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 exponential stability criteria of stochastic partial differential equations. (English) Zbl 0997.60065
Some criteria for the mean square and almost sure exponential stability of the zero solution of the nonlinear stochastic partial differential equation $$X_t=X_0+\int _0^tA(s,X_s) ds + \int _0^tB(s,X_s) dW_s$$ are given, where $A(t,\cdot): V\to V'$, $B(t,\cdot):V\to \Cal L(K,H)$, $V$ is a Banach space and $H,K$ are real separable Banach spaces such that $V\hookrightarrow H=H'\hookrightarrow V'$, where the injections are continuous and dense. The coefficients $A,B$ are assumed to satisfy the usual coercivity, boundedness, monotonicity, hemicontinuity and measurability conditions. The main results obtained by {\it T. Caraballo} and {\it J. Real} [Stochastic Anal. Appl. 12, No. 5, 517-525 (1994; Zbl 0808.93069)] are improved, which in particular concerns the case of non-autonomous equation. Several illustrating examples are also given.

MSC:
60H15Stochastic partial differential equations
WorldCat.org
Full Text: DOI
References:
[1] Caraballo, T.; Real, J.: On the pathwise exponential stability of non-linear stochastic partial differential equations. Stochast. anal. Appl. 12, No. 5, 517-525 (1994) · Zbl 0808.93069
[2] Chow, P. L.: Stability of nonlinear stochastic evolution equations. J. math. Anal. appl. 89, 400-419 (1982) · Zbl 0496.60059
[3] Haussmann, U. G.: Asymptotic stability of the linear itô equation in infinite dimension. J. math. Anal. appl. 65, 219-235 (1978) · Zbl 0385.93051
[4] Mao, X.R., 1994. Exponential Stability of Stochastic Differential Equations. Marcel Dekker, Inc., New York. · Zbl 0806.60044
[5] Pardoux, E., 1975. Equations aux dérivées partielles stochastiques nonlinéaires monotones. Thesis, Université Paris Sud.