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Probability theory and applications. (English) Zbl 0932.60005

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This book contains six lectures series delivered at the IAS/Park City Mathematical Institute Graduate Summer School in June–July 1996.
The lectures on “Stochastic spatial models” by R. Durrett (5–47) can be grouped into two parts. In the first three lectures, the author studies the voter model on the integer lattice by means of its dual random walk. He also considers some recent results about the multitype voter model with mutations. The last three lectures give a brief introduction into a useful method for proving the existence of phase transitions, known as the “block construction”. Furthermore, two techniques making interacting particle systems more tractable are considered: long range limits and rapid stirring limits.
The lectures on “Independent and dependent percolation” by J. T. Chayes, A. L. Puha and T. Sweet (49–166) give an excellent introduction into the theory of percolation. The authors start with an overview of the basic percolation model and of some fundamental inequalities (FKG, BK). The high-density phase of two-dimensional percolation and the correlation length are characterised. This is followed by a discussion of certain “critical exponent inequalities”, and by the proof of two major achievements of percolation theory: Uniqueness of an infinite cluster and the proof that the critical probability for percolation coincides with the critical probability for infinite expected cluster size. The lectures end with a review of Reimer’s new proof of the BK-inequality for arbitrary events in percolation and with a detailed discussion of Potts model and the random cluster model.
The lectures by L. Jensen and H.-T. Yau (167–225) deal with the question of deriving the macroscopic dynamics of large systems of locally interacting particles. The authors state and prove a law of large numbers for the macroscopic evolution of the simple exclusion process. The proof is based on entropy and Dirichlet form estimates. The asymmetric simple exclusion process is analysed by means of the relative entropy method described by H.-T. Yau [Lett. Math. Phys. 22, No. 1, 63-80 (1991; Zbl 0725.60120)]. The lectures end with the discussion of a few new issues related to the Green-Kubo formula.
The notes by D. W. Stroock (227–276)deal with analysis on path spaces over a Riemannian manifold. After reviewing some basic facts about Gaussian measures in an infinite-dimensional setting the author constructs Brownian motion on a Riemannian manifold (RM). The construction is based on a geometrically inspired map known as the rolling map. Basic facts of Riemannian Brownian motion are proved and Bismut’s formula for the gradient of the heat kernel is derived. In the last part Stroock shows how the spaces of Cameron-Martin path over a Riemannian manifold can be regarded as an infinite RM.
In the first part of the lectures by E. P. Hsu ( 277–347), the author deals with Euclidean path and loop spaces. The more demanding second part deals with path and loop spaces over a RM. The author concentrates on various properties of the gradient operator in the path or loop space of a complete RM. The lectures end with a new proof of the logarithmic Sobolev inequalities in the path space.
M. Avellaneda (349–374) gives a brief introduction into the field of mathematical finance. The famous Black-Scholes differential equation is derived. Various risk parameters like Delta, Gamma and Vega are discussed. In the second part the author reviews some of his recent results about uncertain volatility models. Using an entropy approach and solving a constrained optimal control problem, an arbitrage free diffusion model for the evolution of the stock price process is derived. This process minimises a certain distance to a given diffusion reflecting an investor’s prior belief about the dynamics of the volatility process.

MSC:

60-06 Proceedings, conferences, collections, etc. pertaining to probability theory
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
60-02 Research exposition (monographs, survey articles) pertaining to probability theory

Citations:

Zbl 0725.60120
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