Existence of Invariant Measures for Reflected Stochastic Partial Differential Equations

In this article, we close a gap in the literature by proving existence of invariant measures for reflected stochastic partial differential equations with only one reflecting barrier. This is done by arguing that the sequence (u(t,·))t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u(t,\cdot ))_{t \ge 0}$$\end{document} is tight in the space of probability measures on continuous functions and invoking the Krylov–Bogolyubov theorem. As we no longer have an a priori bound on our solution as in the two-barrier case, a key aspect of the proof is the derivation of a suitable Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} bound which is uniform in time.


Introduction and Statement of Theorem
The aim of this paper is to argue existence of invariant measures for reflected stochastic partial differential equations (SPDEs) of the form ∂u ∂t = u + f (x, u(t, x)) + σ (x, u(t, x)) ∂ 2 W ∂t∂ x + η, (1.1) growth in [4]. Finally, Xu and Zhang proved existence and uniqueness for the equation where f and σ satisfy Lipschitz and linear growth conditions in [10]. All of these papers focused on the case where the spatial domain is a finite interval, [0, 1], with Dirichlet conditions imposed on the endpoints. Otobe extended the existence theory to the case when the spatial domain is R in [8], proving uniqueness for the case when σ is constant. Uniqueness has also been shown by Hambly and Kalsi in [5] for the equation on an unbounded domain provided that σ satisfies a Lipschitz condition, with a Lipschitz coefficient which decays exponentially fast in the spatial variable. Some interesting properties of the solutions have been proved. In [2], the contact sets for the solutions are studied in the case where the drift, f , is zero and the volatility, σ , is constant. In particular, it is shown that at all positive times, the solution is equal to zero at most four points almost surely. In [13], Zambotti examines the behaviour of the reflection measure in more detail, showing that it is absolutely continuous with respect to Lebesgue measure in the space variable, and also that for each point x in space these densities can be viewed as renormalised local time processes for (u(t, x)) t≥0 . Zhang proved the strong Feller property of solutions in [14], and together with Xu proved a large deviation result for sequences of solutions to such equations with vanishingly small noise in [10].
In this paper, we are interested in invariant measures for these equations. There are some results on this topic in the literature. Zambotti proved in [12] that the law of the 3D-Bessel bridge is an invariant measure for the equation when σ is constant. Otobe then extended this result to the case where the spatial domain is R in [7], proving that the invariant measure is such that the conditional law in an interval is a 3D Bessel bridge with suitable distributions for the endpoints. For the case when the equation has two reflecting walls, above and below the SPDE solution, existence and uniqueness of invariant measures was proved by Yang and Zhang in [11]. The proof here relied on the a priori bound on the infinity norm of the solution, which is provided by the obstacles. Recently, Xie has proved that invariant measures for the one barrier case, (1.1), are unique when they exist, provided that there exist strictly positive constants c 1 and c 2 such that c 1 ≤ σ (x, u) ≤ c 2 in [9].
To the knowledge of the author, existence of invariant measures in the case where there is only reflection at zero has not been proved in the literature. We close this gap here, under the assumption that the drift and volatility coefficients are bounded. We start by proving an L p bound for our solution when it has been multiplied by an exponential function which dampens the value backwards in time. This control essentially replaces the a priori bound for the two-barrier case in the argument in [11]. We are then able to prove tightness by uniformly controlling the Hölder norm of the solution, adapting the arguments of [3] and [11] in order to do so.
Before stating the main theorem of this paper, we recall the definition of a solution to a reflected SPDE. We work on a complete probability space ( , F , P), with W a space-time white noise on this space. This space is equipped with the filtration generated by W , F W t , which can be written as where N here denotes the P-null sets. We further assume that there exist constants C f , C σ > 0 such that the drift and volatility coefficients, f and σ satisfy the following conditions: (I) For every u, v ∈ R + and every x ∈ [0, 1], (II) For every u, v ∈ R + and every x ∈ [0, 1], Definition 1. 1 We say that the pair (u, η) is a solution the SPDE with reflection (c) η is adapted in the sense that for any measurable mapping ψ: We now state the main result of the paper, which states that reflected SPDEs of form (1.1) have invariant measures.

