I have worked in the following areas. Please select a topic:

- Deterministic Functional Differential Equations on Manifolds
- Stochastic Hereditary Equations
- Regular Linear and Affine Stochastic Hereditary Systems
- Non-linear Stochastic Hereditary Systems
- The Malliavin Calculus. Hypoellipticity.
- Finite-dimensional stochastic flows. The Stable manifold theorem.
- Stochastic Numerics and Finance

- Finite and Infinite-dimensional stochastic dynamical systems

Let $X$ be a manifold. Typically we take $X$ to be a Riemannian manifold (finite or infinte-dimensional) or a Banach manifold with a sufficiently smooth linear connection. Consider a manifold of paths $ P([-r,0],X)$ which inherits its differentiable structure from the ambient manifold $X$. A retarded functional differential equation (RFDE) on $X$ is a continuous map $F:[0,a)\x P([-r,0],X) \to TX$ such that for each $(t,\theta) \in [0,a)\x P([-r,0],X)$ the vector $F(t,\theta) \in T_{\theta (0)} X$, the tangent space to $X$ at $\theta (0)$. A trajectory of $F$ is a $C^1$ path $x:[-r,a)\to X$ such that

x'(t) =F(t,x_t), t\in [0,a)

x_0 = \theta \in P([-r,0],X).

In the above equation $x_t$ stands for the segment $x|[t-r,t]$ of the solution $x$. In Chapter 1, I develop a localization technique (Lemma 1.1, [A 1]) in order to obtain a unique local trajectory for the above initial value problem ([A 1], Theorem 1.2, Chapter 1, p.22). This is done under mild regularity conditions on $F$, assuming that $X$ is a Banach manifold which admits a linear connection and $P=L_1^2$, the space of all Sobolev paths $\theta$ on $X$ with square integrable derivatives. If $X$ carries a Finsler and $F$ satisfies suitable growth conditions, I prove that global trajectories of the hereditary equation exist for all time. Hence one gets a semiflow $R^{+} \x L_1^2 \to L_1^2$ on the space of initial paths $ L_1^2=L_1^2([-r,0],X)$ ([A 1], Theorems (1.3)-(1.5)).

The main objective of Chapter 2 ([A 1]) is to characterize the
topological structure of the critical set $\{\theta : F(\theta)=0\}$
when $F$ is autonomous and $X$ is $n$-dimensional, smooth Riemannian.
It is first proved that solutions of the hereditary equation may reach
equilibrium by converging asymptotically to a constant critical path,
just as for vector fields (Theorem 2.1, [A 1]). The key idea here is to
use parallel transport to show that a smooth hereditary coefficient $F$
pulls back into a a smooth vector field $\xi^{F}$ on $L_1^2$ ([A 1],
Theorem 2.2 and corollary). In spite of the infinite degeneracy of the
critical set $C(F)$, we are able to isolate a class of gradient-like
hereditary
equations for which the critical set is a closed smooth submanifold of
$L_1^2$
with codimension $n$ ([A 1], Proposition 2.4, p. 59). A Morse index
exists
for this class of hereditary equations ([A 1], Proposition 2.5, p.60).
The
index is constant on each connected component of $C(F)$. When $X$ is
compact,
one can count the number of *critical components* in $C(F)$. This
way
I prove Morse inequalities for such hereditary equations ([A 1],
Theorem
(2.4) and corollaries). In particular it follows from these
inequalities
that $F$ has only a finite number of critical components. The number of
critical
components with index $m$ is always greater than or equal to the $m$-th
Betti
number of $X$, the rank of its $m$-th singular homology group ([A 1],
Corollary
2.4.2).

