Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Brownian subordinators and fractional Cauchy problems

Author(s): Boris Baeumer; Mark M. Meerschaert; Erkan Nane
Journal: Trans. Amer. Math. Soc. 361 (2009), 3915-3930.
MSC (2000): Primary 60J65, 60J60, 26A33
Posted: January 28, 2009
MathSciNet review: 2491905
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involves subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes and, consequently, an equivalence between these two families of partial differential equations.

References:

1.: H. Allouba, Brownian-time processes: The pde connection and the corresponding Feynman-Kac formula, Trans. Amer. Math. Soc. 354 (2002), no. 11 4627-4637. MR 1926892 (2003m:60177)
2.: H. Allouba and W. Zheng, Brownian-time processes: The pde connection and the half-derivative generator, Ann. Prob. 29 (2001), no. 2, 1780-1795. MR 1880242 (2002j:60118)
3.: D. Applebaum (2004) Lévy Processes and Stochastic Calculus. Cambridge studies in advanced mathematics. MR 2072890 (2005h:60003)
4.: W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, Birkhäuser-Verlag, Berlin (2001). MR 1886588 (2003g:47072)
5.: L.J.B. Bachelier (1900) Théorie de la Spéculation. Gauthier-Villars, Paris.
6.: E.G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, Eindhoven University of Technology, 2001. MR 1868564 (2002h:34115)
7.: M.T. Barlow (1990) Random walks and diffusion on fractals. In Proceedings of the Internatioal Congress of Mathematicians, Kyoto, Japan 2, 1025-1035. Springer-Verlag. MR 1159287
8.: M.T. Barlow and R.F. Bass (1993) Coupling and Harnack inequalities for Sierpiński carpets. Bull. Amer. Math. Soc. 29 208-212. MR 1215306 (94a:60011)
9.: M.T. Barlow and E.A. Perkins (1988) Brownian motion on the Sierpiński carpet. Probab. Theory Rel. Fields 79 543-623. MR 966175 (89g:60241)
10.: B. Baeumer and M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fractional Calculus Appl. Anal. 4 (2001), 481-500. MR 1874479 (2003d:26006)
11.: R. Bañuelos and R.D. DeBlassie (2006), The exit distribution for iterated Brownian motion in cones. Stochastic Processes and their Applications 116 no. 1, 36-69. MR 2186839 (2007k:60258)
12.: P. Becker-Kern, M.M. Meerschaert and H.P. Scheffler (2004) Limit theorems for coupled continuous time random walks. The Annals of Probability 32, No. 1B, 730-756. MR 2039941 (2004m:60092)
13.: D. Benson, S. Wheatcraft and M. Meerschaert (2000) Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403-1412.
14.: D. Benson, S. Wheatcraft and M. Meerschaert (2000) The fractional-order governing equation of Lévy motion, Water Resources Research 36, 1413-1424.
15.: D. Benson, R. Schumer, M. Meerschaert and S. Wheatcraft (2001) Fractional dispersion, Lévy motions, and the MADE tracer tests. Transport in Porous Media 42, 211-240. MR 1948593
16.: J. Bertoin (1996)
Lévy processes. Cambridge University Press. MR 1406564 (98e:60117)
17.: J. Bisquert (2003) Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous-Time Random Walk. Physical Review Letters 91, No. 1, 602-605.
