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Abstract

Dϕ(u(s))[g(u(s))]dβ(s) u(s)) [g(u(s), g(u(s)] ds .The most frequently used example for which we use Itô's formula is ϕ(u) = u p with Dϕ(u)[v] A.1 Burkholder-Davis-Gundy InequalityIn this subsection, we discuss maximal inequalities of Burkholder-Davis-Gundy type, in order to bound stochastic integrals of convolution type. We consider only bounds for terms like E sup t∈ [0,T ] t 0, where L is an operator given for instance by Assumption 2.1.For a nice review of recent results see [HS01]. We will present some modifications needed for our applications. Especially we have to improve the dependence of the constant on the time T for large times. First we need the following well known theorems (see e.g. [DPZ96] or [HS01])Theorem A.7 (Burkholder-Davis-Gundy) Let β be a Brownian motion, and f some stochastic process adapted to β. Then for all p > 0 there is a constant C > 0 depending only on p such thatA version, which is true for all martingales is the celebrated Doob inequality. A simple lemma, which can also be based on a rescaling argument, is the following. for all ε > 0.Proof. This is a direct consequence of Theorem A.7 and Hölder's inequality.Note that the right hand side in (A.1) is easily bounded byHere we need to improve the dependence of the constant on T > 0, as in our applications the timeis typically very large. Results of this type with a different T -dependence of the constant are well known even for bounded or contraction semigroups. See for example [HS01] or [Tub84]. Nevertheless, we establish here a much better control on the dependence of the constant on T and δ, which was not done before. Here we rely on exponential decay properties of the semigroup. For simplicity of presentation, we do not focus on optimal dependence, but nevertheless the lemma yields a very good result for large T and δ.Proof. We use the celebrated factorisation method of Da Prato and Zabzcyck (cf.[DPZ92]). Based on a stochastic Fubini theorem,for γ ∈ (0, 1) with some constant C γ depending only on γ. Moreover, where C denotes positive constants depending on p, but independent of T, δ, and f . Furthermore, using Lemma A.5Combining all results finishes the proof.The following useful lemma is a moment inequality that generalises Itô's isometry. It follows directly from Theorem A.7, but it is much simpler, and usually used in the proof of Theorem A.7. Lemma A.5 For all p > 0 there is a constant C such that for all Hilbert-space valued stochastic processes f adapted to the filtration of the Brownian motion βSee for example Section 1.7 of [Mao97] for a detailed discussion of moment inequalities and Burkholder-Davis-Gundy inequalities in an SDE setting. A.2 Comparison Argument for ODEsResults of this type are well known. See for example [Hal80]. Nevertheless, we state the result necessary for our applications in Lemma A.6 and give a simple proof. The application in Lemma A.7 will provide the key estimate to use nonlinear stability. In a simple ODE setting we show that after a fixed time, we can bound solutions completely independent o...