Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise
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Communications in
Mathematical Physics
Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise Leonardo Tolomeo Mathematical Institute, Hausdorff Center for Mathematics, Universität Bonn, Bonn, Germany. E-mail: [email protected] Received: 11 July 2019 / Accepted: 25 February 2020 Published online: 7 May 2020 – © The Author(s) 2020
Abstract: In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for d = 1 and the beam equation for d ≤ 3. We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible. 1. Introduction Consider the equation s
u tt + u t + u + (−) 2 u + u 3 =
√ 2ξ,
posed on the d - dimensional torus Td , where ξ is the space-time white noise on R × Td (defined in Sect. 2), and s > d. By expressing this equation in vectorial notation, 0 −1 0 0 u u s ∂t =− − 3 + √ , (1) ut ut u 2ξ 1 1 + (−) 2 from a formal computation, we expect this system to preserve the Gibbs measure 1 1 s 1 dρ(u, u t )“ = " exp − u4 − u 2 + |(−) 2 u|2 exp − u 2t “dudu t ", 4 2 2 where “dudu t " is the non-existent Lebesgue measure on an infinite dimensional vector space (of functions). Heuristically, we expect invariance for this measure by splitting (1) into
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L. Tolomeo
1. ∂t
0 u s =− ut 1 + (−) 2
−1 0
0 u − 3 , ut u
which is a Hamiltonian PDE in the variables u, u t , and so it should preserve the Gibbs measure exp − H (u, u t ) “dudu t ", where H (u, u t ) = 2.
1 4
∂t
u4 +
1 2
s
u 2 + |(−) 2 u|2 +
u 0 =− ut 0
0 1
1 2
u 2t ,
0 u − √ , ut 2ξ
which is the Ornstein - Uhlenbeck process in the variable u t , and so it preserves the spatial white noise 1 exp − u 2t “du t ". 2 For s = 1, up to the damping term exp − 21 u 2t du t , the measure ρ corresponds to the well known 4d model of quantum field theory, which is known to be definable without resorting to renormalisation just for d = 1 (this measure will be rigorously defined - in the case s > d - in Sect. 2). Our goal is to study the global behaviour of the flow of (1), by proving invariance of the measure ρ and furthermore showing that ρ is the unique invariant measure for the flow. Following ideas first appearing in Bourgain’s seminal paper [1] and in the works of McKean–Vasinski [33] and McKean [34,35], there have been many developments in proving invariance of the Gibbs measure for deterministic ispersive PDEs (see for instance [2–7,9,24,38–40]). A natural question that arises when an invariant measure is present is uniqueness of the invariant measure and convergence to equilibrium starting from a “good enough" initial data. This has been extensively studi
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