The Malliavin derivative
Giuseppe Da Prato · Scuola Normale Superiore eBooks · 2014
Let H be an infinite dimensional separable Hilbert space and μ = N Q a non degenerate Gaussian measure. For any ϕ ∈ ℰ(H) (the space of all exponential functions, see Section 3.2) we define the Malliavin derivative of ϕ setting Mφ:= Q 1/2 Dφ, $$M\varphi : = {Q^{1/2}}D\varphi ,$$ where D represents the gradient. The factor Q 1/2 in front of the gradient, has far reaching consequences that we shall explain later (see Remark 3.2 and Chapter 11).