18.: K. Burdzy, Some path properties of iterated Brownian motion, In Seminar on Stochastic Processes (E. Çinlar, K.L. Chung and M.J. Sharpe, eds.), Birkhäuser, Boston (1993), 67-87. MR 1278077 (95c:60075)
19.: K. Burdzy, Variation of iterated Brownian motion, In Workshops and Conference on Measure-valued Processes, Stochastic Partial Differential Equations and Interacting Particle Systems (D.A. Dawson, ed.) Amer. Math. Soc. Providence, RI (1994), 35-53. MR 1278281 (95h:60123)
20.: K. Burdzy and D. Khoshnevisan, The level set of iterated Brownian motion, Séminarie de probabilités XXIX (Eds.: J Azéma, M. Emery, P.-A. Meyer and M. Yor), Lecture Notes in Mathematics, 1613, Springer, Berlin (1995), 231-236. MR 1459464 (98k:60138)
21.: K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack, Ann. Appl. Probabl. 8 (1998), no. 3, 708-748. MR 1627764 (99g:60147)
22.: E. Csáki, M. Csörgö, A. Földes, and P. Révész, The local time of iterated Brownian motion, J. Theoret. Probab. 9 (1996), 717-743. MR 1400596 (97f:60173)
23.: R. D. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Prob. 14 (2004), no. 3, 1529-1558. MR 2071433 (2005f:60172)
24.: A. Einstein (1956) Investigations on the theory of Brownian movement. Dover, New York. MR 0077443 (17:1035g)
25.: L.R.G. Fontes, M. Isope and C.M. Newman (2002) Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization, and aging in one dimension. Ann. Probab. 30, no. 2, 579-604. MR 1905852 (2003e:60229)
26.: T. Funaki, A probabilistic construction of the solution of some higher order parabolic differential equations, Proc. Japan Acad. Ser. A. Math. Sci. 55 (1979), no. 5, 176-179. MR 533542 (80h:60075)
27.: R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto (2001) Fractional calculus and continuous-time finance. III. The diffusion limit. Mathematical finance (Konstanz, 2000), 171-180, Trends Math., Birkhäuser, Basel. MR 1882830
28.: E. Hille and R.S. Phillips (1957) Functional Analysis and Semi-Groups. Amer. Math. Soc. Coll. Publ. 31, American Mathematical Society, Providence, RI. MR 0089373 (19:664d)
29.: Y. Hu, Hausdorff and packing measures of the level sets of iterated Brownian motion. J. Theoret. Probab. 12 (1999), no. 2, 313-346. MR 1684747 (2000f:60054)
30.: Y. Hu., D. Pierre-Loti-Viaud, and Z. Shi, Laws of iterated logarithm for iterated Wiener proceses, J. Theoret. Probabl. 8 (1995), 303-319. MR 1325853 (96b:60073)
31.: N. Jacob (1996) Pseudo-Differential Operators and Markov Processes. Akad. Verlag, Berlin. MR 1409607 (97m:60109)
32.: N. Jacob (1998) Characteristic functions and symbols in the theory of Feller processes. Potential Anal. 8, no. 1, 61-68. MR 1608650 (99b:60121)
33.: N. Jacob and R. Schilling (2001) Lévy-type processes and pseudodifferential operators. Lévy processes, 139-168, Birkhäuser, Boston, MA. MR 1833696 (2002c:60077)
34.: Z. Jurek and J.D. Mason (1993) Operator-Limit Distributions in Probability Theory. Wiley, New York. MR 1243181 (95b:60018)
35.: D. Khoshnevisan and T.M. Lewis, Stochastic calculus for Brownian motion in a Brownian fracture, Ann. Applied Probabl. 9 (1999), no. 3, 629-667. MR 1722276 (2001m:60128)
36.: D. Khoshnevisan and T.M. Lewis, Chung's law of the iterated logarithm for iterated Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 3, 349-359. MR 1387394 (97k:60218)
37.: M.M. Meerschaert and H.P. Scheffler (2001)
Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice.
Wiley Interscience, New York. MR 1840531 (2002i:60047)
38.: M.M. Meerschaert, D.A. Benson, H.P. Scheffler and B. Baeumer (2002) Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 1103-1106. MR 1917983 (2003d:60165)
39.: M.M. Meerschaert, D.A. Benson, H.P. Scheffler and P. Becker-Kern (2002) Governing equations and solutions of anomalous random walk limits. Phys. Rev. E 66, 102R-105R.
