Wiener Integration with Respect to Fractional Brownian Motion

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

The Elements of Fractional Calculus

Let α > 0 (and in most cases below α < 1 though this is not obligatory). Define the Riemann–Liouville left- and right-sided fractional integrals on (a, b) of order α by
$$\left( {I_{a + }^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {f\left( t \right)\left( {x - t} \right)^{^{\alpha - 1} } dt,}$$
and
$$\left( {I_{b - }^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_x^b {f\left( t \right)\left( {t - x} \right)^{^{\alpha - 1} } dt,}$$
respectively.

We say that the function $$f \in D\left( {I_{a + \left( {b - } \right)}^\alpha } \right)$$ (the symbol $$D\left( \cdot \right)$$ denotes the domain of the corresponding operator), if the respective integrals converge for almost all (a.a.) $$x \in \left( {a,b} \right)$$ (with respect to (w.r.t.) Lebesgue measure).

The Riemann-Liouville fractional integrals on R are defined as
$$\left( {I_ + ^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_{ - \infty }^x {f\left( t \right)\left( {x - t} \right)^{\alpha - 1} } dt,$$
and
$$\left( {I_ - ^\alpha f} \right)\left( x \right): = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_x^\infty {f\left( t \right)\left( {t - x} \right)^{\alpha - 1} } dt,$$
respectively.

The function $$f \in D\left( {I_ \pm ^\alpha } \right)$$ if the corresponding integrals converge for a.a.$$x \in R$$. According to (SKM93), we have inclusion $$L_p \left( R \right) \subset D\left( {I_ \pm ^\alpha } \right),1 \le p < \frac{1}{\alpha }.$$. Moreover, the following Hardy–Littlewood theorem holds.

Keywords

Gaussian Process Fractional Derivative Wiener Process Fractional Brownian Motion Maximal Inequality
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
J. Adam, N. Bellomo: A Survey of Models for Tumor-Immune System Dynamics (Birkhäuser, Boston 1997)
2. 2.
J. Adam, S. Maggelakis: Diffusion related growth characteristics of a spherical prevascular carcinoma, Bull. Math. Biol. 52 (1990) pp 549–582
3. 3.
B. Ainseba: Age-dependent population dynamics diffusive systems, Disc. Cont. Dyn. Sys.-Ser. B 4 (2004) pp 1233–1247
4. 4.
B. Ainseba, S. Anita: Local exact controllability of the age-dependent population dynamics with diffusion, Abst. Appl. Anal. 6 (2001) pp 357–368
5. 5.
B. Ainseba, B. M. Langlais: On a population control problem dynamics with age dependence and spatial structure, J. Math. Anal. Appl. 248 (2000) pp 455–474
6. 6.
J. Al-Omari, S. A. Gourley: Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol. 45 (2002) pp 294–312
7. 7.
D. Ambrosi, L. Preziosi: On the closure of mass balance models for tumor growth, Math. Mod. Meth. Appl. Sci. 12 (2002) pp 737–754
8. 8.
A. Anderson: A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol. 22 (2005) pp 163–186
9. 9.
A. Anderson, M. Chaplain: Mathematical modelling, simulation and prediction of tumour-induced angiogenesis, Invasion Metastasis 16 (1996) pp 222–234Google Scholar
10. 10.
A. Anderson, M. Chaplain: Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol. 60 (1998) pp 857–899
11. 11.
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck: One-Parameter Semigroups of Positive Operators (Springer Lecture Notes in Mathematics 1184, New York 1986)
12. 12.
O. Arino: A survey of structured cell-population dynamics, Acta Biotheoret. 43 (1995) pp 3–25Google Scholar
13. 13.
B. Ayati: A variable time step method for an age-dependent population model with nonlinear diffusion, SIAM J. Num. Anal. 37 (2000) pp 1571–1589
14. 14.
B. Ayati: A structured-population model of Proteus mirabilis swarm-colony development, J. Math. Biol. 52 (2006) pp 93–114
15. 15.
B. Ayati, T. Dupont: Galerkin methods in age and space for a population model with nonlinear diffusion, SIAM J. Num. Anal. 40 (2000) pp 1064–1076
16. 16.
B. Ayati, G. Webb, R. Anderson: Computational methods and results for structured multiscale models of tumor invasion, SIAM J. Multsc. Mod. Simul. 5 (2006) pp 1–20
17. 17.
J. Banasiak, L. Arlotti: Perturbations of Positive Semigroups with Applications (Springer, Berlin Heidelberg New York, 2006)
18. 18.
