A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial segment. The relevance of this notion is that if $M$ is an inner model containing all the reals and satisfying $\mathsf{AD}_{\mathbb R}$, then $\Gamma=\mathcal P(\mathbb R)^M$ is a Wadge initial segment and $L(\Gamma,\mathbb R)\models\mathsf{AD}_{\mathbb R}$.

Under appropriate large cardinal assumptions, there is a Wadge initial segment $\Gamma=\Gamma_{min}$ such that $L(\Gamma,\mathbb R)\models\mathsf{AD^+}+\mathsf{AD_{\mathbb R}}+\Gamma=\mathcal P(\mathbb R)$. Moreover, given any inner model $M$ containing all the reals and satisfying $\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$, we have $\Gamma_{min}\subset M$. The mention of $\mathsf{AD}^+$ may well be superfluous here (or, perhaps, we should redefine $\mathsf{AD}_{\mathbb R}$ as $\mathsf{AD^+}+\mathsf{AD_{\mathbb R}}$); the situation does not seem entirely understood otherwise.

Surely $\Gamma_{min}$ admits a purely descriptive set-theoretic description as well (in terms of the complexity of the iteration strategies of the hybrid or hod mice that it captures), but I do not know how to specify it.

I suspect all of this is written up in reasonable detail nowadays. I suggest to read first

MR3362806 Reviewed.
Sargsyan, Grigor.
**Hod mice and the mouse set conjecture** (English summary),
Mem. Amer. Math. Soc. 236 (2015), no. 1111, viii+172 pp. ISBN: 978-1-4704-1692-8

(with all the technical details of the underlying theory) and

MR3087400 Reviewed.
Sargsyan, Grigor(1-RTG).
*Descriptive inner model theory* (English summary),
Bull. Symbolic Logic 19 (2013), no. 1, 1–55

(for a more leisurely introduction).

The result can of course be generalized to other strengthenings of $\mathsf{AD}^+$, but you will eventually run into difficulties, as it is possible that there are incompatible (or ``divergent'') $\mathsf{AD}^+$ models, that is, it is consistent to have sets of reals $A,B$ such that $L(A,\mathbb R)$ and $L(B,\mathbb R)$ are both models of $\mathsf{AD}^+$, but $A$ and $B$ are Wadge-incomparable so $A\notin L(B,\mathbb R)$, $B\notin L(A,\mathbb R)$, and $L(A,B,\mathbb R)$ is not a model of $\mathsf{AD}^+$. In such a setting, it may well be that there is no minimal pointclass $\Gamma$ playing the role of $\Gamma_{min}$ for your strengthening of determinacy. What saves us for $\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$ is that it is a theorem of Woodin that if $A,B$ are as above, and $\Gamma=\mathcal P(\mathbb R)^{L(A,\mathbb R)}\cap\mathcal P(\mathbb R)^{L(B,\mathbb R)}$, then ($\Gamma$ is again a Wadge initial segment, and) $L(\Gamma,\mathbb R)\models\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$.

umby the fact that the intersection of divergent models of AD containing all reals satisfies $\mathsf{AD}_\mathbb{R}$. $\endgroup$ – Trevor Wilson May 7 '17 at 22:57