Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

•An exact low-dimensional partial differential equation with analytical coefficients governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.•The remarkable demonstration on the existence and eligibility of the globally-evolving-based generalized density evolution equation (GE-GDEE) is provided for the case that the original high-dimensional stochastic differential system itself is non-Markovian.•The insights for the physical-mechanism-informed determination of the intrinsic drift coefficient of GE-GDEE is discussed.