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Lectures on Random Interfaces pp 111–124Cite as

KPZ Equation

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Abstract

The KPZ equation is a kind of stochastic PDE which describes the motion of a growing interface with random fluctuation, but known to be ill-posed and to require a renormalization. To give a meaning, we introduce two types of its approximations with diverging renormalization factors. One is simple, in which we replace the space-time white noise by its smeared noise and subtract a proper diverging constant. Then, one can see that the limit is the so-called Cole-Hopf solution of the KPZ equation. Another approximation is suitable for studying the invariant measures. Under the Cole-Hopf transformation, this approximating equation can be rewritten as a linear heat equation with an additional nonlinear term. We show by applying the Boltzmann-Gibbs principle that this nonlinear term can be asymptotically replaced by a linear term and find out that the solution of the approximating KPZ equation of second type converges to the Cole-Hopf solution with the term \(\frac{t} {24}\) added, at least in the stationary situation. This result is generalized to multi-component coupled KPZ equation, which plays an important role in the theory of fluctuating hydrodynamics, due to the paracontrolled calculus.

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Funaki, T. (2016). KPZ Equation. In: Lectures on Random Interfaces. SpringerBriefs in Probability and Mathematical Statistics. Springer, Singapore. https://doi.org/10.1007/978-981-10-0849-8_5

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