N. Bellomo, L. Preziosi: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comp. Mod. 32 (2000) pp 413–452
19. 19.
G. Bell, E. Anderson: Cell growth and division I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J. 7 (1967) pp 329–351Google Scholar
20. 20.
R. Bellman, K. Cooke: Differential Difference Equations (Academic, New York 1963)
21. 21.
A. Bertuzzi, A. Gandolfi: Cell kinetics in a tumor cord, J. Theor. Biol. 204 (2000) pp 587–599Google Scholar
22. 22.
A. Bertuzzi, A. d’Onofrio, A. Fasano, A. Gandolfi: Modeling cell populations with spatial structure: steady state and treatment-induced evolution of tumor cords, Discr. Cont. Dyn. Sys. B 4 (2004) pp 161–186
23. 23.
A. Bressan: Hyperbolic Systems of Conservation Laws, (Oxford University Press, Oxford 2000)
24. 24.
S. Busenberg, M. Iannelli: Separable models in age-dependent population dynamics, J. Math. Biol. 22 (1985) pp 145–173
25. 25.
S. Busenberg, M. Iannelli: A class of nonlinear diffusion problems in age-dependent population dynamics, Nonl. Anal. Theory. Meth. Appl. 7 (1983) pp 501–529
26. 26.
S. Busenberg, K. Cooke: Vertically Transmitted Diseases, (Springer Biomathematics 23, New York 1992)Google Scholar
27. 27.
H. Byrne, M. Chaplain: Free boundary value problems associated with the growth and development of multicellular spheroids, Eur. J. Appl. Math. 8 (1997) pp 639–658
28. 28.
H. M. Byrne, S. A. Gourley: The role of growth in avascular tumor growth, Math. Comp. Mod. 26 (1997) 35–55
29. 29.
H. Byrne, L. Preziosi: Modelling solid tumour growth using the theory of mixtures, Math. Med. Biol. 20 (4) (2003) pp 341–366
30. 30.
C. Castillo-Chavez, Z. Feng: Global stability of an age-structure model for TB and its applications to optimal vaccination, Math. Biosci. 151 (1984) pp 135–154Google Scholar
31. 31.
W. Chan, B. Guo: On the semigroups of age-size dependent population dynamics with spatial diffusion, Manusc. Math. 66 (1989) pp 161–180
32. 32.
W. L. Chan, B. Z. Guo: On the semigroups for age dependent population dynamics with spatial diffusion source, J. Math. Anal. Appl. 184 (1994) pp 190–199
33. 33.
Ph. Clement, H. Heijmans, S. Angenent, C. van Duijn, B. de Pagter: One-Parameter Semigroups (North Holland, Amsterdam 1987)
34. 34.
A. Coale: The Growth and Structure of Human Populations (Princeton University Press, Princeton 1972)Google Scholar
35. 35.
S. Cui, A. Friedman: Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001) pp 636–677
36. 36.
J. Cushing: An Introduction to Structured Population Dynamics (SIAM, Philadelphia 1998)
37. 37.
C. Cusulin, M. Iannelli, G. Marinoschi: Age-structured diffusion in a multi-layer environment, Nonl. Anal. 6 (2005) pp 207–223
38. 38.
E. DeAngelis, L. Preziosi: Advection-dffusion models for solid tumour evolution in vivo and related free boundary problem, Math. Mod. Meth. Appl. Sci. 10 (2000) pp 379–407
39. 39.
M. Delgado, M. Molina-Becerra, A. Suez: A nonlinear age-dependent model with spatial diffusion, J. Math. Anal. Appl. 313 (2006) pp 366–380
40. 40.
Q. Deng, T. Hallam: An age structured population model in a spatially heterogeneous environment: Existence and uniqueness theory, Nonl. Anal. 65 (2206) pp 379–394Google Scholar
41. 41.
G. Diblasio: Non-linear age-dependent population diffusion, J. Math. Biol. 8 (1979) pp 265–284
42. 42.
O. Diekmann, Ph. Getto: Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Dif. Eqs. 215 (2005) pp 268–319
43. 43.
O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz, H. Thieme: On the formulation and analysis of general deterministic structured population models II. Nonlinear theory, J. Math. Biol. 2 (2001) pp 157–189Google Scholar
44. 44.
O. Diekmann, H. Heijmans, H. Thieme: On the stability of the cell size distribution, J. Math. Biol. 19 (1984) pp 227–248
45. 45.
O. Diekmann, H. Heijmans, H. Thieme: On the stability of the cell size distribution II. Time-periodic development rates, Comp. Math. Appl. Part A 12 (1986) pp 491–512Google Scholar
46. 46.
