1$ are provided. These estimates are sharp in the sense they reduce to known estimates in the deterministic or non-delay cases.

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• Spatial Estimates for Stochastic Flows in Euclidean Space" (With M. Scheutzow), The Annals of Probability, Vol. 26, No. 1, (1998), 56-77. (Reprint available upon request.)

#### Abstract:

We study the behavior for large $|x|$ of Kunita-type stochastic flows $\phi (t,\omega, x)$ on $R^d$, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large $|x|$, under very mild regularity conditions on the driving semimartingale random field. In particular we show that the stochastic flow $\phi(s,t,\omega,x)$ grows slower than $|x|(\log |x|)^{\epsilon}$ as $|x|$ goes to infinity, for arbitrarily small positive $\epsilon$. We show by example that this bound is sharp.

We give an example of a one-dimensional s.d.e. with sublinear coefficients but with the underlying stochastic flow growing superlinearly for large $x$. In this example the stochastic flow has a.s. unbounded spatial derivatives, even though the driving martingale has local characteristics with all derivatives globally bounded. It is interesting to note that in this example the driving noise is infinite-dimensional. However the infinite-dimensionality of the driving noise is not the crucial factor. To illustrate this point we provide an example of a s.d.e. driven by one-dimensional Brownian motion, has coefficients with globally bounded derivatives, while its stochastic flow has a.s. unbounded derivatives. This result is surprising since it is in sharp contrast with well-known behavior of deterministic flows driven by vector fields whose derivatives are globally bounded.

For one-dimensional s.d.e.'s, sufficient conditions on the coefficients are given in order for the stochastic flow to have sublinear growth and a.s. bounded derivatives.

It is expected that the results would be of interest for the theory of stochastic flows on non-compact manifolds as well as in the study of non-linear filtering, stochastic functional and partial differential equations.
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• Stochastic Differential Systems with Memory: Theory, Examples and Applications", pp. 91, for Proceedings of the Sixth Oslo-Silivri Workshop, Geilo, Norway, July 29-August 4, 1996 (Invited presentation of six hourly lectures) (preprint is now available upon request), Stochastic Analysis and Related Topics VI. The Geilo Workshop, 1996 , ed. L. Decreusefond, Jon Gjerde, B. Oksendal, A.S. Ustunel, Progress in Probability, Birkhauser (1998), 1-77.
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• The Introduction
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• The Stable Manifold Theorem for Stochastic Differential Equations" (With M. Scheutzow), (preprint, available upon request) ( MSRI Preprint 1998-015 and list of errata, .dvi file), The Annals of Probability, Vol. 27, No. 2, (1999), 615-652.

#### Abstract:

We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Ito-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory.

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The Dirichlet problem for superdegenerate differential operators" (with Denis Bell), C.R. Acad. Sci. Paris (French Academy of Sciences), t. 327, serie I (1998), 81 - 86 ( preprint available upon request).

#### Abstract:

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of Hormander, we show that the Dirichlet problem associated with $L$ has a unique smooth classical solution. The proof uses the Malliavin calculus. At present, there appears to be no proof of this result using classical analytic techniques.

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• The Stable Manifold Theorem for Nonlinear Stochastic Systems with Memory I: Existence of the Semiflow" (with M.K.R. Scheutzow), Journal of Functional Analysis, 205, (2003), 271-305 (communicated by L. Gross) (preprint available upon request).

#### Abstract:

We consider nonlinear stochastic functional differential equations (sfde's) on Euclidean space. We give sufficient conditions for the sfde to admit locally compact smooth cocycles on the underlying infinite-dimensional state space. Our construction is based on the theory of finite-dimensional stochastic flows and a nonlinear variational technique. In Part II of this article, the above result will be used to prove a stable manifold theorem for nonlinear sfde's.

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• The Stable Manifold Theorem for Nonlinear Stochastic Systems with Memory II: The Local Stable Manifold Theorem" (with M.K.R. Scheutzow),  Journal of Functional Analysis, 206, (2004), 253-306 (communicated by L. Gross) (preprint available upon request).

#### Abstract:

We state and prove a Local Stable Manifold Theorem for nonlinear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary solutions of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments.

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• Stochastic Functional Differential Equations on Manifolds " (with R. Leandre), Probab Theory Related Fields 121 (2001) 1, 117-135 ) (preprint available upon request).

#### Abstract:

We prove an existence theorem for solutions of stochastic functional differential equations under smooth constraints in Euclidean space.  The initial states are semimartingales on a compact Riemannian manifold. It is shown that, under suitable regularity hypotheses on the coefficients, and given an initial semimartingale, a sfde on a compact manifold admits a unique solution living on the manifold for all time. We also study the Chen-Souriau regularity of the solution of the sfde in the initial process.

