In the work Dealing with moment measures via entropy and optimal transport, Santambrogio provided an optimal transport approach to study existence of solutions to the moment measure equation, that is: given <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, find u such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ (\nabla u)_{\sharp }e^{-u}\mathcal =\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>♯</mo> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>u</mi> </mrow> </msup> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular he proves that u satisfies the previous equation if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$e^{-u}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>u</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is the minimizer of an entropy and a transport cost. Here we study a modified minimization problem, in which we add a strongly convex regularization depending on a positive <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and we link its solutions to a modified moment measure equation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\nabla u)_{\sharp }e^{-u-\frac{\alpha }{2} \Vert x\Vert ^2}= \mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>♯</mo> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>u</mi> <mo>-</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> </mrow> </msup> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>. Exploiting the regularization term, we study the stability of the minimizers.

Regularized Moment Measures

期刊论文
发表日期：

2026年2月4日

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Alex Delalande,
Sara Farinelli

摘要

In the work Dealing with moment measures via entropy and optimal transport, Santambrogio provided an optimal transport approach to study existence of solutions to the moment measure equation, that is: given $\mu $ $μ$ , find u such that $ (\nabla u)_{\sharp }e^{-u}\mathcal =\mu $ ${(\nabla u)}_{♯} e^{- u} = μ$ . In particular he proves that u satisfies the previous equation if and only if $e^{-u}$ $e^{- u}$ is the minimizer of an entropy and a transport cost. Here we study a modified minimization problem, in which we add a strongly convex regularization depending on a positive $\alpha $ $α$ and we link its solutions to a modified moment measure equation $(\nabla u)_{\sharp }e^{-u-\frac{\alpha }{2} \Vert x\Vert ^2}= \mu $ ${(\nabla u)}_{♯} e^{- u - \frac{α}{2} {‖ x ‖}^{2}} = μ$ . Exploiting the regularization term, we study the stability of the minimizers.