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``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 ét Statistiques Vol. 32,
1, (1996), 69-105.
``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.
(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.
``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.
``Smooth
Densities for Degenerate Stochastic Delay Equations with Hereditary
Drift"
(with Denis Bell),
The Annals of Probability, Vol. 23, No. 4, (1995),
1875-1894.
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 non-anticipating functional defined
on the space of paths in $R^d$ and $g$ is an $nxd$-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.
``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).
``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.)
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.
``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.
``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.
``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).
``
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).
``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).
``
Stochastic Functional Differential Equations on Manifolds " (with
R.
Leandre), Probab Theory Related Fields 121 (2001) 1,
117-135
)
(preprint
available upon request).
Preprint:
``Discrete-time Approximations of
Stochastic
Differential Systems with Memory " (with Y. Hu and F. Yan), (2001) pp.
71 (preprint available
upon request).
Preprint:
``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).
Preprint:
`` 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.
Preprint:
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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
Itô'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 Itô 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.
Preprint:
``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. ``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.Preprint:
`` The Stable Manifold Theorem for Semilinear
Stochastic Evolution Equations and Stochastic Partial Differential
Equations.
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 $C^k$-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.
Preprint:
``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.
Preprint:
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`` A Delayed Black and Scholes Formula"
(with
M. Arriojas, Y. Hu and G. Pap), Journal
of Stochastic Analysis and Applications}, vol. 25, no. 2 (2007),
471 - 492 (preprint
available
upon request).
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``Stochastic
Dynamical Systems in Infinite Dimensions", Preprint 03 (2007),
Mittag-Leffler Preprints Series 2007, to appear in Trends in Stochastic Analysis,
edited by Jochen Blath, Peter Morters and Michael Scheutzow, London
Mathematical Society Lecture Note Series, Cambridge University Press,
(2008), 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).Preprint:
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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:
