In this article, we study the weak coupling limit of the following equation in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>: <Equation ID="Equ17"> <EquationSource Format="TEX">$\begin{aligned} dX_t^\varepsilon =\frac{\hat{\lambda }}{\sqrt{\log \frac{1}{\varepsilon }}}\omega ^\varepsilon (X_t^\varepsilon )dt+\nu dB_t,\quad X_0^\varepsilon =0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <msubsup> <mi>X</mi> <mi>t</mi> <mi>ε</mi> </msubsup> <mo>=</mo> <mfrac> <mover accent="true"> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> <msqrt> <mrow> <mo>log</mo> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mrow> </msqrt> </mfrac> <msup> <mi>ω</mi> <mi>ε</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mi>t</mi> <mi>ε</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>ν</mi> <mi>d</mi> <msub> <mi>B</mi> <mi>t</mi> </msub> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>X</mi> <mn>0</mn> <mi>ε</mi> </msubsup> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Here <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\omega ^\varepsilon =\nabla ^{\perp }\rho _\varepsilon *\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ω</mi> <mi>ε</mi> </msup> <mo>=</mo> <msup> <mi mathvariant="normal">∇</mi> <mo>⊥</mo> </msup> <msub> <mi>ρ</mi> <mi>ε</mi> </msub> <mrow /> <mo>∗</mo> <mi>ξ</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> representing the 2d Gaussian Free Field (GFF) and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\rho _\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> denoting an appropriate identity. <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> denotes a two-dimensional standard Brownian motion, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\hat{\lambda },\;\nu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> <mo>,</mo> <mspace width="0.277778em" /> <mi>ν</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are two given constants. We use the approach from Cannizzaro et al. (Duke Math. J. 172(16), 3013–3104 <CitationRef CitationID="CR5">2023</CitationRef>) to show that the second moment of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X_t^\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>X</mi> <mi>t</mi> <mi>ε</mi> </msubsup> </math></EquationSource> </InlineEquation> under the annealed law converges to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(c(\nu ,\hat{\lambda })^2+2\nu ^2)t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mover accent="true"> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>ν</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> with a precisely determined constant <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$c(\nu ,\hat{\lambda })>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mover accent="true"> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, which implies a non-trivial limit of the drift terms as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\left( \sqrt{\frac{c(\nu ,\hat{\lambda })^2}{2}+\nu ^2}\right) \widetilde{B}_t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <msqrt> <mrow> <mfrac> <mrow> <mi>c</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mover accent="true"> <mi>λ</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <msup> <mi>ν</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfenced> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">~</mo> </mover> <mi>t</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> vanishes, where <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\widetilde{B}_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">~</mo> </mover> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> is a two-dimensional standard Brownian motion.

Weak Coupling Limit of A Brownian Particle in the Curl of the 2D GFF

期刊论文
发表日期：

2026年3月23日

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Huanyu Yang,
Zhilin Yang

摘要

In this article, we study the weak coupling limit of the following equation in $\mathbb {R}^2$ $R^{2}$ : $\begin{aligned} dX_t^\varepsilon =\frac{\hat{\lambda }}{\sqrt{\log \frac{1}{\varepsilon }}}\omega ^\varepsilon (X_t^\varepsilon )dt+\nu dB_t,\quad X_0^\varepsilon =0. \end{aligned}$ $\begin{matrix} d X_{t}^{ε} = \frac{\hat{λ}}{\sqrt{log \frac{1}{ε}}} ω^{ε} (X_{t}^{ε}) d t + ν d B_{t}, X_{0}^{ε} = 0 . \end{matrix}$ Here $\omega ^\varepsilon =\nabla ^{\perp }\rho _\varepsilon *\xi $ $ω^{ε} = \nabla^{⊥} ρ_{ε} * ξ$ with $\xi $ $ξ$ representing the 2d Gaussian Free Field (GFF) and $\rho _\varepsilon $ $ρ_{ε}$ denoting an appropriate identity. $B_t$ $B_{t}$ denotes a two-dimensional standard Brownian motion, and $\hat{\lambda },\;\nu >0$ $\hat{λ}, ν > 0$ are two given constants. We use the approach from Cannizzaro et al. (Duke Math. J. 172(16), 3013–3104 2023) to show that the second moment of $X_t^\varepsilon $ $X_{t}^{ε}$ under the annealed law converges to $(c(\nu ,\hat{\lambda })^2+2\nu ^2)t$ $(c {(ν, \hat{λ})}^{2} + 2 ν^{2}) t$ with a precisely determined constant $c(\nu ,\hat{\lambda })>0$ $c (ν, \hat{λ}) > 0$ , which implies a non-trivial limit of the drift terms as $\varepsilon $ $ε$ vanishes. We also prove that in this weak coupling regime, the sequence of solutions converges in distribution to $\left( \sqrt{\frac{c(\nu ,\hat{\lambda })^2}{2}+\nu ^2}\right) \widetilde{B}_t$ $(\sqrt{\frac{c {(ν, \hat{λ})}^{2}}{2} + ν^{2}}) {\tilde{B}}_{t}$ as $\varepsilon $ $ε$ vanishes, where $\widetilde{B}_t$ ${\tilde{B}}_{t}$ is a two-dimensional standard Brownian motion.