Abstract
We consider the parabolic SPDE
with the Neuman boundary condition
and some initial condition.
We use the Malliavin calculus in order to prove that, if the coefficients ϕ and ψ are smooth and ϕ > 0, then the law of any vector (X(t,x1),..., X(t,xd)), 0 ≤ x1 ≤ ... ≤ xd ≤ 1, has a smooth, strictly positive density with respect to Lebesgue measure.
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Bally, V., Pardoux, E. Malliavin Calculus for White Noise Driven Parabolic SPDEs. Potential Analysis 9, 27–64 (1998). https://doi.org/10.1023/A:1008686922032
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DOI: https://doi.org/10.1023/A:1008686922032