On the Cameron–Martin theorem and almost-sure global existence

Abstract

Abstract. In this note, we discuss various aspects of invariant measures for nonlinear Hamiltonian PDEs. In particular, we show almost sure global existence for some Hamiltonian PDEs with initial data of the form: "a smooth deterministic function + a rough random perturbation", as a corollary to Cameron-Martin Theorem and known almost sure global existence results with respect to Gaussian measures on spaces of functions. Main results1.1. Introduction. In this note, we discuss almost sure global existence results for some nonlinear Hamiltonian partial differential equations (PDEs) as corollaries to Cameron-Martin Theorem [16]. In particular, we show almost sure global existence with initial data of the form u 0 (x; ω) = v 0 (x) + φ(x; ω),(1.1)where v 0 is a deterministic smooth function and φ(ω) is a random function of low regularity. On T, the function φ is given as where {g n } n∈Z is a sequence of independent standard complex-valued Gaussian random variables on a probability space (Ω, F, P ). For both (1.2) and (1.3), we easily see that φ lies almost surely in H 1 We drop the factor of 2π throughout the paper, when it plays no important role. 2) with α = 0 corresponds to the mean-zero Gaussian white noise on T:Let us describe one of the motivations for studying the Cauchy problems with initial data of the form (1.1), namely a smooth deterministic function + a rough random perturbation. (1.5) Given smooth physical data in an ideal situation, we may introduce rough and random perturbations to these data due to the limitations of accuracy in physical observations and storage of such data. Hence, we believe that it is important to study Cauchy problems with initial data of the form (1.5). Initial data (1.1) with (1.2) or (1.3) are the simplest models for (1.5) with rough Gaussian perturbations. One typical random noise we introduce in this kind of situation is the white noise, which appears ubiquitously in the physics literature. The white noise, however, is very rough and we can handle a smooth initial condition perturbed by the white noise only in a limited case.1.2. Invariant Gibbs measures for Hamiltonian PDEs. Given a Hamiltonian flow on R 2n :with Hamiltonian H(p, q) = H(p 1 , · · · , p n , q 1 , · · · , q n ), Liouville's theorem states that the Lebesgue measure n j=1 dp j dq j on R 2n is invariant under the flow. Then, it follows from the conservation of the Hamiltonian H that the Gibbs measures e −βH(p,q) n j=1 dp j dq j are invariant under the dynamics of (1.6), where β > 0 is the reciprocal temperature.In the context of the nonlinear Schrödinger equations (NLS) on T:with the Hamiltonian: |φx| 2 dx dφ.(1.9)2 Throughout the paper, Z denotes various normalizing constants. ON CAMERON-MARTIN THEOREM AND ALMOST SURE GLOBAL EXISTENCE 3Here, dφ denotes the non-existent Lebesgue measure on the infinite dimensional phase space of functions on T, and thus the expression (1.9) is merely formal at this point.Noting that eφx| 2 dx dφ is the Wiener measure on T with variance β −1 , Lebowitz-Rose-Speer showed that such...

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