47 research outputs found
Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment
In this paper, we show that the concept of sigma-convergence associated to
stochastic processes can tackle the homogenization of stochastic partial
differential equations. In this regard, the homogenization problem for a
stochastic nonlinear partial differential equation is studied. Using some deep
compactness results such as the Prokhorov and Skorokhod theorems, we prove that
the sequence of solutions of this problem converges in probability towards the
solution of an equation of the same type. To proceed with, we use a suitable
version of sigma-convergence method, the sigma-convergence for stochastic
processes, which takes into account both the deterministic and random
behaviours of the solutions of the problem. We apply the homogenization result
to some concrete physical situations such as the periodicity, the almost
periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application
Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions
A microscopic heterogeneous system under random influence is considered. The
randomness enters the system at physical boundary of small scale obstacles as
well as at the interior of the physical medium. This system is modeled by a
stochastic partial differential equation defined on a domain perforated with
small holes (obstacles or heterogeneities), together with random dynamical
boundary conditions on the boundaries of these small holes.
A homogenized macroscopic model for this microscopic heterogeneous stochastic
system is derived. This homogenized effective model is a new stochastic partial
differential equation defined on a unified domain without small holes, with
static boundary condition only. In fact, the random dynamical boundary
conditions are homogenized out, but the impact of random forces on the small
holes' boundaries is quantified as an extra stochastic term in the homogenized
stochastic partial differential equation. Moreover, the validity of the
homogenized model is justified by showing that the solutions of the microscopic
model converge to those of the effective macroscopic model in probability
distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200