Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has been added and several inaccuracies have been corrected. To appear in Journal of Theoretical Probabilit

Собрание узаконений и распоряжений правительства, издаваемое при Правительствующем Сенате. 1863. Первое полугодие. № 019. Ст. 90-96

Interest has been revived in the creation of a "bill of rights" for Internet users. This paper analyzes users' rights into ten broad principles, as a basis for assessing what users regard as important and for comparing different multi-issue Internet policy proposals. Stability of the principles is demonstrated in an experimental survey, which also shows that freedoms of users to participate in the design and coding of platforms appear to be viewed as inessential relative to other rights. An analysis of users' rights frameworks that have emerged over the past twenty years similarly shows that such proposals tend to leave out freedoms related to software platforms, as opposed to user data or public networks. Evaluating policy frameworks in a comparative analysis based on prior principles may help people to see what is missing and what is important as the future of the Internet continues to be debated