Via matrix multiplication, the matrix \$$e^{t H}\$$ is applied to the initial vector \$$\Gamma(0)\$$ to get the vector \$$\Gamma(t)\$$.

It is a theorem that whenever \$$H\$$ is infinitesimal stochastic (see [comment #31](https://forum.azimuthproject.org/discussion/comment/22052/#Comment_22052)), the operator \$$e^{t H}\$$ will map stochastic states to stochastic states:

\$e^{t H}: \Delta \rightarrow \Delta\$

Since \$$\Gamma(0) \in \Delta\$$, it follows that \$$\Gamma(t) = e^{t H} \Gamma(0) \in \Delta\$$.

In other words, H being infinitesimal stochastic guarantees that the path of \$$\Gamma\$$ lies within the simplex \$$\Delta\$$ -- which is to be expected, as \$$\Gamma\$$ is supposed to describe the evolution of the probability distribution.