We exhibit various restrictions about the wellposedness of the Schrödinger product <Equation ID="Equ127"> <EquationSource Format="TEX">${\mathcal {L}}:z \longmapsto -\imath \int _0^t e^{\imath s { \partial ^2_x}}\big ( z_s\cdot \Psi _s\big ) ds $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> <mo>:</mo> <mi>z</mi> <mo>⟼</mo> <mo>-</mo> <mi>ı</mi> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msup> <mi>e</mi> <mrow> <mi>ı</mi> <mi>s</mi> <msubsup> <mi>∂</mi> <mi>x</mi> <mn>2</mn> </msubsup> </mrow> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msub> <mi>z</mi> <mi>s</mi> </msub> <mo>·</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mi>s</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <mi>s</mi> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation> refers to the so-called linear solution of the stochastic Schrödinger problem. We focus more specifically on the case where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation> satisfies <Equation ID="Equ1"> <EquationNumber>0.1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} (\imath \partial _t-\partial ^2_x)\Psi =\dot{B}, \quad \Psi _0=0,\quad \quad t\in {\mathbb {R}}, \ x\in \mathbb {T}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>ı</mi> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mo>-</mo> <msubsup> <mi>∂</mi> <mi>x</mi> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> <mo>,</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">Ψ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\dot{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> </math></EquationSource> </InlineEquation> is a white noise in space with fractional time covariance of index <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H>\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.As an consequence of our analysis, we obtain that if H is close to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation> (that is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\dot{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> </math></EquationSource> </InlineEquation> is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem <Equation ID="Equ128"> <EquationSource Format="TEX">$\begin{aligned} (\imath \partial _t-\partial ^2_x)u= |u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in {\mathbb {R}}, \ x\in \mathbb {T}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>ı</mi> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mo>-</mo> <msubsup> <mi>∂</mi> <mi>x</mi> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> <mo>,</mo> <mspace width="1em" /> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">T</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>using only a first-order expansion of the solution (“<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$u=\Psi +z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Ψ</mi> <mo>+</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>”).

On the 1d stochastic Schrödinger product

期刊论文
发表日期：

2025年8月19日

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Aurélien Deya

摘要

We exhibit various restrictions about the wellposedness of the Schrödinger product ${\mathcal {L}}:z \longmapsto -\imath \int _0^t e^{\imath s { \partial ^2_x}}\big ( z_s\cdot \Psi _s\big ) ds $ $L : z ⟼ - ı \int_{0}^{t} e^{ı s \partial_{x}^{2}} (z_{s} \cdot Ψ_{s}) d s$ where $\Psi $ $Ψ$ refers to the so-called linear solution of the stochastic Schrödinger problem. We focus more specifically on the case where $\Psi $ $Ψ$ satisfies 0.1 $\begin{aligned} (\imath \partial _t-\partial ^2_x)\Psi =\dot{B}, \quad \Psi _0=0,\quad \quad t\in {\mathbb {R}}, \ x\in \mathbb {T}, \end{aligned}$ $\begin{matrix} (ı \partial_{t} - \partial_{x}^{2}) Ψ = \dot{B}, Ψ_{0} = 0, t \in R, x \in T, \end{matrix}$ where $\dot{B}$ $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac{1}{2}$ $H > \frac{1}{2}$ .

As an consequence of our analysis, we obtain that if H is close to $\frac{1}{2}$ $\frac{1}{2}$ (that is $\dot{B}$ $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem $\begin{aligned} (\imath \partial _t-\partial ^2_x)u= |u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in {\mathbb {R}}, \ x\in \mathbb {T}, \end{aligned}$ $\begin{matrix} (ı \partial_{t} - \partial_{x}^{2}) u = {| u |}^{2} + \dot{B}, u_{0} = 0, t \in R, x \in T, \end{matrix}$ using only a first-order expansion of the solution (“ $u=\Psi +z$ $u = Ψ + z$ ”).