{"title":"Temporal properties of the stochastic fractional heat equation with spatially-colored noise","authors":"Ran Wang, Yimin Xiao","doi":"10.1090/tpms/1209","DOIUrl":null,"url":null,"abstract":"<p>Consider the stochastic partial differential equation <disp-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartFraction partial-differential Over partial-differential t EndFraction u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis equals minus left-parenthesis negative normal upper Delta right-parenthesis Superscript StartFraction alpha Over 2 EndFraction Baseline u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis plus b left-parenthesis u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis right-parenthesis plus sigma left-parenthesis u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis right-parenthesis ModifyingAbove upper F With dot left-parenthesis t comma bold-italic x right-parenthesis comma t greater-than-or-equal-to 0 comma bold-italic x element-of double-struck upper R Superscript d Baseline comma\">\n <mml:semantics>\n <mml:mrow>\n <mml:mfrac>\n <mml:mi mathvariant=\"normal\">∂</mml:mi>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">∂</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:mrow>\n </mml:mfrac>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mi mathvariant=\"normal\">Δ</mml:mi>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mfrac>\n <mml:mi>α</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n </mml:mrow>\n </mml:msup>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi>b</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>+</mml:mo>\n <mml:mi>σ</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>F</mml:mi>\n <mml:mo>˙</mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mspace width=\"1em\"/>\n <mml:mi>t</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mspace width=\"mediummathspace\"/>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:msup>\n <mml:mo>,</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\begin{equation*} \\frac {\\partial }{\\partial t}u_t(\\boldsymbol {x})= -(-\\Delta )^{\\frac {\\alpha }{2}}u_t(\\boldsymbol {x}) +b\\left (u_t(\\boldsymbol {x})\\right )+\\sigma \\left (u_t(\\boldsymbol {x})\\right ) \\dot F(t, \\boldsymbol {x}), \\quad t\\ge 0,\\: \\boldsymbol {x}\\in \\mathbb R^d, \\end{equation*}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</disp-formula>\n where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus left-parenthesis negative normal upper Delta right-parenthesis Superscript StartFraction alpha Over 2 EndFraction\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>−</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:mi mathvariant=\"normal\">Δ</mml:mi>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mfrac>\n <mml:mi>α</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n </mml:mrow>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">-(-\\Delta )^{\\frac {\\alpha }{2}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denotes the fractional Laplacian with power <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartFraction alpha Over 2 EndFraction element-of left-parenthesis one half comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mfrac>\n <mml:mi>α</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mo>∈</mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mfrac>\n <mml:mn>1</mml:mn>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\frac {\\alpha }{2}\\in (\\frac 12,1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and the driving noise <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper F With dot\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mi>F</mml:mi>\n <mml:mo>˙</mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\dot F</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript t plus epsilon Baseline left-parenthesis bold-italic x right-parenthesis minus u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>t</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>ε</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>−</mml:mo>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">u_{t+{\\varepsilon }}(\\boldsymbol {x})-u_t(\\boldsymbol {x})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> at any fixed <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>t</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">t > 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold-italic x element-of double-struck upper R Superscript d\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\boldsymbol {x}\\in \\mathbb R^d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon down-arrow 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>ε</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">↓</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{\\varepsilon }\\downarrow 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\">\n <mml:semantics>\n <mml:mi>q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-variations of the temporal process <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis right-brace Subscript t greater-than-or-equal-to 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:msub>\n <mml:mi>u</mml:mi>\n <mml:mi>t</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:msub>\n ","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1209","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Consider the stochastic partial differential equation ∂∂tut(x)=−(−Δ)α2ut(x)+b(ut(x))+σ(ut(x))F˙(t,x),t≥0,x∈Rd,\begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*}
where −(−Δ)α2-(-\Delta )^{\frac {\alpha }{2}} denotes the fractional Laplacian with power α2∈(12,1]\frac {\alpha }{2}\in (\frac 12,1], and the driving noise F˙\dot F is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient ut+ε(x)−ut(x)u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x}) at any fixed t>0t > 0 and x∈Rd\boldsymbol {x}\in \mathbb R^d, as ε↓0{\varepsilon }\downarrow 0. As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the qq-variations of the temporal process {ut(x)
考虑随机偏微分方程 ∂ t u t ( x ) = - ( - Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x∈ R d , \begin{equation*}\frac {\partial }{partial t}u_t(\boldsymbol {x})= -(-\Delta )^{frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right ) +\sigma \left (u_t(\boldsymbol {x})\right )\dot F(t, \boldsymbol {x}), \quad t\ge 0,\:\boldsymbol {x}in \mathbb R^d, \end{equation*} 其中 - ( - Δ ) α 2 -(-\Delta )^{frac{alpha}{2}}表示分数拉普拉斯幂 α 2∈ ( 1 2 , 1 ]。 \frac {alpha }{2}\in (\frac 12,1] , 和驱动噪声 F ˙frac {alpha }{2}\in (\frac 12,1]. 驱动噪声 F ˙ \dot F 是一个居中的高斯场,在时间上是白色的,在空间上具有由 Riesz 核给出的同质协方差。我们研究在任意固定的 t > 0 t > 0 且 x∈ R d \boldsymbol {x}\in \mathbb R^d 时,时间梯度 u t + ε ( x ) - u t ( x ) u_{t+{\varepsilon }}(\boldsymbol{x})-u_t(\boldsymbol{x})近似的详细行为,当 ε ↓ 0 {\varepsilon }\downarrow 0 时。作为应用,我们推导出 Khintchin 的迭代对数定律、Chung 的迭代对数定律,以及关于时间过程 { u t ( x ) 的 q q - 变量的结果。