Clarification and Complement to “Mean-Field Description and Propagation of Chaos in Networks of …

In this section, we check that the particle systems and the mean-field limit systems are well posed, and that the components of the processes ( y t i ) <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i200.gif” alt=”View MathML”/> , ( x t i ) <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i201.gif” alt=”View MathML”/> , ( y t α ) <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i202.gif” alt=”View MathML”/> , ( x t α ) <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i203.gif” alt=”View MathML”/> take values in [ 0 , 1 ] <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i204.gif” alt=”View MathML”/> . Then we prove that the particle systems propagate chaos toward the law of the limit systems (5) and (9). Our situation differs from the above mentioned Scheutzow’s counterexamples [4] in the fact that the interaction kernel is globally Lipschitz and the functions F α <img class=”mathimg” src=”http://www.mathematical-neuroscience.com/content/inline/s13408-015-0031-8-i205.gif” alt=”View MathML”/>…


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