Quadratic covariations for the solution to a stochastic heat equation with space-time white noise
Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-05-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-020-02707-9 |
| _version_ | 1818508458059628544 |
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| author | Xichao Sun Litan Yan Xianye Yu |
| author_facet | Xichao Sun Litan Yan Xianye Yu |
| author_sort | Xichao Sun |
| collection | DOAJ |
| description | Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u ( t , x ) $u(t,x)$ . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced. |
| first_indexed | 2024-12-10T22:32:09Z |
| format | Article |
| id | doaj.art-32e0e04edc9e4e6a940902d7c9a95c47 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-10T22:32:09Z |
| publishDate | 2020-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-32e0e04edc9e4e6a940902d7c9a95c472022-12-22T01:31:01ZengSpringerOpenAdvances in Difference Equations1687-18472020-05-012020114210.1186/s13662-020-02707-9Quadratic covariations for the solution to a stochastic heat equation with space-time white noiseXichao Sun0Litan Yan1Xianye Yu2Department of Mathematics, College of Science, Bengbu UniversityDepartment of Statistics, College of Science, Donghua UniversitySchool of Statistics and Mathematics, Zhejiang Gongshang UniversityAbstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u ( t , x ) $u(t,x)$ . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.http://link.springer.com/article/10.1186/s13662-020-02707-9Fractional Brownian motionStochastic heat equationItô formulaQuadratic covariationLocal time |
| spellingShingle | Xichao Sun Litan Yan Xianye Yu Quadratic covariations for the solution to a stochastic heat equation with space-time white noise Advances in Difference Equations Fractional Brownian motion Stochastic heat equation Itô formula Quadratic covariation Local time |
| title | Quadratic covariations for the solution to a stochastic heat equation with space-time white noise |
| title_full | Quadratic covariations for the solution to a stochastic heat equation with space-time white noise |
| title_fullStr | Quadratic covariations for the solution to a stochastic heat equation with space-time white noise |
| title_full_unstemmed | Quadratic covariations for the solution to a stochastic heat equation with space-time white noise |
| title_short | Quadratic covariations for the solution to a stochastic heat equation with space-time white noise |
| title_sort | quadratic covariations for the solution to a stochastic heat equation with space time white noise |
| topic | Fractional Brownian motion Stochastic heat equation Itô formula Quadratic covariation Local time |
| url | http://link.springer.com/article/10.1186/s13662-020-02707-9 |
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