A method is presented for adaptive mesh refinement applied to
stochastic simulations of spatially evolving reaction-diffusion
processes. The propensities for the diffusion process are derived for
meshes that are adaptive, locally refined, and structured.  Convergence of the diffusion process is presented and the fluctuations of the stochastic process are verified.  Furthermore, a refinement criterion is proposed for the evolution of the adaptive mesh. The method is applied to pattern formation problems.  Additionally, a three-dimensional inhomogeneous, stochastic reaction-diffusion equation is used as a model for the simulation of the growth of a glioma in a human brain.  The inhomogeneity of the white and grey matter of the brain is taken into account by a discontinuous diffusion coefficient.  A multi-resolution wavelet-based framework is used to solve the three-dimensional Langevin
equation.  Simulations are shown for the spread of the tumor over the duration of years.  Lastly, a numerical method is proposed for the
simulation of stochastic reaction-diffusion processes subject to
heavy-tailed diffusion.  The method is based on operator splitting of
the diffusion and reaction terms in the master equation. The diffusion
term follows a multinomial distribution governed by a kernel that is the discretized solution of the fractional diffusion equation. Simulations of a nationwide influenza epidemic are shown.