# 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)
Dynamics of stochastic approximation algorithms. (English) Zbl 0955.62085
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 1-68 (1999).

This is the author’s DEA course given at ENS Cachan during 1996-98. From the author’s abstract: The aim of this course is to introduce the reader to the dynamical system aspects of the theory of stochastic approximation.

The general form of stochastic approximation processes is ${x}_{n+1}-{x}_{n}={\gamma }_{n}\left(F\left({x}_{n}\right)+{U}_{n}\right)$, $n\in ℕ$. It is difficult to summarize in few lines all results given in this dense course; every section should be reviewed separately. The basic concept is “asymptotic pseudotrajectories”, whose definition is given in section 3. All other sections give results about the link between asymptotic pseudotrajectories and stochastic approximation.

From the author’s introduction: In section 4 classical results on stochastic approximations are (re)formulated in the language of asymptotic pseudotrajectories. It is shown that, under suitable conditions, the continuous time process obtained by a convenient interpolation of $\left\{{x}_{n}\right\}$ is almost surely an asymptotic pseudotrajectory of the semiflow induced by the associated ODE $dx/dt=F\left(x\right)\cdots$

Section 5 characterizes the limit sets of asymptotic pseudotrajectories. The main result of the section establishes that limit sets of precompact asymptotic pseudotrajectories are internally chain-transitive...

Section 6 applies the abstract results of section 5 in various situations. It is shown how assumptions on the deterministic dynamics can help to identify the possible limit sets of stochastic approximation processes with a great deal of generality....

Section 7 establishes simple sufficient conditions ensuring that a given attractor of an ODE has a positive probability to host the limit set of the stochastic approximation process...

Section 8 considers the question of shadowing...Section 9 focusses on the behavior of stochastic approximation processes near ‘unstable’ sets...

Section 10 introduces the notion of a stochastic process being a weak asymptotic pseudodotrajectory for a semiflow and analyzes properties of its empirical occupation measure.

##### MSC:
 62L20 Stochastic approximation 37L30 Attractors and their dimensions, Lyapunov exponents