Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0918.60040
Arnaudon, Marc; Thalmaier, Anton
Stability of stochastic differential equations in manifolds. (English)
[A] Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 188-214 (1998). ISBN 3-540-64376-1

The authors study SDEs in a smooth closed manifold $M$ endowed with a connection $\nabla^M$ and state suitable assumptions to ensure their stability in the topology of $M$-valued continuous $\nabla$-semimartingales which is proved to coincide with the topology of compact convergence in probability on the set of continuous $\nabla$-martingales. Given two Riemannian manifolds $M$ and $N$ with connections $\nabla^M$ and $\nabla^N$ correspondingly they consider a family of stochastic differential equations of the type $DZ(a)=f(X(a),Z(a))DX(a), a\in I\subset R$, where $f\in \Gamma (T^2(M)^*\otimes T^2(N))$ is a Schwartz morphism and $T^2(M)$ is the second tangent bundle over $M$. If $f$ is smooth enough, then the derivative $\partial_aZ(a)$ is proved to solve the equation $$D\partial_aZ(a)=f'(\partial_aX(a),\partial_a Z(a))D\partial_aZ(a),$$ where $f'$ is a Schwartz morphism between $T^2(TM)$ and $T^2(TN)$. As a consequence the results on differentiability of solutions to SDEs on manifolds are stated. The proof is based on the possibility to treat $f(x,z)=\tau_3\varphi(x,z,x)$ in terms of a Cohen map $ \varphi:M\times N\times M\to N$ where $\tau_3\varphi$ defines the second derivative of $\varphi$ w.r.t the third variable.
[Yana Belopolskaya (St.Peterburg)]

MSC 2000:: *60H10 Stochastic ordinary differential equations

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]