Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is: \begin{align} \tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\sum_{j=1}^d \left([L_j \tilde{\sigma}(t), L_j^*] + [L_j, \tilde{\sigma}(t) L_j^*] \right) dt \\ +\sum_{j=1}^d \left( L_j \tilde{\sigma}(t) + \tilde{\sigma}(t) L_j^* \right)dW_j, \end{align} where $\tilde{\sigma}(t)$ could be viewed as non-normalized density operator, $L_j$ is a Lindblad operator and $dW_j$ is a stochastic differential.
If we drop the stochastic term in Eq (1) then we get the usual master equation in the Lindblad form.
It is possible to obtain the density operator $\tilde{\rho}(t)$ from $\tilde{\sigma}(t)$ as follows: $$ \tilde{\rho}(t) = \tilde{\sigma}(t)/Tr(\tilde{\sigma}(t)). $$
Since Eq (1) is a stochastic differential equation, $\tilde{\rho}(t)$ is a solution which is also random.
Could anyone tell me what's the distribution of $\tilde{\rho}(t)$ is?
Any beginner-level relevant resources are also very much appreciated!
In short, I would like to know the following: $$ \tilde{\rho}(t) \sim \ ? $$
What I have figured out so far:
If we take the expectation of Eq (1) we get the Lindblad master equation, and its solution $\rho(t)$ is the mean of $\tilde{\rho}(t)$, i.e.
$$
\mathbb{E}\left( \tilde{\rho}(t) \right) = \rho(t).
$$
