Large deviations for stochastic porous media equations

In this paper, we establish the Freidlin-Wentzell type large deviation principles for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.


Introduction
In recent years, there has been a lot of interest in porous media equations both in deterministic and random case (see, e.g., [1,4,10,11,28] and the references therein). In this paper, we are interested in the asymptotic behaviour of porous media equations with small multiplicative noise. More precisely, fix any T > 0 and let (Ω, We use E to denote the expectation with respect to P. Fix any N ∈ N, let T N ⊂ R N denote the N−dimensional torus (suppose the periodic length is 1). We are concerned with the following porous media equations with stochastic forcing        du(t, x) = ∆(|u(t, x)| m−1 u(t, x))dt + Φ(u(t, x))dW(t) in T N × (0, T ], u(·, 0) = u 0 (·) ∈ L m+1 (T N ) on T N , (1.1) for m ∈ (1, ∞). Here u : (ω, x, t) ∈ Ω × T N × [0, T ] → u(ω, x, t) := u(x, t) ∈ R is a random field, that is, the equation is periodic in the space variable x ∈ T N , the coefficient Φ : R → R is measurable and fulfills certain conditions specified later, and W is a cylindrical Wiener process defined on a given (separable) Hilbert space U with the form W(t) = k≥1 β k (t)e k , t ∈ [0, T ], where (e k ) k≥1 is a complete orthonormal base in the Hilbert space U. Clearly, equations (1.1) can be viewed as a special case of a class of SPDE of the type du(t, x) = ∆A(u(t, x))dt + Φ(u(t, x))dW(t) in T N × (0, T ], u(·, 0) = u 0 (·) ∈ L m+1 (T N ) on T N .
Having a stochastic forcing term in (1.2) is very natural and important for various modeling problems arising in a wide variety of fields, e.g., physics, engineering, biology and so on. Up to now, the Cauchy problem for the stochastic equations (1.1) has been studied by a lot of papers and different approaches, for example based on monotonicity in H −1 , based on entropy solutions and based on kinetic solutions have been developed. Specifically, in [1,27,28,29,30], a monotone operator approach is employed in the space H −1 . When applied to the Nemytskii type diffusion coefficients, the condition could be verified if Φ are affine linear functions of u, otherwise, the map u → Φ(u) are not known to be Lipschitz continuous in H −1 , even if Φ is smooth. In order to relax the assumptions on Φ, alternative approaches based on L 1 −techniques have been proposed. In the deterministic setting, this has been realized via the theory of accretive operators going back to Crandall-Ligget [9], entropy solutions studied by Otto [26], Kruzkov [22] and kinetic solution by Lions et al. [23] and Chen, Perthame [8]. In the stochastic setting, an entropy solution was first introduced by Kim in [21] when studying the conservation laws driven by additive noise wherein the author proposed a method of compensated compactness to prove the existence of a stochastic weak entropy solution via vanishing viscosity approximation. Moreover, a Kruzkov-type method was used there to prove the uniqueness. Later, Vallet and Wittbold [31] extended the results of Kim to the multi-dimensional Dirichlet problem with additive noise. Concerning the case of the equation with multiplicative noise, for Cauchy problem over the whole spatial space, Feng and Nualart [18] introduced a notion of strong entropy solutions in order to prove the uniqueness of the entropy solution. On the other hand, using a kinetic formulation, Debussche and Vovelle [13] solved the Cauchy problem for stochastic conservation laws in any dimension by making use of a notion of kinetic solutions. In view of the equivalence between kinetic formulation and entropy solution, they obtained the existence and uniqueness of the entropy solutions. The literature concerning the entropy and kinetic solutions to stochastic degenerate parabolic equations (1.2) is quite extensive, let us mention some works. For instance, Bauzet et al. [3] studied the degenerate parabolic-hyperbolic Cauchy problem under the assumptions that A is globally Lipschitz and when Φ is Lipschitz, a behavior A(u) = |u| m−1 u near the origin is allowed only for m > 2. Moreover, by using a kinetic formulation, Gess and Hofmanová [20] showed the global well-posedness of stochastic porous media equations, where the boundedness of A ′ is released, Φ is assumed to be Lipschitz and √ A ′ (u) is γ−Hölder continuous with γ > 1 2 which forces m > 2. Recently, based on a notion of entropy solutions, Dareiotis et al. [11] established the well-posedness of (1.1) in the full range m ∈ (1, ∞) under mild assumptions on the Nemytskii type diffusion coefficient Φ, where the authors proved an L 1 −contraction estimates as well as a generalized L 1 −stability estimates. There are also a lot of interest on the stochastic fast diffusion equation, that is, m ∈ (0, 1] (see [2,29]). To learn invariant measures for the stochastic porous media equations, we can refer readers to [4,10].
