and to draw an experience $e$ following a normal law centered and concentrated on $e'$. Because of the shape of this probability law, individuals tend to explore the set $\mathcal{E}$ close to the elements they already explored (namely, innovating). The concept $c$ is drawn following a probability law centered and concentrated on $k_i^t(e)$.

The learning algorithm finally computes the knowledge-like function at the next time step using a least-square procedure [@CuckerSmale2001] $$k_i^{t+1} \in \operatorname*{arg\,min}_k \sum_{(e,c)\in S_i^t} (k(e)-c)^2.$$

The case of globally shared knowledge: convergence without individual learning {#sec:mathRes}
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In this section, we are interested in the convergence of the learning dynamics with high probability to a common shared conceptualization among individuals, i.e. when everybody carries the same knowledge-like function $k$. This result is obtained assuming no individual learning : in all this section, we shall assume that the rate of individual learning $\tau = 0$. The more realistic case where individuals also learn individually is explored below using numerical simulations.

#### One step idealistic processes.

In order to establish the main result, let us decompose the stochastic process $k^t$ into two other processes that we shall analyze separately.

Let us define the application $\mathcal{T} : \mathcal{F} \times \mathbb{N} \mapsto 