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.


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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