What is the meaning of this normal distribution? Following is a normal distribution equation:
$$q(\mathbf x_t\mid\mathbf x_{t-1})=\mathcal N(\mathbf x_t;\sqrt{1-\beta_t}\mathbf x_{t-1},\beta_t\mathbf I).$$
I understand there should be mean and variance in normal distribution. But what is the meaning of this? It contains three variables.

The semicolon separates off the parameters of the distribution. This function represents the probability density of $x_t$ given those two parameters.
This should be interpreted as the probability density $q$ is defined by this normal distribution.