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• Discrete-time Approximations of Stochastic Differential Systems with Memory " (with Y. Hu and F. Yan), (2001) pp. 71 (preprint available upon request).

#### Abstract:

In this paper, we develop several numerical schemes for solving stochastic differential systems with memory: strong Euler-Maruyama schemes for stochastic delay differential equations (SDDE's) and stochastic functional differential equations (SFDE's) with continuous memory, and a strong Milstein scheme for SDDE's.
The convergence orders of the Euler-Maruyama and Milstein schemes are $0.5$ and $1$ respectively. In order to establish the Milstein scheme, we prove an infinite-dimensional Ito formula for tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus together with the anticipating stochastic analysis of Nualart and Pardoux.

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Discrete-time Approximations of  Stochastic Delay Equations: The Milstein Scheme " (with Y. Hu and F. Yan), The Annals of Probability, 2004, Vol. 32, No. 1A, 265-314 (preprint available upon request).

#### Abstract:

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order $1$. In order to establish the scheme, we prove an infinite-dimensional Ito formula for tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

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 The Stable Manifold Theorem for Semilinear SPDEs" (with Tusheng Zhang and Huaizhong Zhao), (2003) pp. 6 (preprint available upon request).

Abstract:

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic partial differential equations (spde's) near stationary solutions. We first prove general existence theorems for smooth compacting semiflows of semilinear spde's and stochastic evolution equations (see's). We then establish local stable manifold theorems for these infinite-dimensional stochastic dynamical systems. In particular, these results give a random family of Frechet smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The unstable and stable manifolds are stationary, asymptotically invariant under the stochastic semiflow and have fixed finite dimension and codimension, respectively.

Hartman-Grobman theorems along hyperbolic stationary trajectories" (with E. A. Coayla-Teran and P. R. C. Ruffino), Discrete and Continuous Dynamical Systems-Series,  vol. 17, no. 2, 281-292 (2007).

Abstract:

Hartman-Grobman theorems are proved for continuous stochastic dynamical systems near hyperbolic stationary trajectories. A topological conjugacy is established between traveling neighbourhoods of trajectories and and neighbourhoods of the origin in the corresponding tangent bundle. The results apply to stochastic flows generated by sode's.

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A stochastic Calculus for Systems with Memory" (with F. Yan), Stochastic Analysis and Applications, vol. 23, no. 3, 613-657 (2005) (preprint available upon request).

Abstract:

For a given stochastic process X, its segment Xt at time t, represents the "slice" of each path of X over a fixed time-interval [t-r, t], where r is the length of the "memory" of the process. Segment processes are important in the study of stochastic systems with memory (stochastic functional differential equations or sfde's). The main objective of this paper is to study nonlinear transforms of segment processes. Toward this end, we construct a stochastic integral with respect to the Brownian segment process. The difficulty in this construction is the fact that the stochastic integrator is infinite dimensional and is not a (semi)martingale. We overcome this difficulty by employing Malliavin (anticipating) calculus techniques. The segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem. We then prove Ito's formula for the segment of a continuous Skorohod-type process and embed the segment calculus in the theory of anticipating calculus. Applications of the Ito formula include the weak infinitesimal generator for the solution segment of a stochastic system with memory, the associated Feynman-Kac formula, and the Black-Scholes pde for stock dynamics with memory.

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• Absolute Continuity of Stationary Measure-valued Processes Generated by Stochastic Equations with Interaction" (with A. Pilipenko), Theory of Stochastic Process, 2005, vol. 11(27), No.1-2, pp. 17 (preprint available upon request).

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The Weak Euler Scheme for Stochastic Delay Equations" (with E. Buckwar, R. Kuske and T. Shardlow), London Mathematical Society Journal of Computation and Mathematics, 2008, 11, (60-99) (preprint available upon request).

Abstract:

We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

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Large Deviations for Stochastic Systems with Memory" (with Tusheng Zhang), Discrete and Continuous Dynamical Systems-Series B, 2006, vol. 6, No. 4, 881-893 (preprint available upon request).

Abstract:

We establish a large deviations principle for stochastic delay equations driven by small multiplicative white noise. Both upper and lower large deviations estimates are obtained.

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The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations.
Part 1: The Stochastic Semiflow ",
Part 2: Existence of stable and unstable manifolds"
(with Tusheng Zhang and Huaizhong Zhao), pp. 98 (preprint available upon request), Memoirs of the American Mathematical Society, Volume 196, 2008; 105, pp; ISBN-10: 0-8218-4250-1

Abstract:

The main objective of this work is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions.  Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.

