### Abstract

Let X = {X(x, t), x ∈ ℝ^{n}, t ∈ R_{+}} be the R^{2}-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X_{0} = (u_{0}, v_{0}). We discuss the scaling limit of X under suitable conditions on X_{0}. Since the component fields u, v are dependent, even when the initial data u_{0}, v_{0} are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u_{0}, v_{0}) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2] for the single heat equation can be discussed for the two-equation system, in a significant way.

Original language | English |
---|---|

Pages (from-to) | 505-522 |

Number of pages | 18 |

Journal | Stochastic Analysis and Applications |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 May 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Stochastic Analysis and Applications*,

*28*(3), 505-522. https://doi.org/10.1080/07362991003704969

}

*Stochastic Analysis and Applications*, vol. 28, no. 3, pp. 505-522. https://doi.org/10.1080/07362991003704969

**Scaling limits for some P.D.E. systems with random initial conditions.** / Liu, Gi-Ren; Shieh, Narn Rueih.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Scaling limits for some P.D.E. systems with random initial conditions

AU - Liu, Gi-Ren

AU - Shieh, Narn Rueih

PY - 2010/5/1

Y1 - 2010/5/1

N2 - Let X = {X(x, t), x ∈ ℝn, t ∈ R+} be the R2-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X0 = (u0, v0). We discuss the scaling limit of X under suitable conditions on X0. Since the component fields u, v are dependent, even when the initial data u0, v0 are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u0, v0) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2] for the single heat equation can be discussed for the two-equation system, in a significant way.

AB - Let X = {X(x, t), x ∈ ℝn, t ∈ R+} be the R2-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X0 = (u0, v0). We discuss the scaling limit of X under suitable conditions on X0. Since the component fields u, v are dependent, even when the initial data u0, v0 are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u0, v0) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2] for the single heat equation can be discussed for the two-equation system, in a significant way.

UR - http://www.scopus.com/inward/record.url?scp=77951174784&partnerID=8YFLogxK

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U2 - 10.1080/07362991003704969

DO - 10.1080/07362991003704969

M3 - Article

VL - 28

SP - 505

EP - 522

JO - Stochastic Analysis and Applications

JF - Stochastic Analysis and Applications

SN - 0736-2994

IS - 3

ER -