Support Theorem for a Stochastic Cahn-Hilliard Equation

Lijun Bo,Yongjin Wang,Kun Shi

doi:10.1214/ejp.v15-760

Abstract

In this paper, we establish a Stroock-Varadhan support theorem for the global mild solution to a $d$ ($d\leq 3$)-dimensional stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise

Highlights

Introduction and main resultIn this paper, we consider the following stochastic Cahn-Hilliard equation:∂ u/∂ t = −∆ ∆u + f (u) + σ(u)W, u(0) = ψ,∂ u/∂ n = ∂ [∆u] /∂ n = 0, in [0, T ] × D, on [0, T ] × ∂ D, (1.1)where ∆ denotes the Laplace operator, the domain D = [0, π]d (d = 1, 2, 3), and f : R → R is a polynomial of degree 3 with positive dominant coefficient
Where ∆ denotes the Laplace operator, the domain D = [0, π]d (d = 1, 2, 3), and f : R → R is a polynomial of degree 3 with positive dominant coefficient
The Cahn-Hilliard equation (i.e., σ ≡ 0 in (1.1)) has been extensively studied as a well-known model of the macro-phase separation that occurs in an isothermal binary fluid, when a spatially uniform mixture is quenched below a critical temperature at which it becomes unstable

Summary

Introduction and main result

We consider the following stochastic Cahn-Hilliard equation:. where ∆ denotes the Laplace operator, the domain D = [0, π]d (d = 1, 2, 3), and f : R → R is a polynomial of degree 3 with positive dominant coefficient (which is due to the background of the equation from material science). We consider the following stochastic Cahn-Hilliard equation:. We are attempting to establish a support theorem of the law corresponding to the solution to Equation (1.1) in C([0, T ], Lp([D])) for p ≥ 4. The main strategy used in this paper is an approximation procedure by using a space-time polygonal interpolation for the white noise, and we adopt a localization argument, which was used in [7] for studying a support theorem of a Burgers-type equation. The rest of this paper is organized as follows: In the coming section, we give a difference approximation to the (d + 1)–dimensional space-time white noise W (x, t) and study some concrete properties of the approximating noises.

Difference approximation to white noise

Localization framework

Auxiliary lemmas