In Part 1, we establish a general existence and compactness theorem for cocycles of semilinear see's and spde's. Our results cover a large class of semilinear see's as well as certain semilinear spde's with non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. We adopt a notion of stationarity employed in previous work of Mohammed with M. Scheutzow.

In Part 2 of this work, we give a characterization of  the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. The characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's.  These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation.  The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments.

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The Substitution Theorem for Semilinear Stochastic Partial Differential Equations" (with Tusheng Zhang), Journal of Functional Analysis, vol. 253, no. 1, (2007), 122-157.

Abstract:

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinite-dimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating initial conditions.

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A Delayed Black and Scholes Formula" (with M. Arriojas, Y. Hu and G. Pap),  Journal of Stochastic Analysis and Applications+AH0-, vol. 25, no. 2 (2007), 471 - 492 (preprint available upon request).

#### Abstract:

In this article we develop an exlicit formula for pricing European options when the underlying stock price follows a non-linear stochastic functional differential equation. We believe that the proposed model is sufficiently flexible  to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the completeness of the market.

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Stochastic Dynamical Systems in Infinite Dimensions", Trends in Stochastic Analysis, edited by Jochen Blath, Peter Morters and Michael Scheutzow,
London Mathematical Society Lecture Note Series, Cambridge University Press, (2009), pp. 30.

Abstract:

In this article, we summarize some results on the existence and qualitative behavior of stochastic dynamical systems in infinite dimensions. The three main examples covered are stochastic systems with finite memory (stochastic functional differential equations-sfde's), semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's). Due to limitations of space, our summary is by no means intended to be exhaustive: The emphasis is mainly on the local behavior of infinite-dimensional stochastic dynamical systems near hyperbolic equilibria (or stationary solutions).

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Anticipating stochastic differential systems with memory" (with T. S. Zhang), Mittag-Leffler Institute Preprint Series -12, 2007/2008, fall, pp.32.

Abstract:

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfde's) and fully nonlinear sfde's with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Ito-Ventzell formula due to Ocone and Pardoux. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite-dimensionality of the initial condition.

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Anticipating stochastic differential systems with memory" (with T. S. Zhang), Stochastic Processes and their Applications (2009).

Abstract:

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfde's) and fully nonlinear sfde's with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Ito-Ventzell formula due to Ocone and Pardoux. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite-dimensionality of the initial condition.

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Dynamics of Stochastic 2D Navier-Stokes Equations" (with T. S. Zhang), Journal of Functional Analysis, Volume 258, Issue 10, (2010), 3543-3591.

Abstract:

In this paper, we study the dynamics of a two-dimensional stochastic Navier-Stokes equation on a smooth domain, driven by multiplicative white noise. We show that solutions of the 2D Navier-Stokes equation generate a smooth perfect and locally compacting $C^{1,1}$- cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. We give sufficient conditions on the parameters of the Navier-Stokes equation and the geometry of the planar domain which guarantee hyperbolicity of the equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.

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Burgers Equation with Affine Linear Noise: Dynamics and Stability" (with T. S. Zhang) (2011).

Abstract:

It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we compute the discrete non-random Lyapunov spectrum  of the linearized cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. In particular, we construct a countable random family of local asymptotically flow-invariant manifolds, so that on each manifold the cocycle decays with fixed exponential speed. Each local manifold is smooth and has finite fixed codimension. On a global level, we show the existence of a flow-invariant random flag in the energy space. The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift, we also give sufficient conditions on the parameters of the stochastic Burgers equation which guarantee uniqueness of the stationary solution or its hyperbolicity. In the hyperbolic case, we establish a local stable manifold theorem near the zero equilibrium.

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Invariant Manifolds for Stochastic Models in Fluid Dynamics", Stochastics and Dynamics, 2011, vol. 11, Issues: 2-3, (2011), 439-459.

Abstract:

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle. The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao.

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Abstract:

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by affine linear (multiplicative) white noise. It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we compute the discrete non-random Lyapunov spectrum  of the linearized cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. In particular, we construct a countable random family of local asymptotically flow-invariant manifolds, so that on each manifold the cocycle decays with fixed exponential speed. Each local manifold is smooth and has finite fixed codimension. On a global level, we show the existence of a flow-invariant random flag in the energy space. The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift, we also give sufficient conditions on the parameters of the stochastic Burgers equation which guarantee uniqueness of the stationary solution or its hyperbolicity. In the hyperbolic case, we establish a local stable manifold theorem near the zero equilibrium. To appear

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Stochastic Burgers Equation with Random Initial Velocities: A Malliavin Calculus Approach" (with T. S. Zhang),  SIAM Journal of Mathematical Analysis, 45 (4),  (2013), 2396-2420.

