zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Feynman-Kac formula for heat equation driven by fractional white noise. (English) Zbl 1210.60056
The aim of this paper is to obtain a Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. Using an approximation of the Dirac delta function, they show that the stochastic Feynman-Kac functional is a well-defined random variable with exponential integrabiliy. Using again an approximation technique together with Malliavin calculus, they prove that the process defined by the Feynman-Kac functional is a weak solution of the stochastic heat equation. Using the explicit form of the weak solution they prove the Hölder continuity of the solution with respect time and space and they establish the smoothness of the density of the probability law of the solution. Finally, using a wiener chaos technique, they prove that there exists a unique mild solution to the Skorohod-type equation. They also get a Feynman-Kac formula for this solution.

MSC:
60H07Stochastic calculus of variations and the Malliavin calculus
60H15Stochastic partial differential equations
60G17Sample path properties
60G22Fractional processes, including fractional Brownian motion
60G30Continuity and singularity of induced measures (stochastic processes)
35K20Second order parabolic equations, initial boundary value problems
35R60PDEs with randomness, stochastic PDE
WorldCat.org
Full Text: DOI arXiv
References:
[1] Dawson, D. A. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141-180. · Zbl 0439.60051 · doi:10.1016/0047-259X(80)90012-3
[2] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Annals of Mathematics Studies 109 . Princeton Univ. Press, Princeton, NJ. · Zbl 0568.60057
[3] Hinz, H. (2009). Burgers system with a fractional Brownian random force. Preprint, Technische Univ. Berlin. · Zbl 1220.60036
[4] Hu, Y., Lu, F. and Nualart, D. (2010). Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H <1?2. Preprint, Univ. Kansas. · Zbl 1253.60074
[5] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285-328. · Zbl 1152.60331 · doi:10.1007/s00440-007-0127-5
[6] Hu, Y.-Z. and Yan, J.-A. (2009). Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl. Ser. 25 399-414. · Zbl 1186.60034 · doi:10.1007/s10255-008-8808-0
[7] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24 . Cambridge Univ. Press, Cambridge. · Zbl 0743.60052
[8] Le Gall, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. In Séminaire de Probabilités , XXVIII. Lecture Notes in Math. 1583 172-180. Springer, Berlin. · Zbl 0810.60078 · numdam:SPS_1994__28__172_0 · eudml:113872
[9] Mocioalca, O. and Viens, F. (2005). Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222 385-434. · Zbl 1068.60078 · doi:10.1016/j.jfa.2004.07.013
[10] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Springer, Berlin. · Zbl 1099.60003
[11] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403-421. · Zbl 0792.60046 · doi:10.1007/BF01195073
[12] Viens, F. G. and Zhang, T. (2008). Almost sure exponential behavior of a directed polymer in a fractional Brownian environment. J. Funct. Anal. 255 2810-2860. · Zbl 1152.82028 · doi:10.1016/j.jfa.2008.06.020
[13] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060