From statistical mechanics point of view, asymptotic analysis for vanishing the noise force is important and interesting for studying stochastic porous media, in which establishing large deviation principles is a core step for finer analysis as well as gaining deeper insight for the described physical evolutions. There are several works on large deviation principles (LDP) for the stochastic porous media equations, we mention some of them. Röckner et al. [30] established the LDP for a class of generalized stochastic porous media equations for both small noise and short time in the space C([0, T ]; H −1 ) by utilizing the monotonicity of the porous medium operator in H −1 . Later, Liu [24] established LDP for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As application, the author proved the LDP holds for stochastic porous media equations in the space C([0, T ]; H −1 ). The purpose of this paper is to prove that the kinetic solution to the stochastic porous medium-type equations (1.2) satisfies Freidlin-Wentzell type LDP in the space L 1 ([0, T ]; L 1 (T N )), which is a delicate result compared with [30] and [24]. On the other hand, Dong et al. [16] established the LDP for quasilinear parabolic SPDE, where the authors handled the hard term div(B(u)∇u) with B being uniformly positive definite, bounded and Lipschitz. For our model, it holds that ∆A(u) = div(a 2 (u)∇u) with a 2 (r) = A ′ (r), so it has similar structure as the term div(B(u)∇u), but a 2 (r) is neither bounded nor Lipschitz. Thus, our case is much more complex and difficult than [16].
To study the Freidlin-Wentzell's LDP for SPDE, an important tool is the weak convergence approach, which is developed by Dupuis and Ellis in [17]. The key idea of this approach is to prove certain variational representation formula about the Laplace transform of bounded continuous functionals, which then leads to the verification of the equivalence between the LDP and the Laplace principle. In particular, for Brownian functionals, an elegant variational representation formula has been established by Boué, Dupuis in [5] and by Budhiraja, Dupuis in [6]. Recently, a sufficient condition to verify the large deviation criteria of Budhiraja et al. [7] for functionals of Brownian motions is proposed by Matoussi et al. in [25], which turns out to be more suitable for SPDEs arising from fluid mechanics. Thus, in the present paper, we adopt this new sufficient condition.
To our knowledge, the present paper is the first work towards establishing the LDP directly for the kinetic solution to the stochastic porous medium-type equations (1.2). The starting point for our research was the paper of Dareiotis et al. [11], where the global well-posedness of entropy solution to (1.2) was established. According to the equivalence between entropy solution and kinetic solution (see Proposition 2.4 in the below), we firstly deduce the existence and uniqueness of kinetic solution to (1.2). Due to the fact that the kinetic solutions are living in a rather irregular space comparing to various type solutions for parabolic SPDEs, it is indeed a challenge to establish LDP for the stochastic porous media equations with general noise force. In order to prove the LDP holds for the kinetic solution in the space L 1 ([0, T ]; L 1 (T N )), our proof strategy mainly consists of the following procedures. As an important part of the proof, we need to obtain the global well-posedness of the associated skeleton equations. For showing the uniqueness, we establish a general result concerning the stability of the strong solution map on the coefficients by utilizing the doubling of variables method. For showing the existence result, we adopt the similar approach as [11]. To complete the proof of the large deviation principle, we also need to study the weak convergence of the small noise perturbations of the problem (1.2) in the random directions of the Cameron-Martin space of the driving Brownian motions. To verify the convergence of the randomly perturbed equation to the corresponding unperturbed equation in L 1 ([0, T ]; L 1 (T N )), an auxiliary approximating process is introduced and the doubling of variables method is employed.
