ABRAMOVICH STEGUN PDF

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.

References [Enhancements On Off] What's this? Abramovich, I. Anton, Noncollapsing solution below rc for a randomly forced particle , Phys.

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Modern Phys. Chernov and R. Markarian, Introduction to the ergodic theory of chaotic billiards , Second edition. Cornell, M. Swift, and A. Florencio and F. Krieger Publishing Company, Probability and Mathematical Statistics, Vol. Grundlehren der Mathematischen Wissenschaften, Vol. Rational Mech. Hagan , Charles R. Doering , and C. Hwang, J. Jang, and J. Velazquez, Nonuniqueness for the kinetic-Fokker-Planck equation with inelastic boundary conditions.

Il'in and R. Kasminsky, On the equations of Brownian motion , Theory Probab. Kotsev and T. Burkhardt, Randomly accelerated particle in a box: Mean absorption time for partially absorbing and inelastic boundaries , Phys. E 71 , Scuola Norm. Pisa Cl. MR [44] J.

Marshall and E. Watson , A drop of ink falls from my pen. A 18 , no. MR [46] T. Watson , The analytic solutions of some boundary layer problems in the theory of Brownian motion , J. A 20 , no. MR [47] J. Masoliver and J. Letters 75 , no. McKean Jr. Kyoto Univ. Nier , Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries , Mem. An introduction with applications.

Pazy , Semigroups of linear operators and applications to partial differential equations , Applied Mathematical Sciences, vol. Stein , Hypoelliptic differential operators and nilpotent groups , Acta Math. I, II Kyoto, Math. Japan, Tokyo, , pp. MR [56] Elias M. Stein , Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , Princeton Mathematical Series, vol.

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.

Bharucha-Reid, Elements of the theory of Markov processes and their applications , Reprint of the original, Dover Publications, Inc. Chandresekhar, Stochastic problems in physics and astronomy , Rev. Trudinger, Elliptic partial differential equations of second order , Grundlehren der Mathematischen Wissenschaften, Vol.

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.

Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , with the assistance of Timothy S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions , Comm.

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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.

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