In this paper, we study Feynman-Kac semigroups of symmetric <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5201_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5201_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$b|x|^{-\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>b</mi> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>β</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5201_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$b>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5201_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta >\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>></mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>. We obtain two-sided estimates on the densities p(t, x, y) of these semigroups for all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2024_5201_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, along with estimates for the corresponding Green functions.

Fractional Laplacian with Supercritical Killings

期刊论文
发表日期：

2025年1月6日

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Soobin Cho,
Renming Song

摘要

In this paper, we study Feynman-Kac semigroups of symmetric $\alpha $ $α$ -stable processes with supercritical killing potentials belonging to a large class of functions containing functions of the form $b|x|^{-\beta }$ ${b | x |}^{- β}$ , where $b>0$ $b > 0$ and $\beta >\alpha $ $β > α$ . We obtain two-sided estimates on the densities p(t, x, y) of these semigroups for all $t>0$ $t > 0$ , along with estimates for the corresponding Green functions.