Estimates of the transition densities for the reflected Brownian motion on simple nested fractals

We give sharp two-sided estimates for the functions gMt, x, y and gMt, x, y − gt, x, y, where gMt, x, y are the transition probability densities of the reflected Brownian motion on an Mcomplex of order M ∈ Z of an unbounded planar simple nested fractal and gt, x, y are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.

REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

FREE BROWNIAN MOTION AND EVOLUTION TOWARDS ⊞-INFINITE DIVISIBILITY FOR k-TUPLES

Let [Formula: see text] be the space of non-commutative distributions of k-tuples of self-adjoint elements in a C*-probability space. For every t ≥ 0 we consider the transformation [Formula: see text] defined by [Formula: see text] where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on [Formula: see text]. We prove that 𝔹s ◦ 𝔹t = 𝔹s + t, for all s, t ≥ 0. For t = 1, we prove that [Formula: see text] is precisely the set [Formula: see text] of distributions in [Formula: see text] which are infinitely divisible with respect to ⊞, and that the map [Formula: see text] coincides with the multi-variable Boolean Bercovici–Pata bijection put into evidence in our previous paper [1]. Thus for a fixed [Formula: see text], the process {𝔹t(μ)|t ≥ 0} can be viewed as some kind of "evolution towards ⊞-infinite divisibility". On the other hand, we put into evidence a relation between the transformations ⊞t and free Brownian motion. More precisely, we introduce a map [Formula: see text] which transforms the free Brownian motion started at an arbitrary [Formula: see text] into the process {𝔹t(μ)|t ≥ 0} for μ = Φ(ν).

DIFFUSION PROCESSES WITH RESPECT TO FREE BROWNIAN MOTION

We prove the existence, uniqueness and Markov property for SDE of diffusion type in the context of the stochastic analysis on the free Fock space introduced in Ref. 1.