We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we consider Hamilton-Jacobi equations forced by a noise term with small amplitude, and, in increasing the strength of the noise, we observe a sharp transition at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions of stochastically forced Hamilton-Jacobi equations, which are of independent interest.

We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend nontrivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the pathwise equations for which one has been proved. equations driven by W , namely, the characteristic equations corresponding to the first-order part of (1.1), are required to have a stable pathwise theory. More details about this point are given in Sections 2 and 3.The notion of pathwise viscosity solutions for equations like (1.1) was developed by Lions and Souganidis, first for Hamiltonians depending smoothly on the gradient Du [12], and later for nonsmooth Hamiltonians [13]. The comparison principle was proved in [14], and in [15], equations with semilinear noise dependence were considered, that is, Hamiltonians depending linearly on Du and nonlinearly on u. The theory has since been extended to treat Hamiltonians with spatial dependence, as by Friz, Gassiat, Lions, and Souganidis [5], or by the author [19]; these papers use techniques developed by Lions and Souganidis that appear in forthcoming works. Many more details and results are summarized in the notes of Souganidis [21]. The case in which H depends linearly on the gradient has been studied extensively from the point of view of rough path theory by many authors, including, but not limited to, Caruana, Friz, and Oberhauser [3] and Gubinelli, Tindel, and Torrecilla [8]. The problem was also examined by Buckdahn and Ma [1,2] using the pathwise control interpretation.An important feature of the pathwise viscosity theory is the stability of (1.1) with respect to the path in a suitable topology. This leads naturally to a notion of weak solutions. More precisely, it has been shown in various situations that, for suitable C 1 -families {W η } η>0 that satisfy lim η→0 W η = W in an appropriate sense, if u η is the classical viscosity solution of the initial value problem

We would like to thank Professors Serguei Denissov and Alexander Kiselev for their constant support throughout the research and writing processes. We are grateful to them for suggesting the topic and providing us with the opportunity to apply the concepts we have learned throughout our mathematical careers. We thank them both for their excellent advice and encouragement.Abstract. For θ ∈ [0, 2π), consider the rotation matrix R θ and h = λ 0 0 0 , λ > 1.Let Wn(θ) denote the product of m R θ 's and n h's with the condition m ≤ [ǫn] (0 < ǫ < 1). We analyze the measure of the set of θ for which Wn(θ) ≥ λ δn (0 < δ < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.

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