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shminux comments on Can you recognize a random generator? - Less Wrong
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Start with the obvious link, look for hyperbolic and elliptic PDEs and ways to solve them, especially numerically. The wave equation techniques are different from the Laplace equation techniques, though there is some overlap. Anyway, this is getting quite far from the original discussion.
Not quite. For example, the heat/diffusion equation cannot be run backwards, because it fails the uniqueness conditions with the t sign reversed. In a simpler language, you cannot reconstruct the shape of an ink drop once it is dissolved in a cup of water.
I was mainly asking what differences are, in your opinion, important in the context of the present debate.
1) You can run the diffusion equation backwards, only you encounter problems with precision when the solution exponentially grows.
2) Fundamental laws of nature are [second order in time and - edit:that's not true ] symmetric with respect to time reversal.
Well, we are quite a ways from the original context, but I was commenting on that treating time and space on the same footing and saying that future and past are basically interchangeable (sorry for paraphrasing) is often a bad assumption.
In other words, it is ill-posed and cannot be used to recover the initial conditions.
This is a whole other debate on what is fundamental and what is emergent. Clearly, the heat equation is pretty fundamental in many contexts, but its origins can be traced to the microscopic models of diffusion. There are other reasons why the apparent time reversal might not be there. For example, if you take the MWI seriously, the branching process is has a clear time arrow attached to it.
With precise measurement you can. Once started to be solved numerically different initial conditions (for the standard diffusion equation) are all going to yield constant function after some time due to rounding errors, so the information is lost and can't be recovered by time-reversed process. But as a mathematical problem the diffusion equation with reversed time is well defined and has unique solution nevertheless.
From what I recall, the reverse-time diffusion u_t=-u_xx is not well posed, i.e. for a given solution u(t), if we perturb u(0) by epsilon, there is no finite t such that the deviation of the new solution from the old one is bounded by epsilon*e^t. A quick Google search confirms it: (pdf, search inside for "well posed")
I didn't realise that "well-posed" is a term with a technical meaning. The definition of well-posedness I have found says that the solution must exist, be unique and continuously depend on the initial data, I am not sure whether this is equivalent to your definition.
Anyway, the problem with the reverse dissipation equation is that for some initial conditions, namely discontinuous ones, the solution doesn't exist. However, if a function u(x,t) satisfies the diffusion equation on the interval [t1,t2], we can recover it completely from knowledge of not only u(x,t1), but also from u(x,t0) with any fixed t0 lying between t1 and t2.
A small nitpick: The Schrodinger equation is not second order in time.
The Dirac one as well. Corrected.