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.

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.