Abstract:

In this article we prove an existence theorem for solutions of the stochastic Burgers equation (SBE) on the unit interval with Dirichlet boundary conditions and anticipating initial velocities. The SBE is driven by affine (additive + linear) noise. In order to establish the existence theorem, we adopt a somewhat counter-intuitive perspective in which stochastic dynamical systems ideas lead to existence of solutions rather than vice versa. More specifically, our approach uses the Malliavin calculus and is based on the existence and regularity of a perfect cocycle on the energy space for the SBE. The proof of the existence theorem requires Malliavin regularity of the infinite-dimensional initial velocity field together with new spatial estimates on the cocycle, its Fr\'echet and Malliavin derivatives. The existence theorem provides a dynamic characterization of solutions of the {\it non-anticipating} SBE on its unstable invariant manifolds. Furthermore, as a corollary of the existence theorem, we show that random cocycle-invariant points on the energy space correspond to (possibly non-ergodic) stationary pathwise solutions for the SBE.

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Anticipating Stochastic $2D$ Navier-Stokes Equations" (with T. S. Zhang), Journal of Functional Analysis, vol. 264, 6, (2013), 1380-1408.

In this article, we consider  the two-dimensional stochastic Navier-Stokes equation (SNSE) on a smooth bounded domain, driven by affine-linear multiplicative white noise and with random initial conditions and Dirichlet boundary conditions. The random initial condition is allowed to anticipate the forcing noise.  Our main objective is to prove the existence and uniqueness of the solution to the SNSE under sufficient Malliavin regularity of the initial condition. To this end we employ anticipating calculus ideas.

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 Sobolev Differentiable Stochastic Flows for SDE's with Singular Coefficients: Applications to the Transport Equation" (with T. Nilssen and F. Proske), Annals of Probability, Volume 43, Number 3 (2015), 1535-1576, (pp. 41)

Preprint:

In this paper, we establish the existence of a stochastic flow of Sobolev  diffeomorphisms for  an SDE driven by a bounded measurable drift and additive Brownian motion. The result is counter-intuitive, since the dominant  culture' in stochastic (and deterministic) dynamical systems is that the flow inherits' its spatial regularity from the driving vector fields.  The spatial regularity of the stochastic flow yields existence and  uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation.  It is well-known that the  deterministic transport equation does not in general have a solution.

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 Sobolev Differentiable Stochastic Flows for SDE's with Singular Coefficients: Applications to the Transport Equation" (with T. Nilssen and F. Proske) (2014) (pp. 37) (preprint)

Abstract:

In this paper, we establish the existence of a stochastic flow of Sobolev  diffeomorphisms for  an SDE driven by a bounded measurable drift and additive Brownian motion. The result is striking, since the dominant  culture' in stochastic (and deterministic) dynamical systems is that the flow inherits' its spatial regularity from the driving vector fields.  The spatial regularity of the stochastic flow yields existence and  uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation.  It is well-known that the  deterministic transport equation does not in general have a solution. Using stochastic perturbations and our analysis of the above SDE, we establish a deterministic flow of Sobolev diffeomorphisms for classical one-dimensional (deterministic) ODE's  driven by discontinuous vector fields. Furthermore, and as a corollary of the latter result, we construct a Sobolev stochastic flow of diffeomorphisms for one-dimensional SDE's driven by discontinuous diffusion coefficients.

Preprint:

• postscript file (587K, 37 pages)
• pdf file (315K, 37 pages)

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Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives " (with B. Oksendal, E. E. Rose and K. R. Dahl  ) Journal of Functional Analysis, 271, (2016), 289-329 (pp. 41).

Abstract:

In this article we consider a stochastic optimal control problem where the dynamics of the state process, $X(t)$, is a controlled stochastic differential equation with jumps, delay and noisy memory. The term noisy memory is, to the best of our knowledge, new. By this we mean that the dynamics of the state $X(t)$ depends on $\int_{t-\delta}^t X(s) dB(s)$ (where $B(t)$ is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process. We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Oksendal, Sulem and Zhang. The maximum principles also apply to the case where the controller only has partial information, in the sense that the admissible controls are adapted to a sub-$\sigma$-algebra of the natural filtration.

Preprint:

• pdf file (546K, 41 pages)

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May 19, 2016