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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References:

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