The rest of the paper is organised as follows. The mathematical formulation of stochastic porous media equations is presented in Section 2. In Section 3, we introduce the weak convergence method and state our main result. Section 4 is devoted to the study of the associated skeleton equations. The large deviation principle is proved in Section 5.

Preliminaries
Let us first introduce the notations which will be used later on. C b represents the space of bounded, continuous functions and C 1 b stands for the space of bounded, continuously differentiable functions having bounded first order derivative. Let · L p (T N ) denote the norm of Lebesgue space L p (T N ) for p ∈ (0, ∞]. In particular, set H = L 2 (T N ) with the corresponding norm · H . For all a ≥ 0, let H a (T N ) = W a,2 (T N ) be the usual Sobolev space of order a with the norm H −a (T N ) stands for the topological dual of H a (T N ), whose norm is denoted by · H −a (T N ) . Moreover, we use the brackets ·, · to denote the duality between C ∞ c (T N × R) and the space of distributions over T N × R. Similarly, for 1 ≤ p ≤ ∞ and q := p p−1 , the conjugate exponent of p, we denote and also for a measure m on the Borel measurable space Following [11], we impose conditions on the nonlinearity A via assumptions on Ψ, with some constants m > 1, K ≥ 1, which are fixed throughout the whole paper. Precisely, we assume

Hypotheses
Hypothesis H The initial value u 0 satisfies u 0 m+1 L m+1 (T N ) < ∞. The function A : R → R is differentiable, strictly increasing and odd. The function a is differentiable away from the origin and satisfies the bounds (2.4) as well as is a regular function on T N . Denote by g = (g 1 , g 2 , · · ·). More precisely, we assume that g : T N × R → l 2 satisfies the bounds, for x, y ∈ T N , u, v ∈ R. For g = (g 1 , g 2 , · · ·),g = (g 1 ,g 2 , · · ·) as above, we set Remark 1. In order to obtain the large deviations, our assumptions are stronger than those used by [11] to prove the existence and uniqueness of (1.2), that is, the condition (2.7) on g is a special case of Assumption 2.2 in [11] with parameters κ = 1 2 andκ = 1.
Based on the above notations, equation (1.2) can be rewritten as We denote by E(A, g, u 0 ) the Cauchy problem (2.9).
Referring to Theorem 2.1 in [11], the following global well-posedness is proved.
Theorem 2.1. Let (A, g, u 0 ) satisfy Hypotheses H. Then there exists a unique entropy solution to (2.9) with initial condition u 0 . Moreover, ifũ is the unique entropy solution to (2.9) with initial valueũ 0 , then The authors of the work [11] also show the stability of the solution map with respect to the coefficients in the following sense.
Theorem 2.2. Let (A n ) n∈N , (g n ) n∈N satisfy Hypothesis H uniformly in n and the initial values satisfy sup n u 0,n L m+1 (T N ) < ∞. Assume furthermore that A n → A uniformly on compact sets of R, u 0,n → u 0 in L m+1 (T N ) and d(g n , g) → 0, as n → ∞. Let u n , u be the entropy solutions of E(A n , g n , u 0,n ), E(A, g, u 0 ), respectively. Then u n → u in L 1 (Ω × [0, T ] × T N ), as n → ∞.

Kinetic solution and generalized kinetic solution
In this subsection, we pay attention to the definition of kinetic solution. We introduce the kinetic solution to equations (2.9) as follows. Keeping in mind that we are working on the stochastic basis (Ω, F , P, {F t } t∈[0,T ] , (β k (t)) k∈N ).
3. there exists a kinetic measure m such that f := I u>ξ satisfies the following In order to prove the existence of a kinetic solution, the generalized kinetic solution was introduced in [13].
