We consider continuous random dynamical systems with jumps. We estimate the dimension of the invariant measures and apply the results to a model of stochastic gene expression.
REFERENCES
1.
T.
Bielaczyc
, Hausdorff dimension of invariant measures related to Poisson driven stochastic differential equations
, Ann. Polon. Math.
101
, 67
–74
(2011
).2.
T.
Bielaczyc
, K.
Horbacz
, The Hausdorff dimension of invariant measures for random dynamical systems
, J. Math. Anal. Appl.
391
, 298
–311
(2012
).3.
A.
Bobrowski
, Degenerate convergence of semigroups related to a model of eukaryotic gene expression
, Semigroup forum
, 73
, 343
–366
(2006
).4.
K.U.
Frisch
, Wave propagation in random media, stability, Probabilistic Methods in Applied Mathematics
, A. T.
Bharucha-Reid
ed. Academic Press
., 1986
.5.
K.
Horbacz
, Random dynamical systems with jumps
, J. Appl. Probab.
41
, 890
–910
(2004
).6.
K.
Horbacz
, Invariant measures for random dynamical systems
, Dissert. Math.
451
(2008
).7.
K.
Horbacz
, Continuous random dynamical systems
, J. Math. Anal. Appl.
408
, 623
–637
(2013
).8.
K.
Horbacz
and T.
Szarek
, Randomly connected dynamical systems on Banach spaces
, Stoch. Anal. Appl.
19
.4
, 519
–543
(2001
).9.
J.
Jaroszewska
, M.
Rams
, On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS
, Journal of Stat. Phys.
132
, 907
–919
(2008
).10.
T.
Kudo
and I.
Ohba
, Derivation of relativistic wave equation from the Poisson process
, Pramana - J. Phys.
59
, 413
–416
(2002
).11.
A.
Lasota
and M. C.
Mackey
, Cell division and the stability of cellular population
, J. Math. Biol.
38
, 241
–261
(1999
).12.
A.
Lasota
and J.
Myjak
On a dimension of measures
, Bull. Polish Acad. Sci. Math.
50
, 221
–235
(2002
).13.
A.
Lasota
and T.
Szarek
, Dimension of measures invariant with respect to the Ważewska partial differential equation
, J.Differential Equations
2
196
, 448
–456
(2004
).14.
T.
Lipniacki
, P.
Paszek
, A.
Marciniak-Czochra
, A. R.
Brasier
, M.
Kimel
, Transcriptional stochasticity in gene expression
, J. Theor. Biol.
238
, 348
–367
(2006
).15.
J.
Myjak
and T.
Szarek
, On Hausdorff dimension of invariant measures arising from non–contractive iterated function systems
, Ann. Mat. Pura Appl. IV, Ser.
181
, 223
–237
(2002
).16.
G.
Prodi
, Teoremi Ergodici per le Equazioni della Idrodinamica
, C.I.M.E.
, Roma
., 1960
.17.
I.
Werner
, Contractive Markov systems
, J.London Math.Soc.
2
71
, 236
–258
(2005
).
This content is only available via PDF.
© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
