Diiusions on Braided Spaces Ii Preliminaries Ii.1 Braided Spaces

The notion of q-Brownian motion introduced by Majid is extended to braided spaces corresponding to a generic R-matrix, and combined with the theory of quantum probability. This leads to a deenition of diiusions on these spaces. The corresponding heat equations (diierence-diierential equations) are solved in terms of Appell polynomials (i.e. shifted moment systems). Some examples of interest for applications are given. The applications of diiusions in physics go far beyond the description of the physical phenomenon they are named after. Functional integrals can be used to solve partial diierential equations, cf. the celebrated Feynman-Kac formula. Wiener integrals are very close to Feynman path integrals. Another interesting application of diiusions is the stochastic mechanics of Nelson 1]. We will present an approach to diiusions on braided spaces here. Diiusions on man-ifolds are characterized by two properties, the rst being their Markov property, i.e. that at every instant t they start again, and their evolution does not depend on their history, but only on their distribution at time t. This gives rise to a semi-group and this property will also play a central role for the study of diiusions on braided spaces. The other property, i.e. that they have continuous sample paths, does not have a direct counterpart on braided spaces. We replace it by the condition that diiusions can be obtained as the limit of a (simple) random walk. The motivations for our approach come from two directions. First, there is Majid's random walk approach to Brownian motion on the braided line and on anyspaces 2, 3]. We extend his deenition to (pseudo-) diiusions on multi-dimensional braided spaces and, using coalgebraic limit theorems due to Schhrmann 4], give their explicit form. This allows to consider semigroups of functionals and Markovian transition operators, as well as the associated heat equations, and to introduce Appell systems as their polynomial solutions. Heat kernels, i.e. the densities of the functionals are also considered. The second ingredient, M. Schhrmann's theory of quantum LLvy processes, comes into play to assure the existence of the associated processes, e.g. as operators on a Hilbert space. 1 We propose a deenition of *-structures for braided Hopf algebras (diiering from that due to S. Majid), and give several examples that satisfy our axioms. Then we generalize M. Schhrmann's deenitions and a result assuring the positivity of the convolution of positive functionals under certain conditions to braided groups. To distinguish the two …