We'd like to understand how you use our websites in order to improve them. Register your interest. We study the initial-boundary value problem for the Fokker—Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times.

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On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries. Abstract: In this paper we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if. The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point.

In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper Nonuniqueness for the kinetic-Fokker-Planck equation with inelastic boundary conditions to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.

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With the assistance of Timothy S. MR [57] D. Stroock and S. Varadhan , On degenerate elliptic-parabolic operators of second order and their associated diffusions , Comm. Pure Appl. References [1] M. Titulaer, The kinetic boundary layer for the Fokker-Planck equation: selectively absorbing boundaries , J.

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Hagan, Charles R. Doering, and C. MR [43] Alessandra Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in , Ann. Watson, A drop of ink falls from my pen. It comes to earth, I know not when , J.

Watson, The analytic solutions of some boundary layer problems in the theory of Brownian motion , J. Nier, Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries , Mem. Pazy, Semigroups of linear operators and applications to partial differential equations , Applied Mathematical Sciences, vol. Stein, Hypoelliptic differential operators and nilpotent groups , Acta Math.

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## The Fokker–Planck Equation with Absorbing Boundary Conditions

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## NIST Digital Library of Mathematical Functions

Abstract: We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions. There are two at fixed point 2 and two at fixed point 4.