In this talk, I will introduce the "true self repelling motion" (TSRM) which was defined by Bálint Tóth and Wendelin Werner in 1998. It is the continuous counterpart of self repelling random walks and its construction uses a family of coalescing Brownian motions now called the "Brownian Web" which is interesting for its own sake and has been largely studied since.
I will try to explain how to derive properties of the TSRM such as its large deviations or local fluctuations from the Brownian web family. I will also describe how to identify the marginal distributions from a more analytical point of view (using Feynman Kac formulas). It permits in particular to exhibit a surprising shape for the density of the position of the TSRM at time one: Its derivative at the starting point is discontinuous which implies the process strongly remembers its initial position. This last result comes from a joint work with Bálint Tóth.