Motivated by the development of a probabilistic model for growth of biological shapes in the context of large deformations by diffeomorphisms, we present a stochastic perturbation of the Hamiltonian equations of geodesics on shape spaces. We study the finite-dimensional case of groups of points for which we prove that the strong solutions of the stochastic system exist for all time. We extend the model to the space of parameterized curves and surfaces and we develop a convenient analytical setting to prove a strong convergence result from the finite-dimensional to the infinite-dimensional case. We then present some enhancements of the model.