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Is there a universally recognized term for ODEs considered in the sense of distributions used to describe impulsive/discontinuous processes? I noticed that some authors call such ODEs "measure differential equations" while others use the term "differential equations in distributions". But I don't see a major difference between them. Can anyone please make the point clear? Thank you.

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I thought those were jump processes (as described by their stochastic DE)/jump diffusions? Not all stochastic processes are discontinuous, obviously. –  gnometorule Feb 4 '13 at 19:33

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As long as the distributions involved in the equation are (signed) measures, there is no difference and both terms can be used interchangeably. This is the case for impulsive source equations like $y''+y=\delta_{t_0}$.

Conceivably, ODE could also involve distributions that are not measures, such as the derivative of $\delta_{t_0}$. In that case only "differentiable equation in distributions" would be correct. But I can't think of a natural example of such an ODE at this time.

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Thank you, 5PM! Your answer is really helpful. In-depth discussion about difference between measures and distributions can be found at mathoverflow.net/questions/17732/… –  Ivan Matan Feb 6 '13 at 17:17

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