This article presents an extension of Iannis Xenakis's Dynamic Stochastic Synthesis (DSS) called Diffusion Dynamic Stochastic Synthesis (DDSS). This extension solves a diffusion equation whose solutions can be used to map particle positions to amplitude values of several breakpoints in a waveform, following traditional concepts of DSS by directly shaping the waveform of a sound. One significant difference between DSS and DDSS is that the latter includes a drift in the Brownian trajectories that each breakpoint experiences through time. Diffusion Dynamic Stochastic Synthesis can also be used in other ways, such as to control the amplitude values of an oscillator bank using additive synthesis, shaping in this case the spectrum, not the waveform. This second modality goes against Xenakis's original desire to depart from classical Fourier synthesis. The results of spectral analyses of the DDSS waveform approach, implemented using the software environment Max, are discussed and compared with the results of a simplified version of DSS to which, despite the similarity in the overall form of the frequency spectrum, noticeable differences are found. In addition to the Max implementation of the basic DDSS algorithm, a MIDI-controlled synthesizer is also presented here. With DDSS we introduce a real physical process, in this case diffusion, into traditional stochastic synthesis. This sort of sonification can suggest models of sound synthesis that are more complex and grounded in physical concepts.