TY - JOUR
T1 - Scaling limits for some P.D.E. systems with random initial conditions
AU - Liu, Gi Ren
AU - Shieh, Narn Rueih
N1 - Funding Information:
Received February 10, 2009; Accepted October 22, 2009 Research for N.-R.S. partially supported by a Taiwan NSC grant 962115M002005MY3. We thank to the anonymous referee for valuable comments and suggestions. Address correspondence to Narn-Rueih Shieh, Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan; E-mail: [email protected]. edu.tw
PY - 2010/5
Y1 - 2010/5
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.
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U2 - 10.1080/07362991003704969
DO - 10.1080/07362991003704969
M3 - Article
AN - SCOPUS:77951174784
SN - 0736-2994
VL - 28
SP - 505
EP - 522
JO - Stochastic Analysis and Applications
JF - Stochastic Analysis and Applications
IS - 3
ER -