TY - JOUR

T1 - Scaling limits for time-fractional diffusion-wave systems with random initial data

AU - Liu, Gi Ren

AU - Shieh, Narn Rueih

N1 - Funding Information:
The authors are grateful to the inspiring lectures of Professor W. A. Woyczyński at National Taiwan University for the perspective on Mathematical Theory of Fractional P.D.E. We also thank the valuable comments and suggestions of the referee; they make the paper more precise and readable. N. R. S. was partially supported by a Taiwan NSC grant 962115M002005MY3.

PY - 2010/3

Y1 - 2010/3

N2 - Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

AB - Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

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U2 - 10.1142/S0219493710002826

DO - 10.1142/S0219493710002826

M3 - Article

AN - SCOPUS:77951166931

SN - 0219-4937

VL - 10

SP - 1

EP - 35

JO - Stochastics and Dynamics

JF - Stochastics and Dynamics

IS - 1

ER -