In a nutshell, the equation integrated governs a set of scalars, , , , as
where is an isotropic vector-valued Wiener process with independent increments. The statistically stationary solution of the above coupled system of nonlinear stochastic differential equations is the generalized Dirichlet distribution,
provided the coefficients, , , , and , with for , , satisfy
Here , , and are parameters, while for , and . denotes the gamma function. To keep the invariant distribution generalized Dirichlet, the above set of constraints on the coefficients must be satisfied. For more details on the generalized Dirichlet SDE, see https://doi.org/10.1063/1.4822416.
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