Heat content for convolution semigroups

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• 作者：Wojciech Cygan^a ; ¹ ; ^{wojciech.cygan@uwr.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Tomasz Grzywny^b ; ² ; ^{tomasz.grzywny@pwr.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Asymptotic behaviour ; Characteristic exponent ; Heat content ; Isotropic Lé ; vy process ; Perimeter ; Regular variation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1393-1414
• 全文大小：432 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305546&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305546&_rdoc=1&_issn=0022247X&md5=66341877bc62dc165615c500909d4d0a" title="Click to view the MathML source">X={X_t}_t≥0class="mathContainer hidden">class="mathCode">$\mathbf{X}={\{X_t\}}_{t\ge 0}$ be a Lévy process in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305546&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305546&_rdoc=1&_issn=0022247X&md5=190c04315fd656df0f7c7c989d9df803" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ and Ω be an open subset of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305546&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305546&_rdoc=1&_issn=0022247X&md5=190c04315fd656df0f7c7c989d9df803" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ with finite Lebesgue measure. In this article we consider the quantity class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305546&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305546&_rdoc=1&_issn=0022247X&md5=6ce887b4edc3c3da8c493ce44b0aa2d5">class="imgLazyJSB inlineImage" height="19" width="185" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305546-si3.gif">class="mathContainer hidden">class="mathCode">$H(t)={\int }_{\mathrm{\Omega }}{\mathbb{P}}_x(X_t\in {\mathrm{\Omega }}^c)\mathrm{d}x$ related to class="boldFont">X which is called the heat content. We study its asymptotic behaviour as t goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.