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Cahn-Hilliard stochastic equation: Strict positivity of the density. (English) Zbl 1002.60050

The author considers the Cahn-Hilliard stochastic partial differential equation \[ {\partial u \over \partial t} + (\Delta^2 u - \Delta f(u)) = g(u) + \sigma (u) \dot{W}, \] with homogeneous Neumann boundary condition, where \(W\) is a \((d+1)\)-parameter Wiener process. The results of Bally and Pardoux on the existence of density for the solution of this equation when \(g=0\) are extended to vectors of the form \((u(t_0,x_1),\ldots , u(t_0,x_l))\) under a local non-degeneracy condition on \(\sigma\), where \(x_1,\ldots ,x_l\) are pairwise disjoint and \(g\) is a \({\mathcal C}^2\) function with quadratic growth. The proof uses a new lower estimate of the Green kernel. A strict positivity result for the density is also obtained when \(\sigma\) does not vanish.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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