J. Dyson, R. Villella-Bressan, G. Webb: Asymptotic behavior of solutions to abstract logistic equations, Math. Biosci. 206 (2007) pp 216–232
47. 47.
J. Dyson, R. Villella-Bressan, G. Webb: The evolution of a tumor cord cell population, Comm. Pure Appl. Anal. 3 (2004) pp 331–352
48. 48.
J. Dyson, R. Villella-Bressan, G. Webb: The steady state of a maturity structured tumor cord cell population, Discr. Cont. Dyn. Sys. B, 4 (2004) pp 115–134
49. 49.
K-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations, K-J. Engel, R. Nagel, Eds., (Springer Graduate Texts in Mathematics 194, New York, 2000)Google Scholar
50. 50.
G. Fragnelli, L. Tonetto: A population equation with diffusion, J. Math. Anal. Appl. 289 (2004) pp 90–99
51. 51.
W. Feller: On the integral equation of renewal theory, Ann. Math. Stat. 12 (1941) pp 243–267
52. 52.
Z. Feng, W. Huang, C. Castillo-Chavez: Global behavior of a multi-group SIS epidemic model with age structure, J. Diff. Eqs. 218(2) (2005) pp 292–324
53. 53.
Z. Feng, C-C. Li, F. Milner: Schistosomiasis models with density dependence and age of infection in snail dynamics, Math. Biosci. 177–178 (2002) pp 271–286
54. 54.
W. Fitzgibbon, M. Parrott, G. Webb: Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci. 128 (1995) pp 131–155
55. 55.
W. Fitzgibbon, M. Parrott, G. Webb: A diffusive age-structured SEIRS epidemic model, Meth. Appl. Anal. 3 (1996) pp 358–369
56. 56.
A. Friedman: A hierarchy of cancer models and their mathematical challenges, Discr. Cont. Dyn. Sys. B 4 (2004) pp 147–160
57. 57.
A. Friedman, F. Reitich: On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors, Math. Mod. Meth. Appl. Sci. 77 (2001) pp 1–25
58. 58.
M. Garroni, M. Langlais: Age-dependent population diffusion with external constraint, J. Math. Biol. 14 (1982) pp 77–94
59. 59.
J. Goldstein: Semigroups of Linear Operators and Applications (Oxford University Press, New York 1985)
60. 60.
H. Greenspan: Models for the growth of solid tumor by diffusion, Stud. Appl. Math. 51 (1972), pp 317–340
61. 61.
G. Greiner: A typical Perron-Frobenius theorem with applications to an age-dependent population equation, in Infinite Dimensional Systems, Eds. F. Kappel, W. Schappacher (Springer Lecture Notes in Mathematics 1076, 1989) pp 786–100Google Scholar
62. 62.
G. Greiner, R. Nagel: Growth of cell populations via one-parameter semigroups of positive operators in Mathematics Applied to Science, Eds. J.A. Goldstein, S. Rosencrans, and G. Sod, Academic Press, New York (1988) pp 79–104Google Scholar
63. 63.
G. Grippenberg, S-O. Londen, O. Staffans: Volterra Integral and Functional Equations (Cambridge University Press, Cambridge 1990)Google Scholar
64. 64.
M. Gurtin, R. MacCamy: Nonlinear age-dependent population dynamics, Arch. Rat. Mech. Anal. 54 (1974) pp 281–300
65. 65.
M. Gurtin, R. MacCamy: Diffusion models for age-structured populations, Math. Biosci. 54 (1981) pp 49–59
66. 66.
M. Gyllenberg: Nonlinear age-dependent poulation dynamics in continuously propagated bacterial cultures, Math. Biosci. 62 (1982) pp 45–74
67. 67.
H. Heijmans: The dynamical behaviour of the age-size-distribution of a cell population, in The Dynamics of Physiologically Structured Populations (Springer Lecture Notes in Biomathematics 68, 1986) pp 185–202Google Scholar
68. 68.
E. Hille, R. Phillips: Functional Analysis and Semigroups, (American Mathematical Society Colloquium Publications, Providence 1957)Google Scholar
69. 69.
F. Hoppensteadt: Mathematical Theories of Populations: Demographics, Genetics, and Epidemics (SIAM, Philadelphia 1975)
70. 70.
C. Huang: An age-dependent population model with nonlinear diffusion in R n, Quart. Appl. Math. 52 (1994) pp 377–398
71. 71.
