Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Wellposedness of Second Order Master Equations for Mean Field Games with
  Nonsmooth Data   In this paper we study second order master equations arising from mean field
games with common noise over arbitrary time duration. A classical solution
typically requires the monotonicity condition (or small time duration) and
sufficiently smooth data. While keeping the monotonicity condition, our goal is
to relax the regularity of the data, which is an open problem in the
literature. In particular, we do not require any differentiability in terms of
the measures, which prevents us from obtaining classical solutions. We shall
propose three weaker notions of solutions, named as {\it good solutions}, {\it
weak solutions}, and {\it weak-viscosity solutions}, respectively, and
establish the wellposedness of the master equation under all three notions. We
emphasize that, due to the game nature, one cannot expect comparison principle
even for classical solutions. The key for the global (in time) wellposedness is
the uniform a priori estimate for the Lipschitz continuity of the solution in
the measures. The monotonicity condition is crucial for this uniform estimate
and thus is crucial for the existence of the global solution, but is not needed
for the uniqueness in such Lipschitz class. To facilitate our analysis, we
construct a smooth mollifier for functions on Wasserstein space, which is new
in the literature and is interesting in its own right.
  Following the same approach of our wellposedness results, we prove the
convergence of the Nash system, a high dimensional system of PDEs arising from
the corresponding $N$-player game, under mild regularity requirements. We shall
also prove a propagation of chaos property for the associated optimal
trajectories.

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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int