Michael Anshelevich
On a $q$-deformed Fock space, we define multiple $q$-L\'{e}vy processes. Using the partition-dependent stochastic measures derived from such processes, we define partition-dependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of restricted crossings of P.~Biane. In the single variable case, this allows us to define a $q$-convolution for a large class of probability measures. We make some comments on the It\^{o} table in this context, and investigate the $q$-Brownian motion and the $q$-Poisson process in more detail.