In Chapter 3 ([A 1]), I linearize the semiflow of $F$ by
differentiating the canonical vector field $\xi^{F}$ covariantly in
$L_1^2 ([-r,0],X)$.
This linearization defines a compacting semiflow on the tangent bundle
$T
L_1^2 ([-r,0],X)$ ([A 1], Theorem 3.3, p.79). Using semigroup
techniques
along the fibers of $T L_1^2 ([-r,0],X)$ we construct a Whitney direct
sum
splitting of the tangent bundle into two subbundles: the *unstable*
and the * stable *one. Cf. classical results of Hale in the flat
case
$X=R^n$. This splitting is invariant under the linearized semiflow. The
unstable
subbundle is *finite-dimensional*, and on it the linearized
semiflow
can be continued backwards to give a genuine *flow *which is
defined
for all time. Within the stable subbundle the linearized semiflow
decays
exponentially fast in the Sobolev Riemannian metric along each fiber in
$T
L_1^2 ([-r,0],X)$. This is the *Stable Bundle Theorem *([A 1],
Theorem
3.6, p.100).

Vector fields on the ambient manifold $X$ are used in Chapter 4 ([A
1]) to generate examples of FDE's on the manifold. These include
classical vector fields, *differential-delay equations* (DDE's),
the *delayed development* and the *Levin-Nohel equation.*
It is shown in Theorem 4.1 ([A 1], p.109) that a gradient Levin-Nohel
equation on a Riemannian manifold may not
admit non-trivial periodic solutions. I also give a detailed study of
the
*Functional Heat equation* (FHE) in this chapter of the monograph.
The
FHE is shown to correspond to a discontinuous but closed FDE on the
Fr\'echet space of smooth functions on a compact Riemannian manifold.
It is interesting to note here that despite the discontinuity of the
equation and the infinite-dimensionality of the function space, the FHE
still displays dynamical properties very similar to those of continuous
finite-dimensional FDE's. In general, however, the FHE can be solved
forward in time only along a closed Fr\'echet subspace of
the state space. *Backward* solutions of the FHE do exist on the
complementary
subspace in the hyperbolic case. See [A 1], Chapter 4\S 5, pp. 113-133.

In [A 2] I study the stochastic hereditary system

dx(t) = H(t,x_{t}) dt + G(t,x_{t}) dW(t), t > 0

x_{0} = \eta \in C.

where $C=C([-r,0],R^n)$ and $W$ is multidimensional Brownian motion. The coefficients $H,G$ are non-linear functionals on the state space $C$. In [A 2], it is shown that -- under sufficiently general local Lipschitz and global linear growth conditions on the coefficients $H$ and $G$ - the above equation admits a unique continuous solution in $ L^2 (\Omega , C([-r,a], R^{n}))$ for every $a > 0$, ([A 2], Chapter II, Theorem (2.1)). Furthermore, the trajectory field $\{ ^{\eta}x_{t}: t \geq 0, \eta \in C\}$ is a continuous map $R^{+} \times C \to L^2 (\Omega ,C)$ and gives a $C$-valued Feller process ([B 2], [A 2], Theorems III(1.1), p. 51, III(2.1), p. 64, Theorem III(3.1), p. 67). When the delay $r$ is positive, I showed that the semi-group $P_{t}$

P_{t}(\phi )(\eta ) = E \phi [ ^{\eta}x_{t} ], \eta \in C, t \geq 0

is *never* strongly continuous on the Banach space $C_{b}$ of
all bounded uniformly continuous functions $\phi : C \to R$ with the
sup norm (Mohammed [A 2], Chapter (IV), Theorems (2.1), (2.2), (3.2),
pp. 70-97).
The semi-group is however weakly continuous and an explicit formula for
the
weak infinitesimal generator $A: D(A) \subset C_{b} \to C_{b}$ is
established
in ([A 2], Theorems IV(3.2), IV(4.1), IV(4.2), IV(4.3)). Due to the
absence
of non-trivial differentiable functions with bounded support in $C$ and
due to the fact that most tame functions lie outside the domain of the
weak infinitesimal generator, I introduced the class of *quasi-tame
functions* on the state space. This class consists of smooth
functions, is weakly dense and is rich enough to generate the Borel
$\s$-field of $C$. Furthermore the generator $A$ assumes a simple and
concrete form on quasi-tame functions (Definition 4.2, p.105, Theorems
IV (4.2), (4.3) in [A 2]). For each fixed $(v, \eta )$
in the state space $M_{2}:=R^n \x L^2([-r,0],R^n)$, the ``one-point
motion'' $\{ X(t,., (v, \eta )): t \geq 0\}$ in $M_{2}$ -- though a
Markov process -- is *not* a semi-martingale (c.f. the
finite-dimensional stochastic ODE case). This presents some
difficulties in dealing with non-linear transforms $\phi [X(t,., (v,
\eta ))]$ under smooth functions $\phi : M_{2} \to R$. On
the other hand the above analysis shows the existence of the class of
smooth
quasitame functions $\phi : M^{2} \to R$ for which $\phi
[X(t,.,(v,\eta))]$ satisfies an It\^{o}-type formula. It is clear that
this fact will play a significant role in the computation of *Lyapunov
exponents* for specific cases of *singular* hereditary
systems.

