Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs

In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.


Introduction
Consider the following obstacle problems for quasilinear stochastic partial differential equations (SPDEs) in R d : x, U (t, x), ∇U (t, x))dt + f (t, x, U (t, x), ∇U (t, x))dt x, U (t, x), ∇U (t, x))dB j t = −R(dt, dx), (1.1) where B j t , j = 1, 2, ... are independent real-valued standard Brownian motions, the stochastic integral against Brownian motions is interpreted as the backward Ito integral, ∆ is the Laplacian operator, f, g i , h j are appropriate measurable functions specified later, L(t, x) is the given barrier function, R(dt, dx) is a random measure which is a part of the solution pair (U, R). The random measure R plays a similar role as a local time which prevents the solution U (t, x) from falling below the barrier L.
Such SPDEs appear in various applications like pathwise stochastic control problems, the Zakai equations in filtering and stochastic control with partial observations. Existence and uniqueness of the above stochastic obstacle problems were established in [DMZ] based on an analytical approach. Existence and uniqueness of the obstacle problems for quasi-linear SPDEs on the whole space R d and driven by finite dimensional Brownian motions were studied in [MS] using the approach of backward stochastic differential equations (BSDEs). Obstacle problems for nonlinear stochastic heat equations driven by space-time white noise were studied by several authors, see [NP], [XZ] and references therein.
In this paper, we are concerned with the small small noise large deviation principle(LDP) of the following obstacle problems for quasilinear SPDEs: (1.4) Large deviations for stochastic evolution equations and stochastic partial differential equations driven by Brownian motions have been investigated in many papers, see e.g. [DM], [L], [MSS], [RZZ], [CW], [CR], [S], [BDM1], [C] and references therein.
To obtain the large deviation principle, we will adopt the weak convergence approach introduced by Budhiraja, Dupuis and Maroulas in [BDM2] , [BDM1] and ). This approach is now a powerful tool which has been applied by many people to prove large deviation principles for various dynamical systems driven by Gaussian noises, see e.g. , [DM], [L], [MSS], [RZZ], [BDM1].
In order to apply the weak convergence approach to the obstacle problems, we first provide a simple sufficient condition to verify the criteria of Budhiraja-Dupuis-Maroulas. The sufficient condition is particularly suitable for stochastic dynamics generated by stochastic differential equations and stochastic partial differential equations. The important part of the work is to study the so called skeleton equations which, in our case, are the deterministic obstacle problems driven by the elements in the Cameron-Martin space of the Brownian motions. We need to show that if the driving signals converge weakly in the Cameron-Martin space, then the corresponding solutions of the skeleton equations also converge in the appropriate state space. This turns out to be hard because of the singularity caused by the obstacle. To overcome the difficulties, we have to appeal to the penalized approximation of the skeleton equation and to establish some uniform estimate for the solutions of the approximating equations with the help of the backward stochastic differential equation representation of the solutions. This is purely due to the technical reason because primarily the LDP problem has not much to do with backward stochastic differential equations.
The rest of the paper is organized as follows. In Section 2, we introduce the stochastic obstacle problem and the precise framework. In Section 3, we recall the weak convergence approach of large deviations and present a sufficient condition. Section 4 is devoted to the study of skeleton obstacle problems. We will show that the solution of the skeleton problem is continuous with respect to the driving signal. The proof of the large deviation principle is in Section 5.

Obstacle problems
Let H := L 2 R d be the Hilbert space of square integrable functions with respect to the Lebesgue measure on R d . The associated scalar product and the norm are denoted by denote the first order Sobolev space, endowed with the norm and the inner product: V * will denote the dual space of V . When causing no confusion, we also use < u, v > to denote the dual pair between V and V * . Our evolution problem will be considered over a fixed time interval [0, T ]. Now we introduce the following assumptions.
Assumption 2.1. ( x, y, z) and satisfy (ii) There exist constants c > 0, 0 < α < 1 and 0 < β < 1 such that for any (t, where the gradient ∇ and the Laplacian ∆ act on the space variable x. We denote by H T the space of predictable, processes (u t , t ≥ 0) such that u ∈ H T and that denotes the space of realvalued infinitely differentiable functions with compact supports in R + and C ∞ c R d is the space of infinitely differentiable functions with compact supports in R d .
We now precise the definition of solutions for the reflected quasilinear SPDE (1.1): Definition 2.1. We say that a pair (U, R) is a solution of the obstacle problem (1.1) if (1) U ∈ H T , U (t, x) ≥ L(t, x), dP ⊗ dt ⊗ dx-a.e. and U (T, x) = Φ(x), dx − a.e.
(2) R is a random regular measure on [0, T ) × R d , (4) U admits a quasi-continuous versionŨ , and Remark 2.1. We refer the reader to [DMZ] for the precise definition of regular measures and quasi-continuity of functions on the space [0, T ] × R d .
Let us recall the following result from [MS] and [DMZ].

