{"title":"具有无限均值的超布朗运动的绝对连续性","authors":"R. Mamin, L. Mytnik","doi":"10.1214/21-bjps508","DOIUrl":null,"url":null,"abstract":"In this work we prove that for any dimension $d\\geq 1$ and any $\\gamma \\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \\begin{equation*} \\begin{split} \\frac{\\partial v(t,x)}{\\partial t } & = \\frac{1}{2}\\bigtriangleup v(t,x) + v^\\gamma (t,x) ,\\: (t,x) \\in \\mathbb{R}_+\\times \\mathbb{R}^d,\\\\ v(0,x)&= f(x) \\end{split} \\end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \\LL\\ equation. \nWe also prove that when initial datum $v(0,\\cdot)$ is a finite, non-zero measure, then the \\LL\\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Absolute continuity of the super-Brownian motion with infinite mean\",\"authors\":\"R. Mamin, L. Mytnik\",\"doi\":\"10.1214/21-bjps508\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this work we prove that for any dimension $d\\\\geq 1$ and any $\\\\gamma \\\\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \\\\begin{equation*} \\\\begin{split} \\\\frac{\\\\partial v(t,x)}{\\\\partial t } & = \\\\frac{1}{2}\\\\bigtriangleup v(t,x) + v^\\\\gamma (t,x) ,\\\\: (t,x) \\\\in \\\\mathbb{R}_+\\\\times \\\\mathbb{R}^d,\\\\\\\\ v(0,x)&= f(x) \\\\end{split} \\\\end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \\\\LL\\\\ equation. \\nWe also prove that when initial datum $v(0,\\\\cdot)$ is a finite, non-zero measure, then the \\\\LL\\\\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.\",\"PeriodicalId\":51242,\"journal\":{\"name\":\"Brazilian Journal of Probability and Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-12-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Brazilian Journal of Probability and Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1214/21-bjps508\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/21-bjps508","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Absolute continuity of the super-Brownian motion with infinite mean
In this work we prove that for any dimension $d\geq 1$ and any $\gamma \in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \begin{equation*} \begin{split} \frac{\partial v(t,x)}{\partial t } & = \frac{1}{2}\bigtriangleup v(t,x) + v^\gamma (t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \LL\ equation.
We also prove that when initial datum $v(0,\cdot)$ is a finite, non-zero measure, then the \LL\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.
期刊介绍:
The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes.
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