The above results point the way towards the need for a workable theory of parabolic P.D.E.'s in infinite dimensions (c.f. work by Daletskii). If such a theory is established, one hopes to obtain in the long term some results on the existence of invariant probability measures on the Hilbert space $M_{2}$ for the trajectory process $\{(x(t),x_{t}): T \geq 0\}$ (c.f. the finite-dimensional stochastic ODE case $r = 0$).

The *almost sure* dependence of the trajectory $^{\eta} x_t$
on the initial path $\eta$ is a delicate problem. For the
one-dimensional *linear* stochastic delay equation

dx(t)=x(t-r)dW(t), t>0

with a *positive* delay $r$, I obtained the surprising result
that the trajectory random field $ { X(t,\eta ) \equiv ^{\eta}x_{t} : t
\geq
0, \eta \in C }$ has no progressively measurable versions which are
a.s.
* linear* when viewed as operators $C \to C$ ([A 2], [B 7]). To my
knowledge, this fact has never been observed within the context of
linear ordinary differential equations (without memory) whether
stochastic or deterministic. In contrast, one should note the
diffeomorphism property for stochastic o.d.e.'s. See
the works of Kunita, Bismut, Baxendale, Ikeda and Watanabe and
Elworthy.
Note also the continuous (linear) dependence on the initial paths for
solutions of linear FDE's. Consult work by J. K. Hale and others. The
a.s. non-linearity of the sample functions of the random field is
attributed to local unboundedness of these sample functions due to the
presence of ``delayed diffusion'' terms $x(t -r) dW (t) (r > 0)$.
These terms seem to occur in examples and applications where the effect
of noise in the negative feedback loop cannot be neglected. Because of
this highly pathological behavior of the trajectory random field, I
recently introduced a new classification of stochastic differential
equations into *regular* and *singular* types [B 15]. On
the other hand, when
the diffusion coefficients do not look into the past, one gets
extremely regular
almost sure dependence of the trajectory field on the initial paths ([A
2],
Theorem V (2.1) and its corollaries, pp. 121-142). Under a Frobenius
type
condition on the diffusion matrix, I prove the existence of
sufficiently smooth
and a.s. locally compactifying versions of the trajectory field for $t
\geq
r$. In general the compactifying nature of the trajectory field is
shown
to persist in a *distributional sense* even in the singular case
([A
2], Theorems V (4.6) and (4.7)).

In Chapter VI [A 2], I study several classes of stochastic
hereditary
systems. Stochastic equations with several random delays are analyzed
in
VI \S 3 (pp. 167-186). The trajectory field in this case is *not*
a
Markov process in $C$. However, if the delays are essentially bounded
and
independent of the driving white noise, then one can show that the
transition
measures of the trajectory field correspond in the mean to a
measure-valued
process whose values are genuine transition probabilities of a Markov
process
in $C$. See [A 2], Theorem VI (3.1) and Lemma VI (3.3). The latter
result
asserts the stability of the trajectory with respect to random
perturbations
in the delay processes. As a consequence of these results one gets a
sufficient
condition for global asymptotic stability in distribution in terms of
the
corresponding property for the associated system with arbitrary *fixed
deterministic delays* ([A 2], Corollary VI 3.1.2, p. 184).

dx(t) = H(x(t - r_{1}), x(t-r_{2}), ... , x(t - r_{k}), x(t),x_{t})
dt

+ g(x(t)) dW(t)

where $H: (R^{n})^{k} \times M_{2} \to M_{2}$ is continuous linear and $g: R^{n} \to R^{n \times m}$ is linear.

