# Field functions¶

A field function where and is defined by:

(1)¶

with and . A field function transforms a multivariate stochastic process:

where is discretized according to the into the multivariate stochastic process:

such that:

where the mesh is discretized according to the .

A field function also acts on fields and produces fields of possibly different dimension () and mesh ( or ).

## Value function¶

A value function is a particular field function that leaves the mesh of a field invariant and can be defined using a function such that:

(2)¶

Let us note that the input dimension of is still , the dimension of . Its output dimension is equal to . The creation of a value function requires the function and the integer : the dimension of the vertices of the mesh . These data are required to test the compatibility of the dimensions when a composite process is created using the value function.

## Vertex value function¶

A vertex-value function is a particular field function that leaves the mesh of a field invariant and is defined by a function such that:

(3)¶

Let us note that the input dimension of is still , the dimension of . Its output dimension is equal to .