Asymptotic behaviour of a transport equation

Ryszard Rudnicki

Annales Polonici Mathematici (1992)

  • Volume: 57, Issue: 1, page 45-55
  • ISSN: 0066-2216

Abstract

top
We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation u t + b u x + c u = a ∫ 0 a x u ( t , a x - y ) μ ( d y ) . We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.

How to cite

top

Ryszard Rudnicki. "Asymptotic behaviour of a transport equation." Annales Polonici Mathematici 57.1 (1992): 45-55. <http://eudml.org/doc/262498>.

@article{RyszardRudnicki1992,
abstract = {We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation $u_t + bu_x + cu = a∫_0^\{ax\} u(t,ax-y)μ(dy)$. We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.},
author = {Ryszard Rudnicki},
journal = {Annales Polonici Mathematici},
keywords = {Markov operators; asymptotic behaviour; integrodifferential transport equation; stochastic processes},
language = {eng},
number = {1},
pages = {45-55},
title = {Asymptotic behaviour of a transport equation},
url = {http://eudml.org/doc/262498},
volume = {57},
year = {1992},
}

TY - JOUR
AU - Ryszard Rudnicki
TI - Asymptotic behaviour of a transport equation
JO - Annales Polonici Mathematici
PY - 1992
VL - 57
IS - 1
SP - 45
EP - 55
AB - We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation $u_t + bu_x + cu = a∫_0^{ax} u(t,ax-y)μ(dy)$. We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.
LA - eng
KW - Markov operators; asymptotic behaviour; integrodifferential transport equation; stochastic processes
UR - http://eudml.org/doc/262498
ER -

References

top
  1. [1] T. Dłotko and A. Lasota, Statistical stability and the lower bound function technique, in: Semigroups. Theory and Applications, Vol. I, H. Brezis, M. Crandall and F. Kappel (eds.), Longman Scientific & Technical, 1987, 75-95. 
  2. [2] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience Publ., New York 1968. Zbl0128.34803
  3. [3] J. Klaczak, Stability of a transport equation, Ann. Polon. Math. 49 (1988), 69-80. Zbl0673.45009
  4. [4] A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375-384. Zbl0524.28021
  5. [5] K. Łoskot, Stochastic perturbations of dynamical systems, Ann. Polon. Math., to appear. Zbl0876.60050
  6. [6] A. Rényi, Probability Theory, Akadémiai Kiadó, Budapest 1970. 
  7. [7] A. N. Shiryaev, Probability, Nauka, Moscow 1989 (in Russian). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.