As seen http://www.math.unm.edu/~ghuerta/tseries/dlmch2.pdf
"For a Normal DLM, t ∼ N(0, Vt) and ωt ∼ N(0, Wt)."
and Connection between the Kalman filter and the multivariate normal distribution
" Consider at dynamic linear model where $$ \theta_{1} \sim N(\mu_{1}, W_{1}), $$ $$ \theta_{i}=G\theta_{i-1} + w_{i}, w_{i}\sim N(0,W), $$ $$ Y_{i} = F\theta_{i} + v_{i}, v_{i}\sim N(0,V) $$ andand $ \theta_{1}, w_{i}, v_{i} $ all independent random vectors. Let $ \theta_{0:t} : = (\theta_{t}, \theta_{t-1},\ldots, \theta_{0}) $ and $ Y_{1:t}:= (Y_{t},Y_{t-1},\ldots, Y_{1})$.
"
What does the ~ mean?
It is notation for the phrase "distributed as". Example: $\epsilon_t\sim N(\mu,\sigma^2)$ means that $\epsilon_t$ is a normally distributed random variable with mean $\mu$ and variance $\sigma^2$.