In dimension 2, we have the Schramm and Loewner evolutions, very nice processes which are invariant by conformal maps (up to time-changes). 

https://link.springer.com/chapter/10.1007/978-3-540-39982-7_2

https://www.studocu.com/en-gb/document/the-chancellor-masters-and-scholars-of-the-university-of-cambridge/schramm-loewner-evolutions/lecture-notes-schramm-loewner-evolutions-complete/726002