It is proved in [B 11] that the above system is regular in $M_{2}$
and its trajectory random field admits a jointly measurable version $X:
R^{+} \times \Omega \times M_{2} \to M_{2}$ such that, for each $t \in
R^{+}$
and a.a. $\omega \in \Omega$, the map $X(t,\omega,.)$ is a continuous
linear
operator on $M_{2}$. In fact I showed that $X$ is a linear cocycle over
the
standard Brownian shift $\theta$ on path space ([B 11], Theorem 3 \S
3).
In ([B 11 ], Theorem 4) I proved an Oseledec-type multiplicative
ergodic
theorem which gives a countable almost surely non-random Lyapunov
spectrum
for the stochastic flow $X$. The almost sure Lyapunov spectrum is
bounded
above and has no finite accumulation points ([B 11], Theorem 4, \S 4).
I
proved the *Stable Manifold Theorem* for regular linear
hyperbolic systems
in [B 11], Theorem 4, \S 4, Corollary 2, pp. 117-130). The proof of
this
theorem uses deep infinite-dimensional multiplicative ergodic theory
methods
( [B 11], cf. work of Ruelle, Ma\~{n}\'{e}, Thieullen, Flandoli and
Schauml\"offel).

In joint work with M. Scheutzow ([B 14], Theorem 4.2), we proved the existence of a cocycle in $M_{2}$ for the trajectory field of the much more general class of regular stochastic linear hereditary equations:

dx(t) = { \int_{[-r,0]} \mu (t) (ds) x(t + s) } dt

+ dN(t) \int^{0}_{-r} K(t)(s) x(t+s) ds + dL(t) x(t-), t > 0.

In the above hereditary equation, $\mu(t)$ is a stationary (ergodic) measure-valued process, $N(t)$, $L(t)$ are jump semi-martingales with stationary (ergodic) increments. The process $K(t)(s)$ is matrix-valued and stationary in $t$. The increments of $L$ and $N$ may depend on $\mu(t)$ and $K(t)$. The non-delay case was studied by L. Arnold and W. Kliemann when the stationary coefficients are assumed to be independent of the increments of the driving noise. Under fairly general assumptions, we prove that that the above equation has a stochastic flow with a countable set of Lyapunov exponents and a flow-invariant exponential dichotomy in the hyperbolic case ([B 14], Theorems 5.2 and 5.3). In the course of proving these results we develop a new technique for constructing stochastic flows for (linear) stochastic ODE's driven by continuous semimartingales. See [B 14], Theorem 3.1, \S 3.

In joint work with M. Scheutzow ([B 10]), we studied affine linear stochastic hereditary systems of the form

dx(t) = \int^{0}_{-r} \mu (ds) x(t + s) dt + dQ(t).

We proved the regularity of the stochastic flow and gave a detailed study of the Lyapunov spectrum and the existence of stationary solutions of the affine hereditary equation. A summary of these results may be found in my survey article ( [B 15] \S 3 C, Theorems 10, 11, 12, 13). Details are given in [B 10]. Under suitable growth conditions on the driving noise $Q$, the existence of the $p$th moment Lyapunov exponent

g(p) := \lim_{t \to \infty} \frac{1}{t} \log E \parallel x_{t} \parallel^{p}_{\infty} , p \geq 1

is proved in ([B 10]). See also [B 15], \S 3, C, Theorem 14. It is
interesting to note here that the above result asserts the existence of
*only one* $p$-th moment Lyapunov exponent under mild
non-degeneracy conditions. Furthermore the $p$-th moment exponent is
independent of all random (possibly anticipating) initial paths in the
Skorohod space $D([-r,0],R^n)$. This is surprising
if we view the affine system as a *finite-dimensional* stochastic
perturbation of the *infinitely degenerate* deterministic
homogeneous system $(Q \equiv
0)$:

dy(t) = { \int_{[-r,0]} \mu (ds) y (t + s) } dt

with a *countably infinite* Lyapunov spectrum. The latter
spectrum coincides with the set of real parts $\{ ... < \beta_{3}
< \beta_{2} < \beta_{1} \}$ of all roots of the characteristic
equation

\det (\lambda I - \int_{[-r ,0]} e^{\lambda s} \mu (ds)) = 0.

