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Regularization by noise: a McKean-Vlasov case | | Paul-Eric Chaudru de Raynal
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26 Dec 2015 | | Abstract: | In this paper, we prove pathwise uniqueness for stochastic systems of
McKean-Vlasov type with singular drift, even in the measure argument, and
uniformly non-degenerate Lipschitz diffusion matrix. This work extends to the
McKean-Vlasov setting the earlier results obtained by Zvonkin [Zvo74],
Veretennikov [Ver80], Krylov and R"ockner [KR05]. We prove that the noise
prevents the ill-posedness coming from the singularity of the drift, even in
the measure direction. Our proof relies on regularization properties of the
associated PDE, which is stated on the space $[0, T ] imes mathbb{R}^d imes
mathcal{P}_2(mathbb{R}^d)$, where T is a positive number, d denotes the
dimension of the equation and P2(R d) is the space of probability measures on R
d with finite second order moment. In particular, a smoothing effect in the
measure direction is exhibited. Our approach is based on a parametrix expansion
of the transition density of the McKean-Vlasov process. | | Source: | arXiv, 1512.8096 | | Services: | Forum | Review | PDF | Favorites | |
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