No Access

THE STOCHASTIC WAVE EQUATION DRIVEN BY FRACTIONAL BROWNIAN NOISE AND TEMPORALLY CORRELATED SMOOTH NOISE

PETER CAITHAMER

8600 University Blvd., Department of Mathematics, University of Southern Indiana, Evansville, IN 47712, USA

https://doi.org/10.1142/S0219493705001286Cited by:23 (Source: Crossref)

Abstract

The stochastic wave equation in one spatial dimension driven by a class of fractional noises or, alternately, by a class of smooth noises with arbitrary temporal covariance is studied. In either case, the wave equation is explicitly solved and the upper and lower bounds on both the large and small deviations of several sup norms associated with the solution are given. Finally the energy of a system governed by such an equation is calculated and its expected value is found.

Keywords:

AMSC: 60H15, 60G18, 60F10, 35R60

References

E. Alòs, O. Mazet and D. Nualart, Ann. Probab. 29, 766 (2001). Web of Science, Google Scholar
B. P. Belinskiy and P. Caithamer, J. Appl. Probab. 38, 960 (2001). Crossref, Web of Science, Google Scholar
S. K. Biswas and N. U. Ahmed, IEEE Trans. Automat. Control AC-30, 1043 (1985). Web of Science, Google Scholar
E. Cabaña, Z. Wahrscheinlichkeitstheorie verw. Geb. 15, 111 (1970). Crossref, Google Scholar
E. Cabaña, Z. Wahrscheinlichkeitstheorie verw. Geb. 22, 13 (1972). Crossref, Google Scholar
P. Caithamer, J. Appl. Probab. 40, (2003). Google Scholar
J. B. Conway , A Course in Functional Analysis ( Springer-Verlag , 1990 ) . Google Scholar
R. C. Dalang and N. E. Frangos, Ann. Probab. 26, 187 (1998). Web of Science, Google Scholar
G. Da Prato and J. Zabczyk , Stochastic Equations in Infinite Dimensions ( Cambridge Univ. Press , 1992 ) . Crossref, Google Scholar
T. E. Duncan, Y. Hu and B. Pasik-Duncan, SIAM J. Control Optim. 38, 582 (1998). Crossref, Web of Science, Google Scholar
R. Durrett , Probability: Theory and Examples , 2nd edn. ( Duxbury Press , 1995 ) . Google Scholar
R. Durrett , Stochastic Calculus: A Practical Introduction ( CRC Press , 1996 ) . Google Scholar
R. Dudley, J. Funct. Anal. 1, 290 (1967). Crossref, Web of Science, Google Scholar
M. Elshamy, J. Appl. Probab. 32, 417 (1995). Crossref, Web of Science, Google Scholar
M. Elshamy, J. Appl. Probab. 33, 1159 (1996). Crossref, Web of Science, Google Scholar
M. Erraoui, D. Nualart and Y. Ouknine, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet preprint . Google Scholar
L. C. Evans , Partial Differential Equations ( Amer. Math. Soc. , 1998 ) . Google Scholar
X. Fernique, École Été de Probabilitiés de St.-Flour, Lecture Notes in Math. 480 (1975) pp. 1–96. Crossref, Google Scholar
G. Grippenberg and I. Norros, J. Appl. Probab. 33, 400 (1996). Crossref, Web of Science, Google Scholar
C. Houdré , V. Pérez-Abreu and A. S. Üstünel , Choas Expansions, Multiple Wiener–Itô Integrals and Their Applications ( CRC Press , 1994 ) . Google Scholar
Y. Hu, B. Øksendal and T. Zhang, Stochastic partial differential equations driven by multiparameter white noise, in Canadian Math. Soc., Conference Proceedings Vol. 29 (2000), pp. 327–337 . Google Scholar
Y. Hu, Appl. Math. Optim. 43, 221 (2001). Crossref, Web of Science, Google Scholar
M. Ledoux, Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1648 (Springer-Verlag, 1996) pp. 165–294. Crossref, Google Scholar
M. Ledoux and M. Talagrand , Probability in Banach Spaces ( Springer , 1991 ) . Crossref, Google Scholar
W. Li and Q. Shao , Stochastic Processes: Theory and Methods , Handbook of Statistics 19 , eds. C. R. Rao and D. Shanbhag ( ) . Google Scholar
S. J. Lin, Stoch. Stoch. Rep. 55, 121 (1995). Crossref, Google Scholar
B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422 (1968). Crossref, Web of Science, Google Scholar
R. N. Miller, J. Geophys. Res. 95, 482 (1990). Google Scholar
D. Nualart , The Malliavin Calculus and Related Topics ( Springer-Verlag , 1995 ) . Crossref, Google Scholar
D. Nualart and A. Răşcanu, Collect. Math. 53, 55 (2002). Google Scholar
B. Øksendal , Stochastic Differential Equations: An Introduction with Applications ( Springer-Verlag , 2000 ) . Google Scholar
E. Orsingher, Ann. Inst. H. Poincaré B18, 367 (1982). Google Scholar
E. Orsingher, Adv. Appl. Probab. 16, 562 (1984). Crossref, Web of Science, Google Scholar
E. Orsingher, J. Appl. Probab. 26, 182 (1989). Crossref, Web of Science, Google Scholar
E. E. Peters , Fractal Market Analysis ( Wiley , 1994 ) . Google Scholar
P. Protter , Stochastic Integration and Differential Equations: A New Approach ( Springer-Verlag , 1990 ) . Crossref, Google Scholar
K. Sobczyk , Stochastic Differential Equations: With Applications to Physics and Engineering ( Kluwer , 1991 ) . Crossref, Google Scholar
J. B. Walsh, Ecole d'Eté de Probabilitiés de Saint-Flour XIV, Lecture Notes in Math. 1180 (Springer, 1986) pp. 265–439. Crossref, Google Scholar