# Asymptotic Expansion for the Time Evolution of the Probability Distribution by the Brownsche Motion on Semialgebraic Sets

Speaker: Julia Ruppert

Julia Ruppert started her talk with introducing several definitions like the o-Notation (‘middle o-Notation’), asymptotic development and asymptotic set.

By Brownsche Motion the base motivation for her work was given exemplary. The Brownsche Motion is a steady irregular motion which was observed by Robert Brown in 1827 on pollen swimming in a drop of water. Calculating the probability distribution by the Brownsche Motion is challenging on Semialgebraic Sets.

With Semialgebraic Sets it is possible to describes elments like tables, boxes or pacman. But it is not possible to describe a snowflake wich is a fractual element. So Semialgebraic Sets do not contain fractual elements.

Julia Ruppert showed that the Asymptotic Expansion for the Time Evolution of the Probibility Distribution by the Brownsche Motion on Semialgebraic Sets is working for 1-dimensional, 2-dimensional and 3-dimensional cases.

Future work in this topic would be to prove that it is possible for n-dimensional cases too. Analyizing the solutions for 2-dimensional and 3-dimensional cases a pattern for the solution could be recognized which would allow the guess that n-dimensional cases would be solvable.