H-derivative

In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus. In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus. Let i : H → E {displaystyle i:H o E} be an abstract Wiener space, and suppose that F : E → R {displaystyle F:E o mathbb {R} } is differentiable. Then the Fréchet derivative is a map i.e., for x ∈ E {displaystyle xin E} , D F ( x ) {displaystyle mathrm {D} F(x)} is an element of E ∗ {displaystyle E^{*}} , the dual space to E {displaystyle E} . Therefore, define the H {displaystyle H} -derivative D H F {displaystyle mathrm {D} _{H}F} at x ∈ E {displaystyle xin E} by a continuous linear map on H {displaystyle H} . Define the H {displaystyle H} -gradient ∇ H F : E → H {displaystyle abla _{H}F:E o H} by That is, if j : E ∗ → H {displaystyle j:E^{*} o H} denotes the adjoint of i : H → E {displaystyle i:H o E} , we have ∇ H F ( x ) := j ( D F ( x ) ) {displaystyle abla _{H}F(x):=jleft(mathrm {D} F(x) ight)} .