Hitting Distribution of a Correlated Planar Brownian Motion in a Disk
In this article, we study the hitting probability of a circumference <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>R</mi></msub></semantics></math>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/4/536 |
| _version_ | 1797478359634018304 |
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| author | Manfred Marvin Marchione Enzo Orsingher |
| author_facet | Manfred Marvin Marchione Enzo Orsingher |
| author_sort | Manfred Marvin Marchione |
| collection | DOAJ |
| description | In this article, we study the hitting probability of a circumference <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>R</mi></msub></semantics></math></inline-formula> for a correlated Brownian motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mi>B</mi><mo>̲</mo></munder><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>B</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> being the correlation coefficient. The analysis starts by first mapping the circle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>R</mi></msub></semantics></math></inline-formula> into an ellipse <i>E</i> with semiaxes depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels. |
| first_indexed | 2024-03-09T21:31:45Z |
| format | Article |
| id | doaj.art-b72134d5052c4af89bde1e2986ff2966 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:31:45Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b72134d5052c4af89bde1e2986ff29662023-11-23T20:56:15ZengMDPI AGMathematics2227-73902022-02-0110453610.3390/math10040536Hitting Distribution of a Correlated Planar Brownian Motion in a DiskManfred Marvin Marchione0Enzo Orsingher1Department of Statistical Sciences, Sapienza University of Rome, 00185 Rome, ItalyDepartment of Statistical Sciences, Sapienza University of Rome, 00185 Rome, ItalyIn this article, we study the hitting probability of a circumference <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>R</mi></msub></semantics></math></inline-formula> for a correlated Brownian motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mi>B</mi><mo>̲</mo></munder><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>B</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> being the correlation coefficient. The analysis starts by first mapping the circle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>R</mi></msub></semantics></math></inline-formula> into an ellipse <i>E</i> with semiaxes depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.https://www.mdpi.com/2227-7390/10/4/536elliptic coordinatesPoisson kernelGhizzetti transformation |
| spellingShingle | Manfred Marvin Marchione Enzo Orsingher Hitting Distribution of a Correlated Planar Brownian Motion in a Disk Mathematics elliptic coordinates Poisson kernel Ghizzetti transformation |
| title | Hitting Distribution of a Correlated Planar Brownian Motion in a Disk |
| title_full | Hitting Distribution of a Correlated Planar Brownian Motion in a Disk |
| title_fullStr | Hitting Distribution of a Correlated Planar Brownian Motion in a Disk |
| title_full_unstemmed | Hitting Distribution of a Correlated Planar Brownian Motion in a Disk |
| title_short | Hitting Distribution of a Correlated Planar Brownian Motion in a Disk |
| title_sort | hitting distribution of a correlated planar brownian motion in a disk |
| topic | elliptic coordinates Poisson kernel Ghizzetti transformation |
| url | https://www.mdpi.com/2227-7390/10/4/536 |
| work_keys_str_mv | AT manfredmarvinmarchione hittingdistributionofacorrelatedplanarbrownianmotioninadisk AT enzoorsingher hittingdistributionofacorrelatedplanarbrownianmotioninadisk |