Birth-and-growth and nucleation and growth will be used as synonyms in this paper.
i] the spatial location of the j-th nucleus of a site-saturated nucleation process, and by [[THETA].
Then, a space-time nucleation process can be modeled by a marked point process, identifying [T.
In other words, if N is a site-saturated nucleation process, then [LAMBDA](A) represents the mean number of nuclei born in A [subset] [R.
under assumptions of homogeneous nucleation and growth), then [V.
Definition 1 (Causal cone) If N is a nucleation process on a suitable space K, then the causal cone C(t, x) of a point x at time t is the subset of K in which at least one nucleation event has to take place in order to cover the point x at time t.
Then a general expression for the case of nucleation on lower dimensional sets is obtained.
Since, in general, the nucleation is random in time and space, then the transformed region at any time t > 0 will be a random set (Stoyan et al.
As mentioned in the Introduction, we shall consider the case in which all the nucleation takes place at t = 0 (namely, site-saturated case), and the case in which the nucleation takes place in time (namely, time-dependent case).
Site-saturated nucleation processes and space-time dependent nucleation processes can be modeled by point processes and marked point processes, respectively.
We intentionally use the same notation N for the site-saturated nucleation process and for the time-dependent one because the site saturated process may be seen as a particular case of the time-dependent one by assuming [T.
In this paper we shall assume Poissonian nucleation, that is N will be a Poisson process (in [R.