In this talk, I will discuss some techniques originating from Biane's work
"on the free convolution with a semi-circular distribution" and show how they are used
to study free convolution semigroups.
If time permits, I will discuss its connection with compression norm
and application to additivity problem of minimal output entropy of quantum channels.
The second part of the talk is based on a joint work with Benoit Collins
and Motohisa Fukuda.