<p>In this paper, we investigate the propagation of chaos for solutions to the Liouville equation derived from the Linear-Formation particle model. By imposing certain conditions, we derive the rate of convergence between the <i>k</i>-tensor product <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_522_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(f_{t}^{\otimes k}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>f</mi> <mrow> <mi>t</mi> </mrow> <mrow> <mo>⊗</mo> <mi>k</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> of the solution to be Linear-Formation kinetic equation and the <i>k</i>-marginal <i>f</i><Stack> <sub><i>N,k</i></sub> <sup><i>t</i></sup> </Stack> of the solution to the Liouville equation corresponding to the Linear-Formation particle model. Specifically, the following estimate holds in terms of <i>p</i>-Wasserstein (1 ⩽ <i>p</i> &lt; ∞) distance <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_522_Article_Equ1.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="406" /> </MediaObject> <EquationSource Format="TEX">\(W^p_p(f_{t}^{\otimes k},f_{N,k}^{t}) \leqslant C_{1} {{k}\over {N^{\min(p/2,1)}}}\left(1+t^{p}\right){\rm e}^{C_{2}t}, \quad 1\leqslant k\leqslant N.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>W</mi> <mi>p</mi> <mi>p</mi> </msubsup> <mo stretchy="false">(</mo> <msubsup> <mi>f</mi> <mrow> <mi>t</mi> </mrow> <mrow> <mo>⊗</mo> <mi>k</mi> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>f</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>k</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>⩽</mo> <msub> <mi>C</mi> <mrow> <mn>1</mn> </mrow> </msub> <mrow> <mfrac> <mrow> <mi>k</mi> </mrow> <mrow> <msup> <mi>N</mi> <mrow> <mo form="prefix" movablelimits="true">min</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mrow> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow> <mi>p</mi> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mn>1</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mi>N</mi> <mo>.</mo> </math></EquationSource> </Equation></p>

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Mean field limit and propagation of chaos for Linear-Formation model

  • Juntao Wu,
  • Xiao Wang,
  • Yicheng Liu

摘要

In this paper, we investigate the propagation of chaos for solutions to the Liouville equation derived from the Linear-Formation particle model. By imposing certain conditions, we derive the rate of convergence between the k-tensor product \(f_{t}^{\otimes k}\) f t k of the solution to be Linear-Formation kinetic equation and the k-marginal f N,k t of the solution to the Liouville equation corresponding to the Linear-Formation particle model. Specifically, the following estimate holds in terms of p-Wasserstein (1 ⩽ p < ∞) distance \(W^p_p(f_{t}^{\otimes k},f_{N,k}^{t}) \leqslant C_{1} {{k}\over {N^{\min(p/2,1)}}}\left(1+t^{p}\right){\rm e}^{C_{2}t}, \quad 1\leqslant k\leqslant N.\) W p p ( f t k , f N , k t ) C 1 k N min ( p / 2 , 1 ) ( 1 + t p ) e C 2 t , 1 k N .