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From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions

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  • Rang, Guanglin

Abstract

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1+1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [0,1]×R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Itô integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

Suggested Citation

  • Rang, Guanglin, 2020. "From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3408-3444.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3408-3444
    DOI: 10.1016/j.spa.2019.09.018
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    1. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    2. Song, Jian, 2017. "On a class of stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 37-79.
    3. Alberts, Tom & Clark, Jeremy & Kocić, Saša, 2017. "The intermediate disorder regime for a directed polymer model on a hierarchical lattice," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3291-3330.
    4. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
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