# Asymptotic behaviour of a transport equation

Annales Polonici Mathematici (1992)

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

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topRyszard 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] 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] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience Publ., New York 1968. Zbl0128.34803
- [3] J. Klaczak, Stability of a transport equation, Ann. Polon. Math. 49 (1988), 69-80. Zbl0673.45009
- [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] K. Łoskot, Stochastic perturbations of dynamical systems, Ann. Polon. Math., to appear. Zbl0876.60050
- [6] A. Rényi, Probability Theory, Akadémiai Kiadó, Budapest 1970.
- [7] A. N. Shiryaev, Probability, Nauka, Moscow 1989 (in Russian).

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