395 research outputs found
Homogenization of a singular random one dimensional parabolic PDE with time varying coefficients
The paper studies homogenization problem for a non-autonomous parabolic
equation with a large random rapidly oscillating potential in the case of one
dimensional spatial variable. We show that if the potential is a statistically
homogeneous rapidly oscillating function of both temporal and spatial variables
then, under proper mixing assumptions, the limit equation is deterministic and
the convergence in probability holds. To the contrary, for the potential having
a microstructure only in one of these variables, the limit problem is
stochastic and we only prove the convergence in law
On the Poisson equation and diffusion approximation 3
We study the Poisson equation Lu+f=0 in R^d, where L is the infinitesimal
generator of a diffusion process. In this paper, we allow the second-order part
of the generator L to be degenerate, provided a local condition of Doeblin type
is satisfied, so that, if we also assume a condition on the drift which implies
recurrence, the diffusion process is ergodic. The equation is understood in a
weak sense. Our results are then applied to diffusion approximation.Comment: Published at http://dx.doi.org/10.1214/009117905000000062 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media
We establish an averaging principle for a family of
solutions
of a system of
SDE-BSDEwith a null recurrent fast component . Incontrast
to the classical periodic case, we can not rely on aninvariant probability and
the slow forward component cannot be approximated by a
diffusion process.On the other hand, we assume that the coefficients admit a
limit in a\`{C}esaro sense. In such a case, the limit coefficients may
havediscontinuity. We show that we can approximate the
triplet bya
system of SDE-BSDE where is aMarkov diffusion
which is the unique (in law) weak solution of theaveraged forward component and
is the unique solution to the averaged backward component. This is done
with a backward component whosegenerator depends on the variable .
Asapplication, we establish an homogenization result for semilinearPDEs when
the coefficients can be neither periodic nor ergodic. Weshow that the averaged
BDSE is related to the averaged PDE via aprobabilistic representation of the
(unique) Sobolev --solution of the
limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich
are based on stability property and weak convergence of BSDEsin the S-topology
A general comparison theorem for 1-dimensional anticipated BSDEs
Anticipated backward stochastic differential equation (ABSDE) studied the
first time in 2007 is a new type of stochastic differential equations. In this
paper, we establish a general comparison theorem for 1-dimensional ABSDEs with
the generators depending on the anticipated term of .Comment: 8 page
Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators
The aim of the present paper is to study the regularity properties of the
solution of a backward stochastic differential equation with a monotone
generator in infinite dimension. We show some applications to the nonlinear
Kolmogorov equation and to stochastic optimal control
Lyapounov exponent of linear stochastic systems with large diffusion term
AbstractWe study the behaviour of the Lyapounov exponent of the solution of dXtî
AXt+ÎŽâiî
1rBkXtâdWkt, as Ïââ. We obtain the exact behaviour in two cases for arbitrary dimensions, and in most cases for the two dimensional equation
Numerical Schemes for Multivalued Backward Stochastic Differential Systems
We define some approximation schemes for different kinds of generalized
backward stochastic differential systems, considered in the Markovian
framework. We propose a mixed approximation scheme for a decoupled system of
forward reflected SDE and backward stochastic variational inequality. We use an
Euler scheme type, combined with Yosida approximation techniques.Comment: 13 page
Rate of Convergence of Space Time Approximations for stochastic evolution equations
Stochastic evolution equations in Banach spaces with unbounded nonlinear
drift and diffusion operators driven by a finite dimensional Brownian motion
are considered. Under some regularity condition assumed for the solution, the
rate of convergence of various numerical approximations are estimated under
strong monotonicity and Lipschitz conditions. The abstract setting involves
general consistency conditions and is then applied to a class of quasilinear
stochastic PDEs of parabolic type.Comment: 33 page
Homogenization of semi-linear PDEs with discontinuous effective coefficients
20 pagesInternational audienceWe study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in \`{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "viscosity solution" introduced in \cite{CCKS}. We use BSDEs techniques to establish the existence of viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of -viscosity solution
A SIR model on a refining spatial grid: Law of Large Numbers
We study in this paper a compartmental SIR model for a population distributed
in a bounded domain D of , d= 1, 2, or 3. We describe a spatial
model for the spread of a disease on a grid of D. We prove two laws of large
numbers. On the one hand, we prove that the stochastic model converges to the
corresponding deterministic patch model as the size of the population tends to
infinity. On the other hand, by letting both the size of the population tend to
infinity and the mesh of the grid go to zero, we obtain a law of large numbers
in the supremum norm, where the limit is a diffusion SIR model in D
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