Denote by
J
the operator of coefficient stripping. We show that for any free convolution semigroup
{μt:t≥0} with finite
variance, applying a single stripping produces semicircular evolution with nonzero initial
condition,
J[μt]=ρ⊞σ⊞tβ,γ, where
σβ,γ is the semicircular
distribution with mean
β
and variance
γ.
For more general freely infinitely divisible distributions
τ, expressions of
the form
˜ρ⊞τ⊞t arise
from stripping
˜μt,
where
{(˜μt,μt):t≥0}
forms a semigroup under the operation of two-state free convolution. The converse to
this statement holds in the algebraic setting. Numerous examples illustrating these
constructions are computed. Additional results include the formula for generators of
such semigroups.
Keywords
free convolution semigroups, two-state free convolution,
coefficient stripping, subordination distribution