Definition 2.4. (Young measure) Let (X, λ) be a finite measure space. Let P 1 (R) denote the set of all (Borel) probability measures on R. A map ν : X → P 1 (R) is said to be a Young measure on X, if for each φ ∈ C b (R), the map z ∈ X → ν z (φ) ∈ R is measurable. Next, we say that a Young measure ν vanishes at infinity if, for each p ≥ 1, the following holds X R |ξ| p dν z (ξ)dλ(z) < +∞. holds for λ − a.e. z ∈ X,. We say that f is an equilibrium if there exists a measurable function u : X → R such that f (z, ξ) = I u(z)>ξ a.e., or equivalently, ν z = δ u(z) for λ − a.e. z ∈ X.
3. there exists a kinetic measure m such that for ϕ ∈ C 1 Referring to [13], almost surely, any generalized solution admits possibly different left and right weak limits at any point t ∈ [0, T ]. This property is important for establishing a comparison principle which allows to prove uniqueness. Also, it allows us to see that the weak form (2.15) of the equation satisfied by a generalized kinetic solution can be strengthened. We write below a formulation which is weak only respect to x and ξ. The following result is proved in [13]. Proposition 2.3. (Left and right weak limits) Let f 0 be a kinetic initial datum and f be a generalized kinetic solution to (2.9) with initial value f 0 . Then f admits, almost surely, left and right limits respectively at every point t ∈ [0, T ]. More precisely, for any t ∈ [0, T ], there exist kinetic functions f t± on Ω × T N × R such that P−a.s.
In particular, almost surely, the set of t ∈ [0, T ] fulfilling that f t+ f t− is countable.
For a generalized kinetic solution f , define f ± by f ± (t) = f t± , t ∈ [0, T ]. Since we are dealing with the filtration associated to Brownian motion, both f ± are clearly predictable as well. Also f = f + = f − almost everywhere in time and we can take any of them in an integral with respect to the Lebesgue measure or in a stochastic integral. However, if the integral is with respect to a measure-typically a kinetic measure in this article, the integral is not well defined for f and may differ if one chooses either f + or f − .
As discussed above, with the aid of Proposition 2.3, the weak form (2.15) satisfied by a generalized kinetic solution can be strengthened to weak only respect to x and ξ. Concretely, Lemma 2.1. The generalized kinetic solution f satisfying (2.15) can be strengthened to for any t ∈ [0, T ] and Proof. For all t ∈ [0, T ], consider a function α defined by In view of α n ∈ C 1 c ([0, T ]), then (2.15) holds for a test function of the form (x, s, ξ) → ϕ(x, ξ)α n (s), where ϕ ∈ C 1 c (T N × R). Since {α n } n≥1 is countable, we can find a common P full measure setΩ ⊂ Ω such that (2.15) holds for all α n . Now, we focus on taking limitation n → ∞ of (2.15) for any fixed ω inΩ. In the following, we fix ω ∈Ω. For all ϕ ∈ C 1 c (T N × R), the map is continuous. By Fubini theorem, the weak formulation (2.15) for α n is equivalent to Taking into account the fact that both f 0 , ϕ α n (0) and m, ∂ ξ ϕ (α n ) are finite, we obtain that ∂ t g(t) is a Radon measure on (0, T ), i.e., the function g(t) ∈ BV(0, T ). Hence, by (2.18), it yields and α is defined by (2.17). By simple calculation, letting r → 0, we derive that (2.16) holds.
As stated in the introduction, the starting point of this paper is the equivalence between entropy solution and kinetic solution. Now, we give a brief proof.
. For a kinetic solution to (2.9) in the sense of Definition 2.3 is equivalent to be an entropy solution u to (2.9) in the sense of Definition 2.1.
Proof. Let us begin with the proof of a kinetic solution is an entropy solution. To achieve it, we choose test T )) and the convex function η ∈ C 2 (R) with η ′′ > 0 compactly supported. Assume u(x, t) is a kinetic solution to (2.9), then the corresponding kinetic functions can be written as f = I u(x,t)>ξ , f 0 = I u 0 >ξ . From (2.13), we deduce that Similarly, it yields Conversely, we suppose u(x, t) is an entropy solution to (2.9) satisfying (2.11). Then for any non-negative T )) and any convex function η ∈ C 2 (R) with η ′′ > 0 compactly supported, we define a measure m as follows: On the basis of Proposition 2.4, we deduce from Theorem 2.1 that Assume Hypothesis H holds. Then there is a unique kinetic solution u to equation (2.9) with initial datum u 0 .

