Electronic Journal of Probability

Rescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equation

Yu-Ting Chen

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We study some linear SDEs arising from the two-dimensional $q$-Whittaker driven particle system on the torus as $q\to 1$. The main result proves that the SDEs along certain characteristics converge to the additive stochastic heat equation. Extensions for the SDEs with generalized coefficients and in other spatial dimensions are also obtained. Our proof views the limiting process after recentering as a process of the convolution of a space-time white noise and the Fourier transform of the heat kernel. Accordingly we turn to similar space-time stochastic integrals defined by the SDEs, but now the convolution and the Fourier transform are broken. To obtain tightness of these induced integrals, we bound the oscillations of complex exponentials arising from divergence of the characteristics, with two methods of different nature.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 36, 33 pp.

Received: 23 April 2018
Accepted: 2 March 2019
First available in Project Euclid: 9 April 2019

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G15: Gaussian processes 60H15: Stochastic partial differential equations [See also 35R60]

stochastic heat equations stochastic convolutions surface growth models

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Chen, Yu-Ting. Rescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equation. Electron. J. Probab. 24 (2019), paper no. 36, 33 pp. doi:10.1214/19-EJP289. https://projecteuclid.org/euclid.ejp/1554775415

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