Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector.
What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$?
In addition if $\Vert x \Vert_2=1$, what can I say about $\Vert Dx \Vert_2$?
From here, we have that the largest eigenvalue of a stochastic matrix is $1$. Hence, we have $$\Vert Dx \Vert_2 \leq \Vert x \Vert_2$$