40.: M.M. Meerschaert and H.P. Scheffler (2004) Limit theorems for continuous time random walks with infinite mean waiting times. J. Applied Probab. 41, no. 3, 623-638. MR 2074812 (2005f:60105)
41.: M.M. Meerschaert and H.P. Scheffler (2006) Stochastic model for ultraslow diffusion. Stoch. Proc. Appl., 116, no. 9, 1215-1235. MR 2251542
42.: M.M. Meerschaert and E. Scalas (2006) Coupled continuous time random walks in finance. Physica A, 370, 114-118. MR 2263769 (2007e:91097)
43.: R. Metzler and J. Klafter (2000) The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1-77. MR 1809268 (2001k:82082)
44.: R. Metzler and J. Klafter (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Physics A 37, R161-R208. MR 2090004
45.: E. Nane, Iterated Brownian motion in parabola-shaped domains, Potential Analysis, 24 (2006), 105-123. MR 2217416
46.: E. Nane, Iterated Brownian motion in bounded domains in $\mathbb{R}^{n}$ , Stochastic Processes and Their Applications 116 (2006), 905-916. MR 2254664 (2007j:60133)
47.: E. Nane, Higher order PDE's and iterated processes, Transactions of American Mathematical Society 360 (2008), 2681-2692. MR 2373329 (2008j:60202)
48.: E. Nane, Laws of the iterated logarithm for $\alpha$ -time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434-459 (electronic). MR 2223043 (2007c:60087)
49.: E. Nane, Isoperimetric-type inequalities for iterated Brownian motion in $\mathbb{R}^{n}$ , Statistics & Probability Letters 78 (2008), 90-95. MR 2381278 (2008k:60194)
50.: E. Nane, Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$ , Esaim Probab. Stat., March 2007, Vol. 11, pp. 147-160. MR 2299652 (2008a:60207)
51.: E. Orsingher and L. Beghin (2004) Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory Rel. Fields 128, 141-160. MR 2027298 (2005a:60056)
52.: A. Pazy (1983) Semigroups of Linear Operators and Applications to Partial Differential equations. Applied Mathematical Sciences 44, Springer-Verlag, New York. MR 710486 (85g:47061)
53.: T. Prosen and M. Znidaric (2001) Anomalous diffusion and dynamical localization in polygonal billiards. Phys. Rev. Lett. 87, 114101-114104.
54.: W. Rudin (1973) Functional Analysis. 2nd Edition, McGraw-Hill, New York. MR 0365062 (51:1315)
55.: G. Samorodnitsky and M. Taqqu, Stable non-Gaussian Random processes, Chapman and Hall, New York (1994). MR 1280932 (95f:60024)
56.: K.I. Sato (1999)
Lévy Processes and Infinitely Divisible Distributions.
Cambridge University Press. MR 1739520 (2003b:60064)
57.: E. Scalas (2004) Five Years of Continuous-Time Random Walks in Econophysics. Proceedings of WEHIA 2004, A. Namatame (ed.), Kyoto.
58.: R.L. Schilling, Growth and Hölder conditions for sample paths of Feller proceses.
Probability Theory and Related Fields 112, 565-611 (1998) MR 1664705 (99m:60131)
59.: R. Schumer, D.A. Benson, M.M. Meerschaert and S. W. Wheatcraft (2001) Eulerian derivation of the fractional advection-dispersion equation. J. Contaminant Hydrol., 48, 69-88.
60.: Z. Shi and M. Yor, Integrability and lower limits of the local time of iterated Brownian motion, Studia Sci. Math. Hungar. 33 (1997), no. 1-3, 279-298. MR 1454115 (98i:60069)
61.: M. Shlesinger, J. Klafter and Y.M. Wong (1982) Random walks with infinite spatial and temporal moments. J. Statist. Phys. 27, 499-512. MR 659807 (83e:82066)
62.: Y.G. Sinai (1982) The limiting behavior of a one-dimensional random walk in a random medium. Theor. Probab. Appl. 27, 256-268. MR 657919 (83k:60078)
63.: I.M. Sokolov and J. Klafter (2005) From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion. Chaos 15, 6103-6109. MR 2150232 (2006d:82060)
64.: Y. Xiao, Local times and related properties of multi-dimensional iterated Brownian motion, J. Theoret. Probab. 11 (1998), no. 2, 383-408. MR 1622577 (99g:60136)
65.: G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76 (1994), 110-122. MR 1295881 (95h:58120)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|