The measures P m
The operator ∂ t + 1 2 ∆, which represents the main linear part in the equation (1.1), is associated with the Bownian motion in R d . The sample space of the Brownian motion is Ω ′ = C [0, ∞); R d , the canonical process (W t ) t≥0 is defined by W t (ω) = ω(t), for any ω ∈ Ω ′ , t ≥ 0 and the shift operator, θ t : Ω ′ −→ Ω ′ , is defined by θ t (ω)(s) = ω(t + s), for any s ≥ 0 and t ≥ 0. The canonical filtration F W t = σ (W s ; s ≤ t) is completed by the standard procedure with respect to the probability measures produced by the transition function is the Gaussian density. Thus we get a continuous Hunt process Ω ′ , W t , θ t , F , F W t , P x . We shall also use the backward filtration of the future events 0 is a bijection. For each probability measure on R d , the probability P µ of the Brownian motion started with the initial distribution µ is given by In particular, for the Lebesgue measure in R d , which we denote by m = dx, we have We recall the forward and backward stochastic integral defined in [S], [MS] under the measure P m .
We refer the reader to [MS], [S] for more details.

A sufficient condition for LDP
In this section we will recall the criteria obtained in  for proving a large deviation principle and we will also present a sufficient condition to verify the criteria.
Let E be a Polish space with the Borel σ-field B(E). Recall Definition 3.2. (Large deviation principle) Let I be a rate function on E. A family {X ε } of E-valued random elements is said to satisfy a large deviation principle on E with rate function I if the following two claims hold.

A criteria of Budhiraja-Dupuis
The Cameron-Martin space associated with the Brownian motion The set S N endowed with the weak topology is a compact Polish space. SetS N = {φ ∈K; φ(ω) ∈ S N , P-a.s.}.
The following result was proved in .
(a) for every N < +∞ and any family {k ε ; ε > 0} ⊂S N satisfying that k ε converges in law as S N -valued random elements to some element k as Then the family {X ε } ε>0 satisfies a large deviation principle in E with the rate function I given by with the convention inf{∅} = ∞.

A sufficient condition
Here is a sufficient condition for verifying the assumptions in Theorem 3.1 .
s)ds and ρ(·, ·) stands for the metric in the space E (ii) for every N < +∞ and any family {k ε ; ε > 0} ⊂ S N satisfying that k ε converges weakly to some element k as ε → 0, Γ 0 · 0 k ε (s)ds converges to Γ 0 ( · 0 k(s)ds) in the space E. Then the family {X ε } ε>0 satisfies a large deviation principle in E with the rate function I given by with the convention inf{∅} = ∞.
Remark 3.1. . When proving a small noise large deviation principle for stochastic differential equations/stochastic partial differential equations, condition (i) is usually not difficult to check because the small noise disappears when ε → 0.
Proof. We will show that the conditions in Theorem 3.1 are fulfilled. Condition (b) in Theorem 3.1 follows from condition (ii) because S N is compact with respect to the weak topology. Condition (i) implies that for any bounded, uniformly continuous function G(·) on E, Thus, condition (a) will be satisfied if Z ε convergence in law to Γ 0 ( · 0 k(s)ds) in the space E. This is indeed true since the mapping Γ 0 is continuous by condition (ii) and k ε converge in law as S N -valued random elements to k. The proof is complete.

Skeleton equations
Recall K := L 2 ([0, T ], l 2 ). Let k ∈ K and consider the deterministic obstacle problem: Denote by u ε the solution of equation (4.1) with k ε replacing k. The main purpose of this section is to show that u ε converges to u in the space H T if k ε → k weakly in the Hilbert space K. To this end, we need to establish a number of preliminary results.
Consider the penalized equation: For later use, we need to show that for any M > 0, u n → u uniformly over the bounded subset {k; k K ≤ M } as n → ∞. For this purpose, it turns out that we have to appeal to the BSDE representation of the solutions. Let Y n t := u n (t, W t ), Z n t = ∇u n (t, W t ). Then it was shown in [MS] that (Y n , Z n ) is the solution of the backward stochastic differential equation under P m : (4.5) The following result is a uniform estimate for (Y n , Z n ).
Lemma 4.1. For M > 0, we have the following estimate: The proof of this lemma is a repeat of the proof of Lemma 6 in [MS]. One just needs to notice that when applying the Grownwall's inequality, the constant c M on on right of (4.7) only depends on the norm of k which is bounded by M .
We also need the following estimate.
Theorem 4.1. Let Assumptions 2.1 hold. Assume that k ε → k weakly in the Hilbert space K as ε → 0. Then u ε converges to u in the space H T , where u ε denotes the solution of equation (4.1) with k ε replacing k.
Now we are ready to complete the last step of the proof. For any n ≥ 1, we have u ε − u HT ≤ u ε − u ε,n HT + u ε,n − u n HT + u n − u HT . (4.63) For any given δ > 0, by Proposition 4.1 there exists an integer n 0 such that sup ε u ε −u ε,n0 HT ≤ δ 2 and u − u n0 HT ≤ δ 2 . Replacing n in (4.63) by n 0 we get u ε − u HT ≤ δ + u ε,n0 − u n0 HT .
Since the constant δ is arbitrary, the proof is complete.

Large deviations
After the preparations in Section 4, we are ready to state and to prove the large deviation result.
Recall that U ε is the solution of the obstacle problem: For k ∈ K = L 2 ([0, T ], l 2 ), denote by u k the solution of the following deterministic obstacle problem: h j (t, x, u k (t, x), ∇u k (t, x))k j t dt = −ν k (dt, dx), Proof. The existence of a unique strong solution of the obstacle problem (5.1) means that for every ε > 0, there exists a measurable mapping Γ ε (·) : C([0, T ]; R ∞ ) → H T such that U ε = Γ ε (B(·)) .