chap9_The comparison principle.pdf

CHAPTER 9 The comparison principle At the beginning of Chapter 2, we outlined a four-step program for proving large deviation results. The technically most difficult step in the program is usu- ally the verification of the comparison principle. For h1, h2 ∈ B(E), we consider subsolutions of (9.1) (I ? αH?)f = h1 and supersolutions of (9.2) (I ? αH?)f = h2. For h1 = h2 = h, the comparison principle states that if f is a viscosity subsolution of (9.1) and f is a viscosity supersolution (9.2), then f ≤ f . Consequently, to verify the comparison principle, we must be able to bound f from above and f from below. The techniques in this chapter are mainly extensions of those in Crandall, Ishii and Lions [17]. However, these generalizations are non-trivial. The most significant improvement is the treatment for equations with infinite dimensional state space. The method here does not rely on Ekeland’s perturbed optimization principle or its variants. See Section 9.4. In the past, Ekeland’s principle has been a basic point of departure for Hamilton-Jacobi theory in infinite dimensions. See the available theory developed by Crandall and Lions [21], [20] and Tataru [116]. This work applies in a restricted class of Banach spaces for equations with sufficiently regular coefficients. These requirements exclude equations arising for large deviations of interacting particle systems, such as Example 1.14. The method in Section 9.4 allows us to treat a large class of problems in this type of situation. It is based on some techniques in Feng and Katsoulakis [43]. Throughout this chapter, we assume that H? ? M l(E,R)×Mu(E′, R), H? ? Mu(E,R)×M l(E′, R), and H = H?∩H?. It follows that H ? B(E)×B(E). Note that if the comparison principle holds for H, then it holds for the pair (H?,H?). We assume that (f, g) ∈ H? and c ∈ R imply that (f + c, g) ∈ H? and similarly for H?. Recall that by Remark 7.2, if E = E′ and E is compact, then Definition 7.1 is equivalent to Definition 6.1. 9.1. General