An analytical approach to the d-dimensional grain growth, which is a kind of the heterogeneous nucleation-and-growth phase transformation, is offered. The system is assumed to be driven by capillary forces. Another important operative assumption is that the system evolves under preservation of its hypervolume, which results in considering the process as a random walk in the space of grain sizes. A role of the initial condition imposed on the system behaviour, and how does the system behave upon a prescribed initial state, have been examined. A general conclusion appears, which states that this prescription does not affect the asymptotic system behavior, but may be of importance when inspecting the early-time domain more carefully, cf. the Weibull-type initial distribution. This study is addressed to some analogous theoretical descriptions concerning polycrystals as well as bubbles-containing systems. Some comparison to another modelling, in which a crucial role of local material gradients (fluxes) was emphasized, has been attache
