Large time unimodality for classical and free Brownian motions with initial distributions

Takahiro Hasebe, Yuki Ueda
2018 Latin American Journal of Probability and Mathematical Statistics  
We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are shown for a symmetric stable process with index 1 and a positive stable process with index $1/2$. We also prove that free Brownian motion with initial symmetric unimodal distribution is unimodal, and discuss strong unimodality for free convolution.
doi:10.30757/alea.v15-15 fatcat:ugnw2mwoavedthjjerutu2oa34