Freidlin-Wentzell large deviations and statement of the main result
We start with a brief account of notions of large deviations. Let {X ε } ε>0 be a family of random variables defined on a given probability space (Ω, F , P) taking values in some Polish space E.
where A o andĀ denote the interior and closure of A in E, respectively.
Suppose W(t) is a cylindrical Wiener process on a Hilbert space U defined on a filtered probability space Here and in the sequel of this paper, we will always refer to the weak topology on the set S M .
Suppose for each ε > 0, G ε : C([0, T ]; U) → E is a measurable map and let X ε := G ε (W). Now, we list below sufficient conditions for the large deviation principle of the sequence X ε as ε → 0.
Condition A There exists a measurable map G 0 : C([0, T ]; U) → E such that the following conditions hold The following result is due to Budhiraja et al. in [6].
Theorem 3.1. If {G ε } satisfies condition A, then X ε satisfies the large deviation principle on E with the following good rate function I defined by By convention, Recently, a new sufficient condition (Condition B below) to verify the assumptions in condition A (hence the large deviation principle) is proposed by Matoussi, Sabagh and Zhang in [25]. It turns out this new sufficient condition is suitable for establishing the large deviation principle for the stochastic porous media equation.
Condition B There exists a measurable map G 0 : C([0, T ]; U) → E such that the following two items hold (i) For every M < +∞, and for any family {h ε ; ε > 0} ⊂ A M and any δ > 0, and ρ(·, ·) stands for the metric in the space E.

Statement of the main result
In this paper, we are concerned with the following SPDE driven by small multiplicative noise for ε > 0, where u 0 ∈ L m+1 (T N ). Under Hypothesis H, by Theorem 2.5, there exists a unique kinetic solution u ε ∈ L 1 ([0, T ]; L 1 (T N )) a.s.. Therefore, there exists a Borel-measurable function we consider the following skeleton equation The solution u h , whose existence and uniqueness will be proved in next section, defines a measurable mapping . We are ready to proceed with the statement of our main result. and (iii) For all convex function η ∈ C 2 (R) with η ′′ compactly supported and all non- where q η is any function satisfying q ′ η (ξ) = η ′ (ξ)a 2 (ξ).
is said to be a generalized kinetic solution to (3.25) with the initial datum where C m is a positive constant and there exists a measure In the following, we firstly prove the uniqueness of the skeleton equations (3.25). Then, based on the uniqueness, we show the existence.
Similarly to Lemma 2.1, we reformulate (4.30) to a strong version, it yields that for all t ∈ [0, T ], Consider the solutionũ h being the solution of In the following, with the help of (4.31), we prove a comparison theorem for two generalized kinetic solutions of (3.25) and (4.32), respectively.
Proof. Denote by f 1 (x, t, ξ) and f 2 (y, t, ζ) be two generalized solutions to (3.25) and (4.32) with the corresponding kinetic measures m 1 and m 2 .
Next, we prove that u is an entropy solution of (3.25). Utilizing (4.76) and Fatou lemma, we deduce that (i) of Definition 4.1 holds.
Analogously, we define Ψ n, f , q n,η similar to the above with a replaced by a n . For each n, we have Ψ n, f (u n ) ∈ 2 for all r ∈ R, which combined with (4.75) and (4.76) gives that Hence, for a subsequence, we have . With the aid of (4.72), (4.78) and (4.79), we deduce that where we have used f (u n ) → f (u) strongly in L 2 ([0, T ]; H). Hence, (ii) of Definition 4.1 is obtained.
