Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states $$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$ where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate matrix (Q matrix, i.e. each column adds up to zero). For one thing, I'd like to regard master equation as linear dynamical system. But I've no idea how to introduce the property $\sum_ip_i=1$ properly and also that $p_i$ has unique fixed point, i.e. stationary distribution $\pi$ provided $A$ is irreducible. I know the system is restricted on standard $(n-1)$-simplex.

So my question is, are there any suggestions or references about general theory of dynamic systems restricted on simplex? This is different from LDS because such systems have fixed structures and dynamical variable is self correlated. I want directions to help involve structures. I'll appreciate it.

• (Actually, what is the question, exactly?) – Pietro Majer Dec 13 '13 at 14:18

1 Answer

Let $A\in\mathbb{M_n(R)}$. Then, the standard $(n-1)$-simplex $\Delta$ is positively invariant for the linear flow $\exp(tA)$ if and only if:

$$(1)\qquad\quad A_{ij}\ge0\quad \mathrm{for\; all }\quad i\neq j\;$$

and

$$(2)\qquad\qquad\quad \sum_{i=1 }^nA_{ij}=0\quad \mathrm{for\; all }\quad j\; .$$

(The proof is quite elementary; I'll add details at request).