Application of Malliavin calculus to stochastic partial differential equations

The Malliavin calculus is an infinite dimensional calculus on a Gaussian space, which is mainly applied to establish the regularity of the law of nonlinear functionals of the underlying Gaussian process. Suppose that H is a real separable Hilbert space with scalar product denoted by 〈·, ·〉H . The norm of an element h ∈ H will be denoted by ‖h‖H . Consider a Gaussian family of random variables W = {W (h), h ∈ H} defined in a complete probability space (Ω,F , P ), with zero mean and covariance E(W (h)W (g)) = 〈h, g〉H . The mapping h → W (h) provides a linear isometry of H onto a closed subspace of H1 of L(Ω).