## An Introduction to Fronts in Random MediaThis book aims to give a user friendly tutorial of an interdisciplinary research topic (fronts or interfaces in random media) to senior undergraduates and beginning grad uate students with basic knowledge of partial differential equations (PDE) and prob ability. The approach taken is semiformal, using elementary methods to introduce ideas and motivate results as much as possible, then outlining how to pursue rigor ous theorems, with details to be found in the references section. Since the topic concerns both differential equations and probability, and proba bility is traditionally a quite technical subject with a heavy measure theoretic com ponent, the book strives to develop a simplistic approach so that students can grasp the essentials of fronts and random media and their applications in a self contained tutorial. The book introduces three fundamental PDEs (the Burgers equation, Hamilton– Jacobi equations, and reaction–diffusion equations), analysis of their formulas and front solutions, and related stochastic processes. It builds up tools gradually, so that students are brought to the frontiers of research at a steady pace. A moderate number of exercises are provided to consolidate the concepts and ideas. The main methods are representation formulas of solutions, Laplace meth ods, homogenization, ergodic theory, central limit theorems, large deviation princi ples, variational principles, maximum principles, and Harnack inequalities, among others. These methods are normally covered in separate books on either differential equations or probability. It is my hope that this tutorial will help to illustrate how to combine these tools in solving concrete problems. |

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### Contents

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160399_1_En_2_Chapter_OnlinePDFpdf | 22 |

160399_1_En_3_Chapter_OnlinePDFpdf | 53 |

160399_1_En_4_Chapter_OnlinePDFpdf | 68 |

160399_1_En_5_Chapter_OnlinePDFpdf | 93 |

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assumptions asymptotic bistable boundary conditions Burgers equation cell problem central limit theorem Chapter coefﬁcients coercivity convergence convex covariance decay deﬁned denotes deterministic diffusion distribution equal exists exponential Feynman-Kac formula ﬁeld ﬁnite ﬁrst ﬂame follows front proﬁle front solutions front speed Fronts in Random function G-equation Gaussian process Hamilton–Jacobi equation Hamiltonian HJ equation holds implies inequality initial data integral KPP front speed Lagrangian large-deviation Lax formula Legendre transform Lemma linear Lipschitz continuous lower bound Lyapunov exponent maximum principle mean zero method minimal speed Neumann boundary condition nonlinearity nonnegative periodic media perturbation principal eigenvalue probability proof quadratic random media random variables reaction–diffusion satisﬁes scalar scaling shear ﬂow stationary stochastic homogenization subadditive ergodic theorem surely tion traveling fronts traveling-front unbounded uniformly unique upper bound variational formula vector velocity viscous Wiener process