For the affine hereditary system one generically has

\lim_{t \to \infty} \frac{1}{t} \log E \parallel ^{\eta}x_{t} \parallel^{p}_{\infty}

equal to $p \beta_{1}$ for all random (possibly anticipating) initial conditions $\eta \in D([-r ,0], R^{n}$) ([B 10], [B 15], \S 3 C, Theorem 14). Estimates on the second-moment exponent $(p = 2)$ were previously obtained by Mohammed, Scheutzow and Weizs\"{a}cker [B 6] and Mohammed [B 5].

For several examples of one-dimensional linear stochastic hereditary equations, we obtained upper bounds on the top almost sure Lyapunov exponent $\lambda_{1}$ in joint work with M. Scheutzow. Some of this work is still in progress and is currently funded by a collaborative research grant from NATO, with M. Scheutzow. Estimates on $\lambda_{1}$ for the following equations appear in ([B 15], \S 4, Theorems 15, 16, 17):

dx(t) = x((t-1)-) dN(t) t > 0 \\ x_{0} \in D([-1, 0], R)

where $N$ is a compound Poisson process;

dx(t) = \{\nu x(t) + \mu x(t-r)\} dt + { \int^{0}_{-r} x(t+s) \sigma (s) ds } dW(t),

dx(t) = \{\nu x(t) + \mu x(t-r) \} dt + x(t) dM(t), t > 0,

where $W$ is a Wiener process and $M$ is a one-dimensional sample continuous square integrable martingale with stationary ergodic (but not necessarily independent) increments. These estimates can be found in [B 15], \S 4.

For small noise, I proved global asymptotic $ L^{2}$-stability of $X(t, (v,\eta ))$ for stochastic hereditary systems of the form

dx(t) = H(x(t),x_{t})dt + \epsilon G(x(t),x_{t}) dW(t), t > 0

where the deterministic drift $H$ is continuous *linear* and
globally asymptotically stable ([B 5]).

We investigated the linear stochastic hereditary system

dx(t) = H(x_{t})dt + GDW(t)

with constant diffusion matrix $G$ through joint work with H. von Weizs\"{a}cker and M. Scheutzow ([B 6]). In this work, the classical (hyperbolic) splitting of the state space into a stable subspace $ S$ and a finite-dimensional unstable subspace $ U $ is used to determine the $ L^{2}$ asymptotic stability of the random field $\{ X(t,(v, \eta )): t \geq 0, (v, \eta ) \epsilon M_{2}\}$. In particular for each $(v, \eta ) \in S, \lim_{t \to \infty} X(t, (v, \eta ))$ exists in $ L^{2}(\Omega , M_{2})$ and its distribution is an invariant Gaussian measure on $M_{2}$. The convergence has exponential rate which is uniform with respect to the initial state $(v,\eta )$ ([A 2], Theorem 4.2, pp. 208-216). On the other hand, if $(v, \eta ) \in U$, then $\parallel X(t, (v, \eta ))\parallel$ goes to infinity exponentially fast in the $ L^{2}$ sense ([B 8]). Again, the exponential speed of explosion in this case is uniform over $ U$. I applied these results to study the stability of the ``heat-bath'' physical model ([A 2], pp. 223-226).

For the last two mentioned stochastic linear and affine hereditary systems , I proved the existence of a class of unstable distributions within the set of all probability measures on $M_{2}$. This class of unstable distributions is invariant under the adjoint semi-group $P^{*}_{t}$ ([B 8]).

