Journal article 1296 views 193 downloads
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise
Eugene Lytvynov 
,
Irina Rodionova
,
Irina Rodionova
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 21, Issue: 02, Start page: 1850011
Swansea University Authors:
Eugene Lytvynov , Irina Rodionova
-
PDF | Accepted Manuscript
Download (346.26KB)
DOI (Published version): 10.1142/S021902571850011X
Abstract
Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-spa...
| Published in: | Infinite Dimensional Analysis, Quantum Probability and Related Topics |
|---|---|
| ISSN: | 0219-0257 1793-6306 |
| Published: |
2018
|
| Online Access: |
Check full text
|
| URI: | https://cronfa.swan.ac.uk/Record/cronfa36409 |
| first_indexed |
2017-10-30T20:08:52Z |
|---|---|
| last_indexed |
2020-07-28T12:54:52Z |
| id |
cronfa36409 |
| recordtype |
SURis |
| fullrecord |
<?xml version="1.0"?><rfc1807><datestamp>2020-07-28T11:47:14.2838420</datestamp><bib-version>v2</bib-version><id>36409</id><entry>2017-10-30</entry><title>Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise</title><swanseaauthors><author><sid>e5b4fef159d90a480b1961cef89a17b7</sid><ORCID>0000-0001-9685-7727</ORCID><firstname>Eugene</firstname><surname>Lytvynov</surname><name>Eugene Lytvynov</name><active>true</active><ethesisStudent>false</ethesisStudent></author><author><sid>dbec195692a77f629e935ca8f4efa502</sid><firstname>Irina</firstname><surname>Rodionova</surname><name>Irina Rodionova</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2017-10-30</date><deptcode>MACS</deptcode><abstract>Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$.</abstract><type>Journal Article</type><journal>Infinite Dimensional Analysis, Quantum Probability and Related Topics</journal><volume>21</volume><journalNumber>02</journalNumber><paginationStart>1850011</paginationStart><publisher/><issnPrint>0219-0257</issnPrint><issnElectronic>1793-6306</issnElectronic><keywords>Monotone independence, monotone Levy noise, monotone Levy process, Meixner class of orthogonal polynomials.</keywords><publishedDay>18</publishedDay><publishedMonth>6</publishedMonth><publishedYear>2018</publishedYear><publishedDate>2018-06-18</publishedDate><doi>10.1142/S021902571850011X</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics and Computer Science School</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>MACS</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-07-28T11:47:14.2838420</lastEdited><Created>2017-10-30T14:42:27.1149467</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Eugene</firstname><surname>Lytvynov</surname><orcid>0000-0001-9685-7727</orcid><order>1</order></author><author><firstname>Irina</firstname><surname>Rodionova</surname><order>2</order></author></authors><documents><document><filename>0036409-30102017144510.pdf</filename><originalFilename>Monotone_Meixner.pdf</originalFilename><uploaded>2017-10-30T14:45:10.7770000</uploaded><type>Output</type><contentLength>307591</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2019-06-18T00:00:00.0000000</embargoDate><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807> |
| spelling |
2020-07-28T11:47:14.2838420 v2 36409 2017-10-30 Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false dbec195692a77f629e935ca8f4efa502 Irina Rodionova Irina Rodionova true false 2017-10-30 MACS Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$. Journal Article Infinite Dimensional Analysis, Quantum Probability and Related Topics 21 02 1850011 0219-0257 1793-6306 Monotone independence, monotone Levy noise, monotone Levy process, Meixner class of orthogonal polynomials. 18 6 2018 2018-06-18 10.1142/S021902571850011X COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2020-07-28T11:47:14.2838420 2017-10-30T14:42:27.1149467 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Eugene Lytvynov 0000-0001-9685-7727 1 Irina Rodionova 2 0036409-30102017144510.pdf Monotone_Meixner.pdf 2017-10-30T14:45:10.7770000 Output 307591 application/pdf Accepted Manuscript true 2019-06-18T00:00:00.0000000 true eng |
| title |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| spellingShingle |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise Eugene Lytvynov Irina Rodionova |
| title_short |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| title_full |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| title_fullStr |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| title_full_unstemmed |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| title_sort |
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise |
| author_id_str_mv |
e5b4fef159d90a480b1961cef89a17b7 dbec195692a77f629e935ca8f4efa502 |
| author_id_fullname_str_mv |
e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov dbec195692a77f629e935ca8f4efa502_***_Irina Rodionova |
| author |
Eugene Lytvynov Irina Rodionova |
| author2 |
Eugene Lytvynov Irina Rodionova |
| format |
Journal article |
| container_title |
Infinite Dimensional Analysis, Quantum Probability and Related Topics |
| container_volume |
21 |
| container_issue |
02 |
| container_start_page |
1850011 |
| publishDate |
2018 |
| institution |
Swansea University |
| issn |
0219-0257 1793-6306 |
| doi_str_mv |
10.1142/S021902571850011X |
| college_str |
Faculty of Science and Engineering |
| hierarchytype |
|
| hierarchy_top_id |
facultyofscienceandengineering |
| hierarchy_top_title |
Faculty of Science and Engineering |
| hierarchy_parent_id |
facultyofscienceandengineering |
| hierarchy_parent_title |
Faculty of Science and Engineering |
| department_str |
School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics |
| document_store_str |
1 |
| active_str |
0 |
| description |
Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$. |
| published_date |
2018-06-18T04:16:21Z |
| _version_ |
1821286943956140032 |
| score |
11.390808 |