W. Huyer: A size-structured population model with dispersion, J. Math. Anal. Appl. 181 (1994) pp 716–754
72. 72.
W. Huyer: Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion, Semigroup Forum 49 (1994) pp 99–114
73. 73.
M. Iannelli: Mathematical Theory of Age-Structured Population Dynamics (Giardini Editori e Stampatori, Pisa 1994)Google Scholar
74. 74.
M. Iannelli, M. Marcheva, F. A. Milner: Gender-Structured Population Modeling, (Mathematical Methods, Numerics, and Simulations, SIAM, Philadelphia 2005)
75. 75.
H. Inaba: Mathematical Models for Demography and Epidemics, (University of Tokyo Press, Tokyo 2002)Google Scholar
76. 76.
W. Kermack, A. McKendrick: Contributions to the mathematical theory of epidemics III. Further studies on the problem of endemicity, Proc. Roy. Soc. 141 (1943) pp 94–122Google Scholar
77. 77.
78. 78.
K. Kubo, M. Langlais: Periodic solutions for a population dynamics problem with age-dependence and spatial structure, J. Math. Biol. 29 (1991) pp 363–378
79. 79.
K. Kunisch, W. Schappacher, G. Webb: Nonlinear age-dependent population dynamics with diffusion, Inter. J. Comput. Math. Appl. 11 (1985) pp 155–173
80. 80.
M. Langlais: A nonlinear problem in age-dependent population diffusion, SIAM J. Math. Anal. 16 (1985) pp 510–529
81. 81.
M. Langlais: Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Biol. 26 (1988) pp 319–346
82. 82.
M. Langlais, F. A. Milner: Existence and uniqueness of solutions for a diffusion model of host-parasite dynamics, J. Math. Anal. Appl 279 (2003) pp 463–474
83. 83.
H. Levine, S. Pamuk, B. Sleeman, M. Nilsen-Hamilton: Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol. 63 (2001) pp 801–863Google Scholar
84. 84.
H. Levine, B. Sleeman: Modelling tumor-induced angiogenesis, in Cancer Modelling and Simulation, Ed. L. Preziosi (Chapman and Hall/CRC, Boca Raton, 2003)Google Scholar
85. 85.
A. Lotka: The stability of the normal age-distribution, Proc. Natl. Acad. Sci. USA 8 (1922) pp 339–345Google Scholar
86. 86.
A. Lotka: On an integral equation in population analysis, Ann. Math. Stat. 10 (1939) pp 1–35
87. 87.
R. MacCamy: A population problem with nonlinear diffusion, J. Diff. Eqs. 39 (21) (1981) pp 52–72
88. 88.
P. Magal: Compact attractors for time-periodic age-structured population models, Elect. J. Dif. Eqs. 65 (2001) pp 1–35Google Scholar
89. 89.
P. Magal, H. R. Thieme: Eventual compactness for semiflows generated by nonlinear age-structured models, Comm. Pure Appl. Anal. 3 (2004) pp 695–727
90. 90.
P. Marcati: Asymptotic behavior in age dependent population diffusion model, SIAM J. Math. Anal. 12 (1981) pp 904–914
91. 91.
R. Martin: Nonlinear Operators and Differential Equations in Banach Spaces, (Wiley-Interscience, New York 1976)
92. 92.
A. McKendrick: Applications of mathematics to medical problems, Proc. Edin. Math. Soc. 44 (1926) pp 98–130Google Scholar
93. 93.
J. Metz, O. Diekmann: The Dynamics of Physiologically Structured Populations (Springer Lecture Notes in Biomathematics 68, New York 1986)
94. 94.
R. Nagel: One-Parameter Semigroups of Positive Operators, Ed. R. Nagel (Springer, Berlin Heidelberg New York 1986)Google Scholar
95. 95.
J. Nagy: The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng. 2 (2005) pp 381–418
96. 96.
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer Series in Applied Mathematical Sciences, New York 1983)
97. 97.
J. Pollard: Mathematical Models for the Growth of Human Populations, (Cambridge University Press, Cambridge 1973)
98. 98.
J. Prüss: Equilibrium solutions of age-specific population dynamics of several species, J. Math. Biol. 11 (1981) pp 65–84
99. 99.
J. Prüss: On the qualitative behaviour of populations with age-specific interractions, Comp Math. Appl. 9 (1983) pp 327–339
100. 100.
J. Prüss: Stability analysis for equilibria in age-specific population dynamics, Nonl. Anal. 7 (1983) pp 1291–1313
101. 101.
A. Rhandi: Positivity and stability for a population equation with diffusion on L 1, Positivity 2 (1998) pp 101–113
102. 102.
