Support of the Brown Measure of Free Multiplicative Brownian Motions with Non-Negative Initial Condition and Its Three-Parameter Family

We study a family of free multiplicative Brownian motions parameterized by a real variance and a complex covariance. In the case variance and covariance are the same, we identify a region such that the Brown measure is vanishing outside of the closure of this region except possibly at the origin. In the general case, we construct a mapping and define the support as the complement of the image of the complement of the first region under the mapping. Then the Brown measure is again zero outside this support except possibly at the origin.