An L p Bound for Solutions to Reflected SPDEs
The aim of this section is to prove the following Theorem.
Such a bound will later enable us to obtain uniform Hölder-type estimates for the functions u(t, ·). The first step towards obtaining this bound is understanding the equation satisfied byũ(t, x) := e −α(T −t) u(t, x).
Proof This can be shown by testing the equation and a change of variables.
We now present some estimates for the heat kernel. We will then be able to bound the solutions to our SPDEs by first writing them in mild form and then applying these estimates, together with Burkholder's inequality and Hölder's inequality.
Proof We have the following expression for G Calculating, we have that

3.
For every x, y ∈ [0, 1] Proof We note that We can write this as where L is a smooth function which vanishes at t = 0. To control the contributions of the first three terms, see the Proof of Proposition A.1 and Proposition A.4 in [5] for details. Note that the constants will not depend on t for this case. The residual component L can also be controlled by differentiating under the sum.
Equipped with these heat kernel estimates, we can now prove the following bound on the white noise term which will appear in the mild form forũ. Then for p ≥ 1 and γ ∈ (0, 1), Assume without loss of generality that s ≤ t. We have that Applying Burkholder's inequality to each of these terms allows us to bound the right hand side by We focus on the first of these terms and note that the arguments for the other two are essentially the same, the difference being in which inequality from Proposition 2.4 we apply. Since σ is bounded and s, t ∈ [n, n +1], we have that e −α(T −r ) σ 2 (z, u(r , z)) ≤ σ 2 ∞ e −α(T −(n+1)) for (r , z) ∈ [0, t] × [0, 1]. This gives that: u(r , y))dzdr By Hölder's inequality we have that almost surely, where X n is a positive random variable such that Now suppose that 0 ≤ s ≤ t ≤ T , x, y ∈ [0, 1], and that there exists n < m ∈ N such that s ∈ [n, n + 1] and t ∈ [m, m + 1]. We then have that where we use the convention that the sum is zero if m = n + 1. By applying (2.3), we then obtain that (2.5) is at most Since n < m, we have that |t − m| ≤ |t − s| and |(n + 1) − s| ≤ |t − s|. In addition, if m ≥ n + 2, we have that |t − s| ≥ 1. Therefore, (2.6) can be bounded by Altogether we have shown that for any s, t ∈ [0, T ] and x, y ∈ [0, 1], where q = p/( p − 1). By (2.4), we obtain that this is at most Importantly, this is independent of T . By (2.7), we then have that Taking the supremum over T > 0 concludes the proof.
Proof Note that I T 2 (t, 0) = 0 for every t ∈ [0, T ] almost surely. Therefore, for x ∈ (0, 1], By taking the supremum on the left hand side and then taking the L p ( )-norm, we see that the result follows from Proposition 2.5.
We are now in position to prove the main Theorem for this section. Throughout the proof, we will denote the infinity norm on We then have, by Theorem 1.4 in [6], that ũ ∞,T ≤ 2 ṽ ∞,T almost surely. Writing It follows that Bounding the I T 1 term, we obtain by applying Proposition 2.3: with Dirichlet boundary conditions v(t, 0) = v(t, 1) = 0 and zero initial data. We now examine the Hölder continuity ofṽ. Writingṽ in mild form, we have that y,ũ(s, y))dyds s, x, y)σ (s, y,ũ(s, y))W (dy, ds) =: Fix γ ∈ (0, 1). By Proposition 2.5, we have that for p ≥ 1 there exists X T ∈ L p ( ) such that, for every s, t ∈ [0, T ] and every x, y ∈ [0, 1] almost surely, with the following uniform bound on the X T : We now control the Hölder norm of I T 1 . We have that For the first of these terms, we note that for n ∈ N ands,t ∈ [n, n + 1], we have that where we make use of the bound |f (r , z,ũ(r , x)| ≤ C f e −α(T −r ) + αũ(r , x) ≤ (C f + α)e −α/2(T −r ) (1 + e α/2(T −r )ũ (r , z)).