Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following mapping: $$ T(\mathbf{p}) = \mathbf{p} \lim_{m \to \infty} P_{\mathbf{p}}^m. $$ Note that $T(\mathbf{p})$ can be seen as the stationary distribution of the Markov chain with transition matrix $P_\mathbf{p}$. I would like to show that the Jacobian matrix of this mapping, that I define as $$ (*) \;\; T'_{ij} = \frac{\partial T_j}{\partial p_i}, $$ is also row-stochastic. So far, I managed to show that the Jacobian matrix of the (simpler) mapping: $$ \tilde{T}(\mathbf{p}) = \mathbf{p} \ P_{\mathbf{p}} $$ is row-stochastic.

Question: is there some way I can use this knowledge in my (bigger) problem above? Maybe some clever application of a chain rule for derivatives?

One approach I've been thinking about is an induction on $m$, but it seems quite tedious.

### Edit on 06.10

Slightly changed my definition of Jacobian matrix $(*)$, using the transpose now.