Still do not understand difference between diagonal and scalar noise

I have system of N coupled stochastic differential equations, that looks like


here F is the function that mixes u_i with u_j. The quantity \eta_i is so-called white noise with properties \langle \eta_i(t)\rangle =0, \langle \eta_i(t)\eta_j(t')\rangle=2D\delta(t-t')\delta_{ij}. I try to solve this system in Julia. I define two functions,

function deterministic_u!(du, u, p, t)
    du .= ...
function noise!(du, u, p, t)
    du = sqrt(2)

When I define noise! function I assume that it coresponds to D=1 case. Then I construct SDEProblem with this functions and a given initial conditions,

prob = SDEProblem(deterministic_u!, noise!, u0, tspan, p);

However, I obtain results that contradict my computations by hand. I have checked this topic but I still do not understand properly: when I want to add white noise should I use diagonal noise or scalar noise?

It depends on your model. Both of those are adding white noise to the model. The question is how many independent white noise processes do you have. Do you have one white noise process, or do you have one per ODE?

For the system you describe above, it looks like you have \eta_i and so you have one white noise per u_i, and each of them are independent. Therefore it’s diagonal noise.


Yes, indeed in my set-up its diagonal noise. I have tested and compared cases with diagonal and scalar noise. Thank you for this reply