123 research outputs found
Martingale representations for diffusion processes and backward stochastic differential equations
In this paper we explain that the natural filtration of a continuous Hunt
process is continuous, and show that martingales over such a filtration are
continuous. We further establish a martingale representation theorem for a
class of continuous Hunt processes under certain technical conditions. In
particular we establish the martingale representation theorem for the
martingale parts of (reflecting) symmetric diffusions in a bounded domain with
a continuous boundary. Together with an approach put forward in Lyons et
al(2009), our martingale representation theorem is then applied to the study of
initial and boundary problems for quasi-linear parabolic equations by using
solutions to backward stochastic differential equations over the filtered
probability space determined by reflecting diffusions in a bounded domain with
only continuous boundary.Comment: 28 page
Markov semi-groups generated by elliptic operators with divergence-free drift
In this paper we construct a conservative Markov semi-group with generator
on , where is a divergence-free
vector field which belongs to with . The
research is motivated by the question of understanding the blow-up solutions of
the fluid dynamic equations, which attracts a lot of attention in recent years.Comment: 12 page
Parabolic equations with divergence-free drift in space
In this paper we study the fundamental solution of
the parabolic operator , where for
every , is a divergence-free vector field, and we consider the
case that belongs to the Lebesgue space
. The regularity of
weak solutions to the parabolic equation depends critically on the
value of the parabolic exponent . Without the
divergence-free condition on , the regularity of weak solutions has been
established when , and the heat kernel estimate has been obtained
as well, except for the case that . The regularity of weak
solutions was deemed not true for the critical case
for a general
, while it is true for the divergence-free case, and a written proof can be
deduced from the results in [Semenov, 2006]. One of the results obtained in the
present paper establishes the Aronson type estimate for critical and
supercritical cases and for vector fields which are divergence-free. We
will prove the best possible lower and upper bounds for the fundamental
solution one can derive under the current approach. The significance of the
divergence-free condition enters the study of parabolic equations rather
recently, mainly due to the discovery of the compensated compactness. The
interest for the study of such parabolic equations comes from its connections
with Leray's weak solutions of the Navier-Stokes equations and the Taylor
diffusion associated with a vector field where the heat operator
appears naturally.Comment: 31 page
Optimal probabilities and controls for reflecting diffusion processes
A solution to the optimal problem for determining vector fields which
maximize (resp. minimize) the transition probabilities from one location to
another for a class of reflecting diffusion processes is obtained in the
present paper. The approach is based on a representation for the transition
probability density functions. The optimal transition probabilities under the
constraint that the drift vector field is bounded by a constant are studied in
terms of the HJB equation. In dimension one, the optimal reflecting diffusion
processes and the bang-bang diffusion processes are considered. We demonstrate
by simulations that, even in this special case, the optimal diffusion processes
exhibit an interesting feature of phase transitions. We also solve an optimal
stochastic control problem for a class of stochastic control problems involving
diffusion processes with reflection.Comment: 20 Pages, 2 figure
Large deviation principle for fractional Brownian motion with respect to capacity
We show that fractional Brownian motion(fBM) defined via Volterra integral
representation with Hurst parameter is a quasi-surely
defined Wiener functional on classical Wiener space,and we establish the large
deviation principle(LDP) for such fBM with respect to -capacity on
classical Wiener space in Malliavin's sense
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