Transient Fokker-Planck-Kolmogorov equation solved with smoothed particle hydrodynamics method

Probabilistic theories aim at describing the properties of systems subjected to random excitations by
means of statistical characteristics such as the probability density function (pdf). The time evolution
of the pdf of the response of a randomly excited deterministic system is commonly described with the
transient Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation
of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a
generalized formalism for the resolution of the transient FPK equation using the well-known mesh-free
Lagrangian method, Smoothed Particle Hydrodynamics (SPH).
Numerical implementation shows notable advantages of this method in an unbounded state space:
(i) the conservation of total probability in the state space is explicitly written, (ii) no artifact is required
to manage far- eld boundary conditions , (iii) the positivity of the pdf is ensured and (iv) the extension
to higher dimensions is straightforward.
Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial
conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori
knowledge of the stationary distribution; just a precise representation of the initial distribution is
required.