On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...

Regularity of the distribution (continued)
For σ = 1, the ”main” term is ɛ1−d/4 since 2 − d/2 > 1 − d/4.
Thus the truncated sequence ũn is such that for t > 0,
� t �
|Dz,r ũn(x, t)| 2 dzdr > 0 a.s.
(iv) For σ which does not vanish, write
0
O
Dz,r ũn(x, t) = σ(ũn(x, t))vn(z, r)
where vn(z, r) behaves as the previous Malliavin derivative (when σ = 1)
for t > 0,
� t
0
�
O
|Dz,r ũn(x, t)| 2 dzdr > 0 a.s.
ũn(x, t) has a density (Bouleau-Hirsch criterion)
u = ũn on Ωn and P(Ωn) → 1 implies that u(x, t) has a density.
A. Millet (SAMM, Paris 1 and PMA) Cahn-Hilliard/Allen-Cahn equation Workshop Stochastic PDEs 18 / 18