Given a hyperbolic PDE, the *domain of influence* of a spacetime point $x$, say $I\_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them *geometric* ($I\_x^G$) and the other *analytical* ($I\_x^A$). In Lorentzian geometry, the geometric domain of influence consists of the interior of the cone of null geodesics emanating from $x$ (let me not bother about whether to include the boundary in the definition of $I\_x^G$ or not). In general, a similar definition can be given using characteristic cones instead of null cones. The analytical domain of influence can be defined as the set of all spacetime points $y$ such that for every neighborhood neighborhood $O$ of $x$ there exist two solutions $u\_1$ and $u\_2$ satisfying the condition $u\_1(x')=u\_2(x')$, for all $x'$ on a Cauchy surface passing through $x$ except for $x'\in O$, and also the condition $u\_1(y)\ne u\_2(y)$. The latter one is the definition used in Lax's book on Hyperbolic PDEs.


Similar defintions can be given for the geometric and analytical domain of dependence, say $D\_S^G$ and $D\_S^A$. Such a definition should capture the desired equality $D\_K = I\_{S\setminus K}$, for a Cauchy surface $S$ and $K\subset S$ (once again, being sloppy with boundaries). I know that energy methods can be used to establish that $D^G\_K \subseteq D^A\_K$ (the analytical domain of dependence is at least as large as the geometric one). Hence, by duality, the same methods establish $I\_K^A \subseteq I\_K^G$ (that the geometric domain of influence is at least as large as the analytical one).


My question is about the reverse inclusion, $I\_K^G \subseteq I\_K^A$ or by duality $D\_K^A \subseteq D\_K^G$. Maybe it's too much to ask for the analytical and geometric definitions to coincide. But when they do, what methods are used to establish that? When they don't what methods can identify the obstruction? Except briefly in Lax's book, I don't know what references discuss this problem explicitly, so those would also be appreciated!



Methods for determining domains of influence
I highly doubt the result you actually asked for is true. 


Consider the **linear wave equation** on $(1+3)$ Minkowski space. The *analytic domain of influence* of a point $x$ as Lax defined it, which morally says that $y$ is in the analytic domain only if one can find perturbations in arbitrary small neighborhoods of $x$ that change $y$ (if I interpret your question statement correctly), actually consists of only the null cone emanating from $x$ and nothing more, since strong Huygen's principle holds. 


The same is true for the linear wave equation on $(1+(2k+1))$ Minkowski spaces. 


The opposite conclusion can be drawn on $(1+2k)$ dimensional Minkowski spaces, where the Green's function have support inside the cone. 


For more general situations, you may want to consult the classical result of Atiyah-Bott-Garding on the existence of [Petrowsky lacunae](http://en.wikipedia.org/wiki/Petrovsky_lacuna). For any linear hyperbolic equation that admits a lacuna, the analytic domain of influence cannot cover the entirety of the geometric one. 


But for the result that you seem to actually want, where you should replace the analytic domain of dependence by a suitable "convex" envelope of it, I don't know if such a result is proven anywhere, but my guess is that, at least for the "local" version one can approach it using some sort of geometric optics construction. 


For possible references (I haven't actually finished reading either, so they may not contain what you want), maybe you want to look at Michael Beals' book on propagation of singularities (sorry, the title escapes me at the moment) or Rauch's notes on *Hyperbolic PDEs and Geometric Optics* which I think you can find floating around on the internet. 