A. Rhandi, R. Schnaubelt: Asymptotic behaviour of a non-autonomous population equation with diffusion in L 1, Discr. Cont. Dyn. Sys. 5 (1999) pp 663–683
103. 103.
I. Segal: Nonlinear semigroups, Ann. Math. 78 (1963) pp 339–364Google Scholar
104. 104.
F. Sharpe, A. Lotka: A problem in age-distribution, Philos. Mag. 6 (1911) pp 435–438Google Scholar
105. 105.
J. Sherratt, M. Chaplain: A new mathematical model for avascular tumour growth, J. Math. Biol. 43 (2001) pp 291–312
106. 106.
J. Sinko, W. Streifer: A new model for age-size structure of a population, Ecology 48 (1967) pp 910–918Google Scholar
107. 107.
B. Sleeman, A. Anderson, M. Chaplain: A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett. 12 (1999) pp 121–127
108. 108.
J. W.-H So, J. Wu, X. Zou: A reaction−diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains, Proc. Roy. Soc. 457 (2001) pp 1841–1853Google Scholar
109. 109.
H. R. Thieme: Semiflows generated by Lipschitz perturbations of non-densely defined operators, Dif. Int. Eqs. 3 (1990) pp 1035–1066
110. 110.
H. R. Thieme: Analysis of age-structured population models with an additional structure, in Mathematical Population Dynamics, Proceedings of the 2nd International Conference, Rutgers University 1989, Eds. O. Arino, D.E. Axelrod, M. Kimmel (Lecture Notes in Pure and Applied Mathematics 131, 1991) pp 115–126Google Scholar
111. 111.
H. R. Thieme: Quasi-compact semigroups via bounded perturbation, in Advances in Mathematical Population Dynamics: Molecules, Cells and Man, Eds. O. Arino, D. Axelrod, M. Kimmel (World Scientific, Singapore 1997) pp 691–711Google Scholar
112. 112.
H. R. Thieme: Positive perturbation of operator semigroups: Growth bounds, essential compactness, and asynchronous exponential growth, Disc. Cont. Dyn. Sys. 4 (1998) pp 735–764
113. 113.
H. Thieme: Balanced exponential growth of operator semigroups, J. Math. Anal. Appl. 223 (1998) pp 30–49
114. 114.
H. R. Thieme: Mathematics in Populations Biology, (Princeton University Press, Princeton 2003)Google Scholar
115. 115.
H. R. Thieme, J. I. Vrabie: Relatively compact orbits and compact attractors for a class of nonlinear evolution equations, J. Dyn. Diff. Eqs. 15 (2003) pp 731–750
116. 116.
S. Tucker, S. Zimmerman: A nonlinear model of population-dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math. 48 (1988) pp 549–591
117. 117.
H. Von Foerster: Some remarks on changing populations, in The Kinetcs of Cellular Proliferation, Ed. F. Stohlman (Grune and Stratton, New York 1959)Google Scholar
118. 118.
C. Walker: Global well-posedness of a haptotaxis model with spatial and age structure, Differ. Integral Equations Athens 20 (9) (2007) pp 1053–1074Google Scholar
119. 119.
C. Walker, G. Webb: Global existence of classical solutions to a haptotaxis model, SIAM J. Math. Anal. 38 (5) (2007) pp 1694–1713
120. 120.
G. Webb: Lan age-dependent epidemic model with spatial diffusion, Arch. Rat. Mech. Anal. 75 (1980) pp 91–102
121. 121.
G. Webb: Diffusive age-dependent population models and a application to genetics, Math. Biosci. 61 (1982) pp 1–16
122. 122.
G. Webb: Theory of Nonlinear Age-Dependent Population Dynamics (Marcel Dekker 89, New York 1985)
123. 123.
G. Webb: Logistic models of structured population growth, J. Comput. Appl. 12 (1986) pp 319–335
124. 124.
G. Webb: An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc. 303 (1987) pp 751–763
125. 125.
G. Webb: Periodic and chaotic behavior in structured models of cell population dynamics, in Recent Developments in Evolution Equations, Pitman Res. Notes Math. Ser. 324 (1995) pp 40–49Google Scholar
126. 126.
G. Webb: The steady state of a tumor cord cell popultion, J. Evol Eqs. 2 (2002) pp 425–438
127. 127.
G. Webb: Structured population dynamics, in Mathmatical Modelling of Population Dynamics (Banach Center Publications 63, Institute of Mathematics, Polish Academy of Sciences, Warsaw 2004)Google Scholar 