How to use symmetry of transition rate matrix in a continuous-time Markov chain? This is part of a bigger question, so I have to change the question a bit to focus on the point. We have a continuous- time Markov chain with the following transition rate matrix:
$$Q=
\begin{pmatrix}
0 & \lambda & \lambda \\
\lambda & 0 & \lambda \\
\lambda & \lambda & 0
\end{pmatrix}
$$
Let $p_{ij}(t)=P(X(t)=j|X(0)=i)$. I have already found $p_{11}(t)$, $p_{21}(t)$ and $p_{31}(t)$ by solving Kolmogorov equations. The question asks to find the rest of $p_{ij}$ using the symmetry of $Q$.
It is already known that $p_{21}(t)=p_{31}(t)$. My feeling is that $p_{ii}(t)=p_{11}(t)$ for all $i$ and $p_{ij}(t)=p_{21}(t)=p_{31}(t)$ for all $i\neq j$. This is because for example going from $2$ to $3$, everything is the same as going from $2$ to $1$. But I don't know how to mathematically justify that.
 A: By definition either the rows or columns of the transition rate matrix $Q$ should sum to zero (depending on whether row or column vectors are used for the state probability vectors). Since your transition rates are symmetric, either convention gives the same result in this case.
I.e.
$$Q=
\begin{pmatrix}
-2 \lambda & \lambda & \lambda \\
\lambda & -2 \lambda & \lambda \\
\lambda & \lambda & -2 \lambda
\end{pmatrix}.
$$
From this more precise statement of $Q$ you can symbollically compute the entire matrix $P(t)$ via the matrix exponential 
$$
P(t) = e^{t Q}
$$
which can be symbolically evaluated as:
$$
\begin{pmatrix}
\frac{2}{3} e^{-3 \lambda t} + \frac{1}{3} & \frac{1}{3} - \frac{1}{3}e^{-3 \lambda t} & \frac{1}{3} - \frac{1}{3}e^{-3 \lambda t} \\
\frac{1}{3} - \frac{1}{3}e^{-3 \lambda t} & \frac{2}{3} e^{-3 \lambda t} + \frac{1}{3} & \frac{1}{3} - \frac{1}{3}e^{-3 \lambda t}  \\
\frac{1}{3} - \frac{1}{3}e^{-3 \lambda t}  & \frac{1}{3} - \frac{1}{3}e^{-3 \lambda t}  & \frac{2}{3} e^{-3 \lambda t} + \frac{1}{3}
\end{pmatrix}
$$
e.g. using Maple. This analytic expression proves the desired symmetry result and also provides a precise and computationally efficient closed-form solution for evaluating the elements of $P(t)$ for any finite set times $\{t_i : t_i \geqslant 0 \}_{i=1}^{N}$ where $N \in \mathbb{N}$.