Part of my research efforts are directed towards investigating the a.s. local behavior of stochastic flows of regular non-linear autonomous stochastic hereditary systems. This project is joint work with M. Scheutzow and supported by two NSF grants DMS-9206785 (1992-1995) and DMS-9503702 (1995-1997).

dx(t) = g(x(t-r)) dW(t)

with a positive delay $r > 0$. Such systems were previously
studied by Kusuoka and Stroock in the case when $g$ is smooth and *bounded
uniformly away* from zero. Under the above strong non-degeneracy
condition, S.
Kusuoka and D. Stroock proved that the solution of the above equation
has
a smooth density with respect to Lebesque measure on Euclidean space.
Needless
to say the Kusuoka-Stroock result *excludes* the singular *linear*
case. In recent joint work with D. Bell, we proved that the solution of
the above equation admits a smooth density for fixed $t$ and $(v, \eta
)$,
even if the vector field $g$ has several polynomial-type degeneracies
([B
12 ], [B 13]). This result is obtained using the Malliavin calculus and
new
probabilistic bounds on the segment $x_{t}$ of the solution (e.g., [B
12],
Lemma 4, \S 4, [B 17], Theorem 1 \S 2).

I worked with D. Bell on the relationship between degenerate
stochastic ODEs, elliptic parabolic partial differential operators ([B
18]). This work appeared in *Duke Mathematical Journal*. We
establish a ``maximal" extension of H\"ormander's classical
hypoellipticity theorem, whereby H\"ormander's general condition is
allowed to fail on a collection of hypersurfaces in
Euclidean domains. The proof is based on the Malliavin calculus and
involves
the derivation of new probabilistic estimates for multidimensional
time-dependent
degenerate diffusion processes. We allow several *moving*
degeneracy
hypersurfaces of *infinite* (exponential) order in the diffusion
covariance.
These degeneracy surfaces are called *non-H\"ormander sets (of
parabolic
type)* . H\"ormander's general Lie algebra condition fails for these
classes
of operators. See Theorems 1.1, 1.2 in \S 1 [B 18]. In particular
Theorem
1.1 ([B 18]) establishes parabolic hypoellipticity of the partial
differential operator when the second-order coefficient matrix can have
a degeneracies of exponential order $p$, with $p \in (0,1)$. These
degeneracies may occur anywhere on a finite set of moving hypersurfaces
in Euclidean space. Furthermore the range $(0,1)$ for $p$ is *optimal.*
As far as I know these results cannot be obtained using classical PDE
techniques, e.g. weighted Sobolev spaces.
In the course of proving the main theorems, we obtain several results
concerning
the existence of smooth densities for time-dependent degenerate
stochastic
hereditary and ordinary differential equations ([B 18], [B 19], \S 2,
Theorems
2.1-2.3).

In joint work with M. Scheutzow, 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 ([B 23]). Both Stratonovich and It\^o-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.

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 It\^ o 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. Given the

methods appears to be novel. These results have appeared in {\it The Annals of

Probability}.

In [B 29] (joint work with M. Arriojas, Y. Hu and Y. Pap) we develop a

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.

Stochastic ordinary differential equations (sodes) on finite-dimensional manifolds generate stochastic flows on the manifold. One objective of the research is to construct invariant manifolds for such flows near stationary solutions, under suitable regularity and growth conditions on the driving vector fields. In particular, this yields the existence of stable, unstable and center manifolds near each stationary point on the manifold.

An important class of infinite-dimensional semiflows on
Hilbert space
is generated by dissipative semilinear stochastic partial
differential equations (spdes) on smooth compact manifolds or
smooth bounded Euclidean domains. For these semiflows, we
construct 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 ([B 33], [B 34], [B 35]). This is joint
work with T. S. Zhang
and H. Zhao. Important examples of spdes covered by this analysis
are
Burger's equation, affine linear stochastic evolution equations
and
stochastic reaction-diffusion equations. The results of the research
reveal
new features of the stochastic dynamics of these
well-studied
models.

One encounters models of stochastic systems with memory (sfdes)
in many engineering and physical applications. Deterministic
smooth constraints on the solutions of such models lead naturally
to sfdes on (compact) Riemannian manifolds. The article [B 25] is
joint work with R. L\'eandre. In this article

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.

NSF
Highilghts (pdf file)

To download, click
on the following links:

February 20, 2007