Stochastic equations with singular drift driven by fractional Brownian motion
We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $...
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| creator | Butkovsky, Oleg Lê, Khoa Mytnik, Leonid |
| description | We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$
where the drift $b$ is either a measure or an integrable function, and $W^H$ is
a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,
$d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$
we show weak existence of solutions to this equation under the condition $$
\frac{d}p |
| doi_str_mv | 10.48550/arxiv.2302.11937 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_2302_11937</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2302_11937</sourcerecordid><originalsourceid>FETCH-LOGICAL-a677-e7e6f55145602b0c4f9ba6cd22ec605086a0037a82f6be4723d9a1185c23f5f83</originalsourceid><addsrcrecordid>eNotj8tOwzAURL1hgQofwAr_QMK1HT-ybKsWkIpY0H1049itpdQpjtvSv0cpbGak0dFIh5AnBmVlpIQXTD_hXHIBvGSsFvqefHzlwe5xzMFS933CHIY40kvIezqGuDv1mGiXgs9Tnl2k7ZX6hHbisKeLNFxiwEgPw7Q8kDuP_ege_3tGtuvVdvlWbD5f35fzTYFK68Jpp7yUrJIKeAu28nWLynacO6tAglEIIDQa7lXrKs1FVyNjRlouvPRGzMjz3-3NpzmmcMB0bSav5uYlfgHiUEkO</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Stochastic equations with singular drift driven by fractional Brownian motion</title><source>arXiv.org</source><creator>Butkovsky, Oleg ; Lê, Khoa ; Mytnik, Leonid</creator><creatorcontrib>Butkovsky, Oleg ; Lê, Khoa ; Mytnik, Leonid</creatorcontrib><description>We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$
where the drift $b$ is either a measure or an integrable function, and $W^H$ is
a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,
$d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$
we show weak existence of solutions to this equation under the condition $$
\frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-R\"ockner condition
(2005) to the fractional case. We construct a counter-example showing
optimality of this condition. If $b$ is a Radon measure, particularly the delta
measure, we prove weak existence of solutions to this equation under the
optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of
solutions to this equation under certain conditions. To establish these
results, we utilize the stochastic sewing technique and develop a new version
of the stochastic sewing lemma.</description><identifier>DOI: 10.48550/arxiv.2302.11937</identifier><language>eng</language><subject>Mathematics - Probability</subject><creationdate>2023-02</creationdate><rights>http://creativecommons.org/licenses/by/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2302.11937$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2302.11937$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Butkovsky, Oleg</creatorcontrib><creatorcontrib>Lê, Khoa</creatorcontrib><creatorcontrib>Mytnik, Leonid</creatorcontrib><title>Stochastic equations with singular drift driven by fractional Brownian motion</title><description>We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$
where the drift $b$ is either a measure or an integrable function, and $W^H$ is
a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,
$d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$
we show weak existence of solutions to this equation under the condition $$
\frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-R\"ockner condition
(2005) to the fractional case. We construct a counter-example showing
optimality of this condition. If $b$ is a Radon measure, particularly the delta
measure, we prove weak existence of solutions to this equation under the
optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of
solutions to this equation under certain conditions. To establish these
results, we utilize the stochastic sewing technique and develop a new version
of the stochastic sewing lemma.</description><subject>Mathematics - Probability</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotj8tOwzAURL1hgQofwAr_QMK1HT-ybKsWkIpY0H1049itpdQpjtvSv0cpbGak0dFIh5AnBmVlpIQXTD_hXHIBvGSsFvqefHzlwe5xzMFS933CHIY40kvIezqGuDv1mGiXgs9Tnl2k7ZX6hHbisKeLNFxiwEgPw7Q8kDuP_ege_3tGtuvVdvlWbD5f35fzTYFK68Jpp7yUrJIKeAu28nWLynacO6tAglEIIDQa7lXrKs1FVyNjRlouvPRGzMjz3-3NpzmmcMB0bSav5uYlfgHiUEkO</recordid><startdate>20230223</startdate><enddate>20230223</enddate><creator>Butkovsky, Oleg</creator><creator>Lê, Khoa</creator><creator>Mytnik, Leonid</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230223</creationdate><title>Stochastic equations with singular drift driven by fractional Brownian motion</title><author>Butkovsky, Oleg ; Lê, Khoa ; Mytnik, Leonid</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a677-e7e6f55145602b0c4f9ba6cd22ec605086a0037a82f6be4723d9a1185c23f5f83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Probability</topic><toplevel>online_resources</toplevel><creatorcontrib>Butkovsky, Oleg</creatorcontrib><creatorcontrib>Lê, Khoa</creatorcontrib><creatorcontrib>Mytnik, Leonid</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Butkovsky, Oleg</au><au>Lê, Khoa</au><au>Mytnik, Leonid</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Stochastic equations with singular drift driven by fractional Brownian motion</atitle><date>2023-02-23</date><risdate>2023</risdate><abstract>We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$
where the drift $b$ is either a measure or an integrable function, and $W^H$ is
a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,
$d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$
we show weak existence of solutions to this equation under the condition $$
\frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-R\"ockner condition
(2005) to the fractional case. We construct a counter-example showing
optimality of this condition. If $b$ is a Radon measure, particularly the delta
measure, we prove weak existence of solutions to this equation under the
optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of
solutions to this equation under certain conditions. To establish these
results, we utilize the stochastic sewing technique and develop a new version
of the stochastic sewing lemma.</abstract><doi>10.48550/arxiv.2302.11937</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Probability |
| title | Stochastic equations with singular drift driven by fractional Brownian motion |
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