ISSN:
1572-929X

Keywords:
Stochastics partial differential equations
;
space time noise
;
Malliavin calculus

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics

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

Type of Medium:
Electronic Resource

Permalink