Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula


Author: Hassan Allouba
Journal: Trans. Amer. Math. Soc. 354 (2002), 4627-4637
MSC (2000): Primary 60H30, 60J45, 60J35; Secondary 60J60, 60J65
DOI: https://doi.org/10.1090/S0002-9947-02-03074-X
Published electronically: June 4, 2002
MathSciNet review: 1926892
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.


References [Enhancements On Off] (What's this?)

  • 1. H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator. Ann. of Probab. 29 no. 4, (2001), 1780-1795.
  • 2. H. Allouba, Measure-valued Brownian-time processes: the PDE connection. (2001). In preparation.
  • 3. H. Allouba, Brownian-time type processes: the PDE connection III. (2001). In preparation
  • 4. R. Ash, Real analysis and probability. Academic Press, Inc., New York, 1972. MR 55:8280
  • 5. R. Bass, Probabilistic techniques in analysis. Springer-Verlag, New York, 1995. MR 96e:60001
  • 6. R. Bass, Diffusions and elliptic operators. Springer-Verlag, New York, 1997. MR 99h:60136
  • 7. K. Burdzy, Some path properties of iterated Brownian motion. Seminar on Stochastic Processes 1992, Birkhäuser, (1993), 67-87. MR 95c:60075
  • 8. K. Burdzy, Variation of iterated Brownian motion. Workshop and conference on measure-valued processes, stochastic PDEs and interacting particle systems. CRM Proceedings and Lecture Notes 5, (1994), 35-53. MR 95h:60123
  • 9. K. Burdzy and D. Khoshnevisan, The level sets of iterated Brownian motion. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1613, (1995), 231-236. MR 98k:60138
  • 10. K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack . Ann. Appl. Probab. 8 no. 3, (1998), 708-748. MR 99g:60147
  • 11. R. Durrett, Stochastic calculus, a practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL., 1996. MR 97k:60148
  • 12. K. D. Elworthy, Stochastic differential equations on manifolds, Cambridge University Press, 1982. MR 84d:58080
  • 13. A Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR 31:6062
  • 14. T. Funaki, Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad. Ser. A Math. Sci. 55 no. 5, (1979), 176-179. MR 80h:60075
  • 15. K. Hochberg and E. Orsingher, Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9, no. 2, (1996), 511-532. MR 97f:60181
  • 16. I. Karatzas and S. Shreve, Brownian motion and stochastic calculus. Springer-Verlag, 1988. MR 89c:60096
  • 17. D. Khoshnevisan and T. Lewis Iterated Brownian motion and its intrinsic skeletal structure. Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996), 201-210, Progr. Probab., 45, Birkhäuser, Basel, 1999. MR 2001m:60183
  • 18. H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press, 1990. MR 91m:60107
  • 19. J-F. Le Gall, Solutions positives de $\Delta u =u^2$dans le disque unité. C. R. Acad. Sci. Paris Série I. 317, (1993), 873-878. MR 94h:35059
  • 20. J-F. Le Gall, A path-valued Markov process and its connections with partial differential equations. First European Congress of Mathematics, Vol. II (Paris, 1992). Progr. Math. 120, Birkhäuser, (1994), 185-212. MR 96m:60169
  • 21. J-F. Le Gall, The Brownian snake and solutions of $\Delta u=u\sp 2$ in a domain. Probab. Theory Related Fields. 102 no. 3, (1995), 393-432. MR 96c:60098
  • 22. T. Lyons, and W. Zheng, On conditional diffusion processes. Proc. Roy. Soc. Edinburgh. 115, (1990), 243-255. MR 91m:60148
  • 23. W. Zheng, Conditional propagation of chaos and a class of quasilinear PDE's. Ann. Probab. 23 no. 3, (1995), 1389-1413. MR 96m:60183

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60H30, 60J45, 60J35, 60J60, 60J65

Retrieve articles in all journals with MSC (2000): 60H30, 60J45, 60J35, 60J60, 60J65


Additional Information

Hassan Allouba
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-7106
Address at time of publication: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240-0001
Email: allouba@indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03074-X
Keywords: Brownian-time processes, initially perturbed fourth order PDEs, Brownian-time Feynman-Kac formula, iterated Brownian motion
Received by editor(s): November 5, 2001
Received by editor(s) in revised form: April 9, 2002
Published electronically: June 4, 2002
Additional Notes: Supported in part by NSA grant MDA904-02-1-0083
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society