In this paper, we build a procedure that allows to establish regularity and controls in time for probabilistic solutions to PDEs. Probabilistic approaches to global wellposedness problems usually provide ensemble bounds on the solutions. These bounds are the main tools to ensure convergence procedures yielding the existence and uniqueness of global solutions. A question of interest consists in transforming such ensemble bounds into individual controls on the flow ; this, among other uses, gives valuable information on the long-time behavior of the solutions. Toward such question of bounds transformation, Bourgain initiated a successful procedure that exploited the local wellposedness of the PDE, with an estimate of the time of size-doubling. In this note, we construct an estimation procedure which relies on a different local requirement. It turns out that this substitute is flexible enough to be possible to fulfill with the help of the ensemble bound itself. For applications of the procedure, we are able to provide new pathwise controls on solutions to NLS equations.
In the analysis of differential equations, a key tool consists in changing one form of an inequality into another more suitable one. Such kind of trade gave outstanding usefulness to the famous Gronwall inequality and related estimates.
In a context where the differential equation is subject to randomness, the time variable is accompanied by an ensemble structure that, in principle, generates ensemble bounds on the solutions. The main question of this work concerns the change of these ensemble bounds into individual (pathwise) ones : bounds on the time evolution of the realizations.
Individual bounds on solutions can be instrumental in proving fine properties of the evolution, ranging from globalization arguments to long-time behavior properties. For instance, the so-called weak turbulence hypothesis in dispersive PDEs posed on bounded domains can be analyzed through the lens of the growth of Sobolev norms. For problem of the growth of norms, exponential bounds can be obtained from an iteration of a uniform local increment property. More interesting are the polynomial bounds obtained via bilinear estimations [9], the high-low method [2] or the I-method [3]. Closer to our subject of study, logarithmic bounds have been obtained in probabilistic settings, precisely through an individualization of ensemble bounds.
In the Gibbs measures theory for PDEs, the need of such change of the nature of bounds has been a barrier to establish global existence of solutions. Bourgain [1] introduced a clever exploitation of local increment property of the flow to successfully transform Gaussian type ensemble bounds into logarithmic controls on trajectories (see also the construction of [11]).
Several works have been able to establish unique global solutions to PDEs via probabilistic strategies which do not rely on local wellposedness; however, in the best of our knoweledge, none of these established individual controls on the time-evolution of the solutions (see e.g. the well-known fluctuation-dissipation method [8,6,7,4,5]).
The aim of the present work is to build a procedure employing a different local condition, making it flexible enough to apply to various situations not necessarily built upon the framework of [1]. Here are two main features of the procedure:
• Any ensemble control on an increasing function of the norm is admissible for obtaining individual bounds ; • The local increment requirement can in practice be established by exploiting the ensemble bound itself. This implies important flexibility, as probabilistic estimates, which are the decisive ingredients of the procedure, find a new utilization.
As for applications of the estimation procedure, we will consider in Section 3 the following two situations :
(1) Context of an inaccessible local theory -cubic NLS after Kuksin-Shirikyan [6]. Using probabilistic and compactness arguments, the authors of [6] constructed unique global strong solutions to
satisfying an ensemble bound
Our task will be to change this bound with a control on the trajectories.
(2) Context of a weak ensemble bound [10]. It was build a probabilistic global wellposedness for
The solutions satisfy the ensemble bounds :
where ξ -1 : R + → R + is a convex function. In [10], the estimate (1.5) has been exploited within the Bourgain procedure to produce individual bounds. However, due to the relatively weak control of (1.4), no individual H s bound was established. In section 3, we will perform such bound transformation using the procedure that will be deployed. We obtain the following new estimates : Theorem 1.1. We have that :
• For the 3D case, solutions of the cubic nonlinear Schrödinger equation obtained in [6] satisfy the individual bound :
• Solutions of the energy supercritical nonlinear Schrödinger equation obtained in [10] satisfy the individual bound :
Below, we present some tools, of independent interest, that form the basis of the procedure presented in Section 3 giving the proof of Theorem 1.1. Let (Ω, F, P) be a complete probability space, and let (X, ∥ • ∥ X ) be a Banach space. We have Theorem 1.2. Let u : Ω × R + → X be a stochastic process valued in the Banach space X. Assume that
(1) for P-a. a. ω, u ω has the local increment
with A ω < ∞ and ϕ : R + → R + an increasing function ;
(2) u satisfies the following ensemble bound
for increasing functions ψ, λ : R + → R + . Then, for P-almost all ω, there is
In the particular case of stationary processes, we have the following.
Corollary 1.3. Let u : Ω × R + → X be a stationary stochastic process valued in the Banach space X. Assume that
(1) for P-a. a. ω, u ω has the local increment
with A ω < ∞ and ϕ : R + → R + an increasing function ;
(2) u satisfies the following ensemble bound Eψ(∥u(0)∥ X ) < +∞, (1.12) for an increasing function ψ : R + → R + . Then, for P-almost all ω, there is C ω < +∞ such that
From the increment condition (1.8), we could use an integration argument to find the bound
This bound wou
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