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These lecture notes are woven around the subject of Burgers' turbulence/KPZ model of interface growth, a study of the nonlinear parabolic equation with random initial data. The analysis is conducted mostly in the space-time domain, with less attention paid to the frequency-domain picture. However, the bibliography contains a more complete information about other directions in the field which over the last decade enjoyed a vigorous expansion. The notes are addressed to a diverse audience, including mathematicians, statisticians, physicists, fluid dynamicists and engineers, and contain both rigorous and heuristic arguments. Because of the multidisciplinary audience, the notes also include a concise exposition of some classical topics in probability theory, such as Brownian motion, Wiener polynomial chaos, etc.

Partial differential equations --- Interfaces (Physical sciences) --- Turbulence --- Burgers equation --- Differential equations, Parabolic --- Mathematical models --- Mathematical Theory --- Atomic Physics --- Physics --- Mathematics --- Physical Sciences & Mathematics --- Differentiaalvergelijkingen [Parabolische ] --- Differential equations [Parabolic] --- Diffusion equation [Nonlinear ] --- Equations differentielles paraboliques --- Heat flow equation [Nonlinear ] --- Interface (Physical sciences) --- Partial differential equations. --- Probabilities. --- Partial Differential Equations. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Burgers equation. --- Mathematical models. --- Diffusion equation, Nonlinear --- Heat flow equation, Nonlinear --- Nonlinear diffusion equation --- Nonlinear heat flow equation --- Heat equation --- Navier-Stokes equations --- Interfaces (Physical sciences) - Mathematical models --- Turbulence - Mathematical models