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)
On a stochastic partial differential equation with a fractional Laplacian operator. (English) Zbl 1254.60068
Summary: We consider the regularity of the solution of $$du(t,x)= (\Delta^{{\alpha\over 2}} u(t,x)+ f(t,x))\,dt+ \sum^m_{i=1} g^i(t, x)\,dw^i_t,\quad u(0,x)= u_0(x).$$ We adopt the framework given in some works of Krylov which are related to the theory of stochastic partial differential equations with the Laplace operator. We construct the important estimates for the theory and prove regularity theorems using them.

MSC:
60H15Stochastic partial differential equations
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI
References:
[1] Bergh, J.; Lofstrom, J.: Interpolation spaces, an introduction, (1976)
[2] Bogdan, K.; Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. math. Phys. 271, No. 1, 179-198 (2007) · Zbl 1129.47033 · doi:10.1007/s00220-006-0178-y
[3] Caffarelli, L.; Silvestre, L.: An extenstion problem related to the fractional Laplacian, Comm. partial differential equations 32, No. 7--9, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[4] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, Graduate texts in mathematics 113 (1991) · Zbl 0734.60060
[5] I. Kim, K.H. Kim, A generalization of the Littlewood--Paley inequality for the fractional Laplacian, arXiv:1006.2898v1 [math.FA], 15 June, 2010.
[6] Krylov, N. V.: Introduction to the theory of diffusion processes, Translations of mathematical monographs 142 (1995) · Zbl 0844.60050
[7] Krylov, N. V.: An analytic approach to spdes, stochastic partial differential equations: six perspectives, Math. surv. Monogr. 64, 185-242 (1999) · Zbl 0933.60073
[8] Krylov, N. V.: A generalization of the Littlewood--Paley inequality and some other results related to stochastic partial differential equations, Ulam quart. 2, No. 4, 16-26 (1994) · Zbl 0870.42005
[9] Krylov, N. V.: Lectures on elliptic and parabolic equations in Sobolev spaces, (2008) · Zbl 1147.35001
[10] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, NJ, 1970. · Zbl 0207.13501