The role of weak convergence in probability theory. (English) Zbl 1186.60019

Varadhan, S. R. S.

Babbitt, Donald (ed.) et al., Symmetry in mathematics and physics. Conference in honor of V. S. Varadarajan’s 70th birthday, Los Angeles, CA, USA, January 18–20, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4731-2/pbk). Contemporary Mathematics 490, 3-10 (2009).

The author gives a survey on the ideas of weak convergence of probabilities and of stochastic processes, with applications to Markov processes, large deviations, scaling limits of large systems of interacting processes and large deviations of interacting particle systems.
Probabilities and weak convergence on topological spaces are motivated by investigations of convergence of stochastic processes identified by probabilities on path-spaces endowed with a topology. See e.g. [the author, Sankhyā, Ser. A 24, 213–238 (1962; Zbl 0113.34101); Bull. Am. Math. Soc. 67, 276–280 (1961; Zbl 0107.12402); K. R. Parthasarathy, R. Ranga Rao and the author, Ill. J. Math. 7, 337–369 (1963; Zbl 0129.10902)].
In Section 2 the author explains the advantages of considering processes as probabilities on path spaces by the martingale approach to Markov processes. See e.g. the pioneering work of D. Stroock and the author [Commun. Pure Appl. Math. 22, 345–400 (1969; Zbl 0167.43903); Commun. Pure Appl. Math. 22, 479–530 (1969; Zbl 0167.43904)].
The following sections are concerned with the large deviations problem and similarities to weak convergence, with scaling limits for large systems of interacting processes [M. Z. Guo, G. C. Papanicolau and the author, Commun. Math. Phys. 118, No. 1, 31–59 (1988; Zbl 0652.60107)] and large deviations and a problem of interacting particle systems.
For the entire collection see [Zbl 1168.22001].

Reviewer: Wilfried Hazod (Dortmund)

MSC:

60F05	Central limit and other weak theorems
60B10	Convergence of probability measures
60B99	Probability theory on algebraic and topological structures
60F10	Large deviations
60G44	Martingales with continuous parameter
60J25	Continuous-time Markov processes on general state spaces
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

weak convergence; martingale problem; large deviations; scaling limits

Citations:

Zbl 0113.34101; Zbl 0107.12402; Zbl 0129.10902; Zbl 0167.43903; Zbl 0167.43904; Zbl 0652.60107