Let $\boldsymbol{u}=[u_{1}~~u_{2}~~\dots~~u_{n}]$ and $\boldsymbol{v}=[v_{1}~~v_{2}~~\dots~~v_{n}]$ be two known row vectors of proportions. If $\boldsymbol{P}_{(n\times n)}=[p_{ij}]$ is a known stochastic matrix such that the sum of each row of $\boldsymbol{P}$ is 1 and $p_{ij}\ge0$, then for which class of $\mathscr{M}$, the following functional relation holds: $$f(g(\boldsymbol{u},\boldsymbol{P}),g(\boldsymbol{v,\boldsymbol{P}}))=f(\boldsymbol{u},\boldsymbol{v}),$$ where the function $f(\cdot,\cdot)$ is: $$f(\boldsymbol{x},\boldsymbol{y})=\frac{1}{2}\sum\limits_{i=1}^{n}|y_{i}-x_{i}|.$$ ($\boldsymbol{P}\in\mathscr{M}$ and $\boldsymbol{P}$ should not be an identity or Permutation matrix and $g(\cdot,\cdot)$ maybe a linear or nonlinear function).
Note I want to show the condition where $f(\boldsymbol{x},\boldsymbol{y})$ is stochastic matrix $\boldsymbol{P}$ invariant.
For simplicity, I have tried to solve the above question for a three--dimensional case first by taking a numerical example as follows.
Suppose $\boldsymbol{u}=[5/10~~3/10~~2/10]$, $\boldsymbol{v}=[7/10~~2/10~~1/10]$ and $$\boldsymbol{P}= \begin{bmatrix} 1 & 0 & 0\\ 1/3 & 2/3 & 0\\ 1/2 & 0 & 1/2 \end{bmatrix}. $$
Thus, the distance measure for $\boldsymbol{u}$ and $\boldsymbol{v}$ is $f(\boldsymbol{u},\boldsymbol{v})=1/2\sum_{i=1}^{3}|v_{i}-u_{i}|=5/6$. Now I am assume that $$g(\boldsymbol{u},\boldsymbol{P})=\boldsymbol{u}\boldsymbol{P}=[7/10~~2/10~~1/10]$$ and $$g(\boldsymbol{v},\boldsymbol{P})=\boldsymbol{v}\boldsymbol{P}=[49/60~~4/30~~1/20].$$ Thus, the distance measure $f(g(\boldsymbol{u},\boldsymbol{P}),g(\boldsymbol{v},\boldsymbol{P}))=7/6.$ Clearly, $f(\boldsymbol{u},\boldsymbol{v})\ne f(g(\boldsymbol{u},\boldsymbol{P}),g(\boldsymbol{v},\boldsymbol{P})).$
My objective is to find $g(\boldsymbol{u},\boldsymbol{P})$ (which should be a function of $\boldsymbol{u}$ and $\boldsymbol{P}$) and $g(\boldsymbol{v},\boldsymbol{P})$ (which should be a function of $\boldsymbol{v}$ and $\boldsymbol{P}$) for which $f(\boldsymbol{u},\boldsymbol{v})= f(g(\boldsymbol{u},\boldsymbol{P}),g(\boldsymbol{v},\boldsymbol{P})).$ Now I got stuck.
Which other $g(\boldsymbol{u},\boldsymbol{P})$ and $g(\boldsymbol{v},\boldsymbol{P})$ should I try? Note $g(\cdot,\cdot)$ can be a nonlinear function.
Any help will be greatly appreciated.
Thanks