I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit. On one hand, in the literature it is not hard to find books or papers that deal with the theoretical and numerical aspects of Deterministic Riemann problem, i.e.

$$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} \end{cases} $$

Here $u_l , u_r$ are constants. For this problem we could find a lot things. On the other hand, let's say that we add some randomness in the problem (1). For example, let's say we have (Stochastic Riemann problem):

$$(2) \hspace{1cm} \begin{cases} u_t+f(u)_x=g(u)W(t) \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} \end{cases} $$

Here u $\in \mathbb{R}^n$ and W(t) is a white noise. In this case we added some randomness in the source term on the right-hand side of the system. Similarly, we could add randomness in initial conditions, in the flux function, in the coefficients,...

**My question is this:** Does anyone know any reference where I could find more about **theory which concerns Stochastic Riemann problems**?

The only thing that maters is that it studies Riemann discontinuous initial conditions theoretically and that it has stochastic in any form mentioned above (I am interested in stochastic sources - problem (2) mainly but any other type of added stochastic in the problem could help me).

So far I've found a few papers/books that mention Stochastic Riemann problems and solving them numerically but I haven't found any paper that deals with theory. All papers given bellow deal with random initial conditions and have no source terms:

- Poette, Despres, Lucor - Uncertainty quantification for systems of conservation laws, 2009
- Tryoen, Le Maitre, Ndjinga, Ern - Roe solver with entropy corrector for uncertain hyperbolic systems, 2010
- Bijl, Lucor, Mishra, Schwab - Uncertainty quantification in computational fluid dynamics, 2013

Also if someone knows some additional papers that deal with this problem numerically write it down please.