Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets

Julien Bect (1) , Hana Baili (1) , Gilles Fleury (1)

1 E3S - Supélec Sciences des Systèmes

This paper is concerned with the generalized Fokker-Planck equation for a class of stochastic hybrid processes, where diffusion and instantaneous jumps at the boundary are allowed. The state of the process after a jump is defined by a deterministic reset map. We establish a partial differential equation for the probability density function, which is a generalisation of the usual Fokker-Planck equation for diffusion processes. The result involves a non-local boundary condition, which accounts for the jumping behaviour of the process, and an absorbing boundary condition on the non-characteristic part of the boundary. Two applications are given, with numerical results obtained by finite volume discretization.

Mots clés: Instantaneous jumps at the boundary; Boundary hitting resets; Stochastic Hybrid Systems; Fokker-Planck equation; Kolmogorov's forward equation; Stochastic differential equations

Domaines: Probabilités [math.PR]; Automatique / Robotique; Equations aux dérivées partielles [math.AP]

Julien Bect, Hana Baili, Gilles Fleury. Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets. MTNS 2006, Jul 2006, Kyoto, Japan. pp.1360-1367. ⟨hal-00016373⟩