  In this work, we present and analyze a system of PDEs, which models tumor
growth by considering chemotaxis, active transport, and random effects. The
stochasticity of the system is modelled by random initial data and Wiener
noises that appear in the tumor and nutrient equations. The volume fraction of
the tumor is governed by a stochastic phase-field equation of Cahn-Hilliard
type, and the mass density of the nutrients is modelled by a stochastic
reaction-diffusion equation. We allow a variable mobility function and
non-increasing growth functions, such as logistic and Gompertzian growth. Via
approximation and stochastic compactness arguments, we prove the existence of a
probabilistic weak solution and, in the case of constant mobilities, the
well-posedness of the model in the strong probabilistic sense. Lastly, we
propose a numerical approximation based on the Galerkin finite element method
in space and the semi-implicit Euler-Maruyama scheme in time. We illustrate the
effects of the stochastic forcing in the tumor growth in several numerical
simulations.
