Given a selfadjoint random variable x 0 and a unitary random variable u, different from Haar unitary, free from the free circular Brownian motion c t and the free multiplicative Brownian motion b t , we use the Hamilton-Jacobi method to compute the Brown measures of free circular Brownian motion x 0 + c t and the free multiplicative Brownian motion ub t with probabilistic initial point, extending the recent work [13] by Driver-Hall-Kemp. We find that the supports of the Brown measures are related to the subordination functions of the free additive and multiplicative convolutions. The density of the Brown measure of x 0 + c t is constant along the vertical direction in the additive case, and the density of the Brown measure of ub t has the form, in polar coordinates, of (1/r 2 )w t (θ). The densities of the Brown measures are closely related to the laws of x 0 + s t and uu t , where s t is the free semicircular Brownian motion and u t is the free unitary Brownian motion, both freely independent of x 0 and u.

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If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -The purpose of this paper is to demonstrate consumers' voluntary behaviors on Facebook through exploring how members' community participation affects consumer citizenship behaviors toward the brand. The study also provided further insight into the mediating effect by considering brand trust and community identification. Design/methodology/approach -This research begins by developing a framework to describe and examine the relationship among Facebook participants, brand trust, community identification, and consumer citizenship behaviors. Furthermore, it tests the mediating effects of brand trust and community identification on the relationship between Facebook participation and consumer citizenship behaviors. The model and hypotheses in this study employ structural equation modeling with survey data. Findings -First, this study reveals consumers' community participation on Facebook has directly positive and significant effects on brand trust and community identification. Second, this research confirms that brand trust has directly positive and significant effects on community identification. Third, this study found that brand trust and community identification play a mediating role between Facebook participation and consumer citizenship behaviors.Research limitations/implications -The sample comprised primarily young adults, which may not be completely generalizable to the population at large. This study examined a specific form of virtual community, Facebook, so the results cannot be ascribed to other formats of brand community. Originality/value -The issue of consumer' voluntary behavior on social networking sites has become more and more important. This study proposed an exclusive model of the process by which the paper can consider consumers' voluntary behaviors on Facebook from participation to consumer citizenship behavior toward the brand. This finding can be viewed as pioneering, setting a benchmark for further research.

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The purpose of this study was to demonstrate how to manage digital customer relationships (i.e. relationships with the brand, the product, the company, and other fans) on social media based community (i.e. Facebook brand fan-pages) and to influence post-purchase intentions (i.e. word-of-mouth and re-purchase intentions). This study used partial least squares to test the hypotheses and analyze the data. The results of this study indicated that all these four customer-community relationships can enhance post-purchase behaviors by improving individual community participation or identification. The findings are of benefit to both academics and practitioners and this research is one of the first to demonstrate how to manage digital customer relationships on social media brand community.