Malliavin Calculus for White Noise Driven Parabolic SPDEs

Abstract

We consider the parabolic SPDE

$$\partial _t X\left( {t,x} \right) = \partial _{_x }^2 X\left( {t,x} \right) + \psi \left( {X\left( {t,x} \right)} \right) + \varphi \left( {X\left( {t,x} \right)} \right)\dot W\left( {t,x} \right),\left( {t,x} \right) \in R_ + \times \left[ {0,1} \right]$$

with the Neuman boundary condition

$${{\partial x}}\left( {t,0} \right) = \frac{{\partial X}} {{\partial x}}\left( {t,1} \right) = 1$$

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,x_d)), 0 ≤ x₁ ≤ ... ≤ x_d ≤ 1, has a smooth, strictly positive density with respect to Lebesgue measure.