We investigate the zero-noise limit for SDE’s driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^1_{loc}((0,T];BV(\mathbb {T}^d;\mathbb {R}^d))\cap L^q((0,T);L^p(\mathbb {T}^d;\mathbb {R}^d))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <mi>B</mi> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where p and q satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [<CitationRef CitationID="CR12">12</CitationRef>] lies in this class and we show that for almost every initial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.

On the zero-noise limit for SDE’s singular at the initial time

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2025年10月10日

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Jules Pitcho

摘要

We investigate the zero-noise limit for SDE’s driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb {T}^d;\mathbb {R}^d))\cap L^q((0,T);L^p(\mathbb {T}^d;\mathbb {R}^d))$ $L_{loc}^{1} ((0, T]; B V (T^{d}; R^{d})) \cap L^{q} ((0, T); L^{p} (T^{d}; R^{d}))$ where p and q satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [12] lies in this class and we show that for almost every initial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.