Fractal Transformation of Krein–Feller Operators
DOI:
https://doi.org/10.15407/mag19.02.482Анотація
Ми розглядаємо фрактально перетворений броунiвський рух з подвiйним вiдбиттям з простором станiв, що є множиною, подiбною до канторової. Застосовуючи теорiю фрактальних перетворень, розвинуту Барнслi та iн., а також узагальнений вираз Тейлора, ми доводимо, що його iнфiнiтезимальний генератор задається в термiнах геометричної похiдної другого порядку за мiрою $\frac{d}{d\mu}\frac{d}{d\mu}$, яку було розглянуто Фрайберґом i Целе. Крiм того, ми дослiджуємо його зв’язок з добре вiдомим класичним оператором Крейна–Феллера $\frac{d}{d\mu}\frac{d}{dx}$, який є генератором так званої “щiлинної дифузiї” “gap-diffusion”.
Mathematical Subject Classification 2020: 26A24, 26A30, 28A25, 28A80, 47A05, 60J35, 60J60
Ключові слова:
геометричний оператор мiри Крейна–Феллера, множини, подiбнi до канторової, iнфiнiтезимальний генератор, фрактальне перетворення, щiлинна дифузiя (gap-diffusion)Посилання
P. Arzt, Eigenvalues of Measure Theoretic Laplacians on Cantor-like Sets, PhD thesis, Universität Siegen, 2014.
C. Bandt, M. Barnsley, M. Hegland, and A. Vince, Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces, preprint, https://arxiv.org/abs/1409.3309.
G. Burkhardt, Über Quasidiffusionen als Zeittransformationen des Wienerschen Prozesses, PhD thesis, TU Dresden, 1983.
E.B. Dynkin, Markov Processes}, I, II, Springer-Verlag Berlin, Heidelberg, 1965. https://doi.org/10.1007/978-3-662-00031-1
T. Ehnes, Eigenschaften einer fraktaltransformierten doppelt-reflektierten Brownschen Bewegung, Master's thesis, Universität Stuttgart, 2017.
T. Ehnes, Stochastic Partial Differential Equations on Cantor-like Sets, PhD thesis, Universität Stuttgart, 2020.
E. Ekström, D. Hobson, S. Janson, and J. Tysk, Can time-homogeneous diffusions produce any distribution?, Probab. Theory Related Fields 155 (2013), 493--520.
K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math. 1 (1957), 459--504. https://doi.org/10.1215/ijm/1255380673
U. Freiberg, Analytical properties of measure geometric Krein-Feller-operators on the real line, Math. Nachr. 260 (2003), 34--47. https://doi.org/10.1002/mana.200310102
U. Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets, Forum Math. 17 (2005), 87--104. https://doi.org/10.1515/form.2005.17.1.87
U. Freiberg, Dirichlet forms on fractal subsets of the real line, Real Anal. Exchange 30, 2004/2005, 589--604. https://doi.org/10.14321/realanalexch.30.2.0589
U. Freiberg and M. Zähle, Harmonic calculus on fractals - A measure geometric approach I, Potential Anal. 16 (2002), 265--277. https://doi.org/10.1023/A:1014085203265
T. Fujita, A fractional dimension, self-similarity and a generalized diffusion operator, Probabilistic Methods in Mathematical Physics, Proc. of Taniguchi International Symp., 1987, 83--90.
J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713--747. https://doi.org/10.1512/iumj.1981.30.30055
K. Itô and H. P. McKean, Diffusion processes and their sample paths, 2$^{nd}$ ed., Springer, Berlin, Heidelberg, New York, 1974.
I.S. Kac and G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl., 103, 1974, 19--102. https://doi.org/10.1090/trans2/103/02
M. Kesseböhmer, A. Niemann, T. Samuel, and H. Weyer, Generalised Krein-Feller operators and Liouville Brownian motion via transformation of measure spaces, preprint, arXiv{1909.08832v3}.
M. Kesseböhmer, T. Samuel, and H. Weyer, A note on measure-geometric Laplacians, preprint, arXiv{1411.2491v1}
H. Kunze, D. La Torre, F. Mendivil, and E. R. Vrscay, Differential Equations Using Generalized Derivatives on Fractals, Recent Developments in Mathematical, Statistical and Computational Sciences (Eds. D.M. Kilgour, H. Kunze, R. Makarov, R. Melnik, X. Wang), Springer, 2019, 81--91.
H. Kunze, D. La Torre, F. Mendivil, and E. R. Vrscay, Self-similarity of solutions to integral and differential equations with respect to a fractal measure, Fractals 27 (2019), 1950014. https://doi.org/10.1142/S0218348X19500142
L. Minorics, Eigenwertapproximation von Krein-Feller-Operatoren bezüglich singulärer invarianter Wahrscheinlichkeitsmaße, Master's thesis, Universität Stuttgart, 2016.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 2005.
R.L. Schilling and L. Partzsch, Brownian Motion: An Introduction to Stochastic Processes, DeGruyter, Berlin-Boston, 2012. https://doi.org/10.1515/9783110278989
A. Winter and J. Swart, Markov Processes: Theory and Examples, Lecture notes, 2013. Available from: https://www.uni-due.de/~hm0112/teaching/markovprocesses-19ss/sw20.pdf
