This is yet another indispensable volume for all probabilists and collectors of the Saint-Flour series, and is also of great interest for mathematical physicists. It contains two of the three lecture courses given at the 32nd Probability Summer School in Saint-Flour (July 7-24, 2002). Tsirelson's lectures introduce the notion of nonclassical noise produced by very nonlinear functions of many independent random variables, for instance singular stochastic flows or oriented percolation. Werner's contribution gives a survey of results on conformal invariance, scaling limits and properties of some two-dimensional random curves. It provides a definition and properties of the Schramm-Loewner evolutions, computations (probabilities, critical exponents), the relation with critical exponents of planar Brownian motions, planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
The study reported here presents the design, the findings and the conclusions of a research project involving researchers from seven countries. The project was conducted by a working groups led by the German Institute for International Educational Research (DIPF). The findings are based on reports of the school systems in Canada, England, Finland, France, the Netherlands and Sweden." - p. 7.
Since the fundamental work of Walras (1874), markets have received particular attention by economists because they lead to an efficient allocation of goods and services. However, the proper functioning of markets rests on certain assumptions. For instance, the good or ser vice which is to be traded must be clearly defined. This elementary requirement is often violated in reality, in particular when services are concerned. Consider the example of railway workers who are hired to lay tracks. A labour contract which stipulates a fixed wage and defines the workers' task as "laying tracks" is rather unspecific. Workers may profit from this vagueness by reducing effort to a comfortable amount -as long as tracks are laid, they do not violate contract conditions. Thus, an im precise definition of the service can result in inefficiently low efforts. An obvious solution to this problem is a clearer definition of the ser vice, but often this way is barred: To specify, for instance, all actions which are involved in laying tracks and which may vary with weather, surface and other conditions is far too complicated and too costly. In deed, labour contracts seldom give a detailed account of the task of a worker. Alternatively to a more precise task description, the wage of the worker could be conditioned on information about the worker's performance. For example, the railway workers might be paid by the length of tracks laid so that they are motivated to exert more effort.
This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.
This is yet another indispensable volume for all probabilists and collectors of the Saint-Flour series, and is also of great interest for mathematical physicists. It contains two of the three lecture courses given at the 32nd Probability Summer School in Saint-Flour (July 7-24, 2002). Tsirelson's lectures introduce the notion of nonclassical noise produced by very nonlinear functions of many independent random variables, for instance singular stochastic flows or oriented percolation. Werner's contribution gives a survey of results on conformal invariance, scaling limits and properties of some two-dimensional random curves. It provides a definition and properties of the Schramm-Loewner evolutions, computations (probabilities, critical exponents), the relation with critical exponents of planar Brownian motions, planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
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