Nu-transform

In the theory of stochastic processes, a ν-transform is an operation that transforms a measure or a point process into a different point process. Intuitively the ν-transform randomly relocates the points of the point process, with the type of relocation being dependent on the position of each point.

For measures
Let $$ \delta_x $$ denote the Dirac measure on the point $$ x $$ and let $$ \mu $$ be a simple point measure on $$ S $$. This means that
 * $$ \mu= \sum_k \delta_{s_k} $$

for distinct $$ s_k \in S$$ and $$ \mu(B)< \infty $$ for every bounded set $$ B $$ in $$ S $$. Further, let $$ \nu $$ be a Markov kernel from $$ S $$ to $$ T $$.

Let $$ \tau_k $$ be independent random elements with distribution $$ \nu_{s_k}=\nu(s_k,\cdot) $$. Then the point process
 * $$ \zeta = \sum_{k} \delta_{\tau_k} $$

is called the ν-transform of the measure $$ \mu $$ if it is locally finite, meaning that $$ \zeta(B) < \infty $$ for every bounded set $$ B $$

For point processes
For a point process $$ \xi $$, a second point process $$ \zeta $$ is called a $$ \nu $$-transform of $$ \xi $$ if, conditional on $$ \{ \xi=\mu\} $$, the point process $$ \zeta $$ is a $$ \nu $$-transform of $$ \mu $$.

Stability
If $$ \zeta $$ is a Cox process directed by the random measure $$ \xi $$, then the $$ \nu $$-transform of $$ \zeta $$ is again a Cox-process, directed by the random measure $$ \xi \cdot \nu $$ (see Transition kernel)

Therefore, the $$ \nu $$-transform of a Poisson process with intensity measure $$ \mu $$ is a Cox process directed by a random measure with distribution $$ \mu \cdot \nu $$.

Laplace transform
It $$ \zeta $$ is a $$ \nu $$-transform of $$ \xi $$, then the Laplace transform of $$ \zeta $$ is given by
 * $$ \mathcal L_{\zeta}(f)= \exp \left( \int \log \left[ \int \exp(-f(t)) \mu_s(\mathrm dt)\right] \xi(\mathrm ds)\right)$$

for all bounded, positive and measurable functions $$ f $$.