The question is as in the title, but let me explain a bit.


Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm curious if there is a similarly natural inner model for AD$\_\mathbb{R}$.


Now, ZF + $V=L(\mathcal{P}(\mathbb{R}))$ + large cardinals proves AD$\_\mathbb{R}$ if I recall correctly; however, this is somewhat misleading as $L(\mathcal{P}(\mathbb{R}))$ is never a model of AD, let alone AD$\_\mathbb{R}$, since in it the reals are well-ordered.


So I'm curious: assuming large cardinals, is there a reasonably canonical inner model of AD$\_\mathbb{R}$ which is "easy to describe"?



Is there a natural inner model of AD$_\mathbb{R}$?
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. 




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



> 
> [MR3362806](http://www.ams.org/mathscinet-getitem?mr=3362806) Reviewed. 
>  Sargsyan, Grigor.
>  **[Hod mice and the mouse set conjecture](http://www.ams.org/books/memo/1111/)** (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](http://www.ams.org/mathscinet-getitem?mr=3087400) Reviewed. 
>  Sargsyan, Grigor(1-RTG).
>  *[Descriptive inner model theory](https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/descriptive-inner-model-theory/06641B957E4E983C4BE9DD59987658DF)* (English summary), 
>  Bull. Symbolic Logic 19 (2013), no. 1, 1–55
> 
> 
> 


(for a more leisurely introduction). 




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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) $L(\Gamma,\mathbb R)\models\mathsf{AD}^++\mathsf{AD}\_{\mathbb R}$.


