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FROM NONLINEAR FOKKER–PLANCK EQUATIONS TO SOLUTIONS OF DISTRIBUTION DEPENDENT SDE
We construct weak solutions to the McKean–Vlasov SDE d X ( t ) = b ( X ( t ) , d L X ( t ) d x ( X ( t ) ) ) d t + σ ( X ( t ) , d L X ( t ) d t ( X ( t ) ) ) d W ( t ) on R d for possibly degenerate diffusion matrices σ with X(0) having a given law, which has a density with respect to Lebesgue meas...
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| Published in: | The Annals of probability 2020-07, Vol.48 (4), p.1902-1920 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We construct weak solutions to the McKean–Vlasov SDE
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X
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=
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on R
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for possibly degenerate diffusion matrices σ with X(0) having a given law, which has a density with respect to Lebesgue measure, dx. Here, L
X(t)
denotes the law of X(t). Our approach is to first solve the corresponding non-linear Fokker–Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE. |
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| ISSN: | 0091-1798 2168-894X |
| DOI: | 10.1214/19-AOP1410 |