<p>We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \Subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⋐</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with general—strictly positive and temporally constant—Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an interval in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, we prove that such a solution is a gradient flow—curve of maximal slope—within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${{\overline{\Omega }}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi mathvariant="normal">Ω</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an interval in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, we find a formula for the descending slope of this geodesically nonconvex functional.</p>

Variational structures for the Fokker–Planck equation with general Dirichlet boundary conditions

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2025年12月6日

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Filippo Quattrocchi

摘要

We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in $\Omega \Subset \mathbb {R}^d$ $Ω ⋐ R^{d}$ with general—strictly positive and temporally constant—Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where $\Omega $ $Ω$ is an interval in $\mathbb {R}^1$ $R^{1}$ , we prove that such a solution is a gradient flow—curve of maximal slope—within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\partial \Omega $ $\partial Ω$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure ${{\overline{\Omega }}}$ $\bar{Ω}$ . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $\Omega $ $Ω$ is an interval in $\mathbb {R}^1$ $R^{1}$ , we find a formula for the descending slope of this geodesically nonconvex functional.