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<p>While the nondegenerate case is well known, there are only few results on the existence of strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker–Planck(–Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x_0\in {\mathbb {R}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. For these kind of equations the existence of a Schwartz-distributional solution <i>u</i> is well-known. We show that there exists a unique strong solution to the associated degenerate McKean–Vlasov SDE with time marginal law densities <i>u</i>. In particular, every weak solution to this equation with time marginal law densities <i>u</i> can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities <i>u</i>.</p>

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Strong solutions to McKean–Vlasov SDEs associated to a class of degenerate Fokker–Planck equations with coefficients of Nemytskii-type

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  • 发表日期：

    2025年10月9日

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  • Sebastian Grube

摘要

While the nondegenerate case is well known, there are only few results on the existence of strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker–Planck(–Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from \(L^1\) L 1 , and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point \(x_0\in {\mathbb {R}}\) x 0 ∈ R . For these kind of equations the existence of a Schwartz-distributional solution u is well-known. We show that there exists a unique strong solution to the associated degenerate McKean–Vlasov SDE with time marginal law densities u. In particular, every weak solution to this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities u.