SPDEs with rough noise in space: Hölder continuity of the solution

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• 作者：Raluca M. Balan^a ; ^{rbalan@uottawa.ca" class="auth_mail" title="E-mail the corresponding author} ; Maria Jolis^b ; ^{mjolis@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; Lluí ; s Quer-Sardanyons^b ; ^{quer@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60H15 ; secondary ; 60H05
• 刊名：Statistics & Probability Letters
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：119
• 期：Complete
• 页码：310-316
• 全文大小：397 K

文摘

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301687&_mathId=si1.gif&_user=111111111&_pii=S0167715216301687&_rdoc=1&_issn=01677152&md5=7d68a4be96297834192c51d668fcc0b3">class="imgLazyJSB inlineImage" height="21" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301687-si1.gif">class="mathContainer hidden">class="mathCode">$H\in (\frac{1}{4},\frac{1}{2})$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301687&_mathId=si2.gif&_user=111111111&_pii=S0167715216301687&_rdoc=1&_issn=01677152&md5=3daf29107c75c83536f9e4a665bfdb53" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$H$, and Hölder continuous in time of order smaller than class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301687&_mathId=si3.gif&_user=111111111&_pii=S0167715216301687&_rdoc=1&_issn=01677152&md5=2077eb77b705fe1ce54c02fb0016617e" title="Click to view the MathML source">γclass="mathContainer hidden">class="mathCode">$\gamma$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301687&_mathId=si4.gif&_user=111111111&_pii=S0167715216301687&_rdoc=1&_issn=01677152&md5=2e55f59b2aa8a515acc68aac1cf24f90" title="Click to view the MathML source">γ=Hclass="mathContainer hidden">class="mathCode">$\gamma =H$ for the wave equation and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301687&_mathId=si5.gif&_user=111111111&_pii=S0167715216301687&_rdoc=1&_issn=01677152&md5=5a9eec30205bcba6927fc8c61cc5ed5e" title="Click to view the MathML source">γ=H/2class="mathContainer hidden">class="mathCode">$\gamma =H/2$ for the heat equation.