In the following, we will show (iii) in Definition 4.1 holds. Let η and φ be as in (iii). By integration by parts, we get On the basis of (4.70)-(4.73), (4.78) and (4.79), we have Setf (r) = η ′′ (r). Notice that ∂ i Ψ n,f (u n ) = η ′′ (u n )∂ i Ψ n (u n ). As before, we have (after passing to a subsequence if necessary) Hence, taking lim inf in both sides of (4.80), we by choosing an appropriate subsequence, we obtain that u satisfies (iii) of Definition 4.1. We complete the proof. In

The continuity of the skeleton equation
In this part, we aim to prove the continuity of the mapping G 0 . Namely, let u h ε denote the kinetic solution of (3.25) with h replaced by h ε and we will show that u h ε converges to the kinetic solution u h of the skeleton equation (3.25) . To achieve it, we need the auxiliary approximating process (A n , g n , u 0,n ) defined by (4.69)-(4.73).
For any family {h ε , ε > 0} ⊂ S M with h ε (t) = k≥1 h ε k (t)e k , let us consider the following approximation (4.81) Referring to Section 4 in [16], for each n, the Cauchy problem E(A n , g n , u 0,n ) has a unique solution u h ε n ∈ C([0, T ]; H) ∩ L 2 ([0, T ]; H 1 ) satisfying the following energy estimates uniformly in n: for all p ≥ 2, where we have used u 0,n ≤ u 0 . Since a n ≥ 2 n > 0, we have |∇u h ε n | ≤ N(n)|∇Ψ n (u h ε n )|, then by (4.82), it yields With the above approximation process (4.81), for any ε > 0 and n ≥ 1, we have In order to establish the continuity of the skeleton equations, several steps are involved. Firstly, we show the uniform convergence of the sequence {u h n , n ≥ 1} to u h over h ∈ S M . Proof. As discussed above, under assumption, we know that (A n , g n , u 0,n ) satisfies Hypotheses H. According to (1) of Theorem 4.2, we have that for all γ, δ ∈ (0, 1), λ ∈ [0, 1], α ∈ (0, 1 ∧ m 2 ) and a.e. t ∈ [0, T ], where E 0 (γ, δ), E t (γ, δ) → 0 as γ, δ → 0, the constant N 0 is independent of γ, δ, λ and R λ is defined by For n ≥ 4, we can choose λ = 4 n , by (4.72), we deduce that R λ ≥ n. Furthermore, let γ = n − 1 2 and δ = γ 3 2α , it follows that Note that In the following, we prove the compactness of {u h ε n , ε > 0}. For simplicity, we set u ε n := u h ε n . As in [19], we introduce the following space. Let Y be a separable Banach space with the norm · Y . Given p > 1, β ∈ (0, 1), let W β,p ([0, T ]; Y) be the Sobolev space of all functions u ∈ L p ([0, T ]; Y) such that which is then endowed with the norm The following result can be found in [19]. To obtain the compactness, we also need the following lemma.  Proof. From (4.81), u ε n can be written as Clearly, I ε 1 H ≤ C 1 . Next, where u h n is the solution of (4.81) with h ε replaced by h. This will be achieved if we show that for any sequence ε m → 0, one can find a subsequence ε m k → 0 such that Clearly, we also have h ε m k → h weakly in L 2 ([0, T ]; U). Thus, we only need to prove u * = u h n . From (4.81), we know that for a test function φ ∈ C ∞ (T N ), it holds that Due to (4.87), we get Using (4.87) and the Lipschitz property of A n , we deduce that Note that With the aid of (4.87), we get t 0 k≥1 Thus, let k → ∞ in (4.89) to obtain which means that u * is the solution to (4.81) with h ε replaced by h. By the uniqueness of (4.81), we conclude u * = u h n . Thus, (4.86) is proved, which implies (4.85) holds. Note that for any ε > 0 and n ≥ 1, we have For any ι > 0, by Proposition 4.5, there exists N 0 such that for all ε > 0, Letting n = N 0 , we deduce from (4.90) that Using (4.85), we know that there are ε 0 > 0 such that for any ε ≤ ε 0 , it holds that Thus, we conclude that lim ε→0 u h ε − u h L 1 ([0,T ];L 1 (T N )) ≤ ι.
Since the constant ι is arbitrary, we obtain the desired result.

Large deviations
For any family {h ε ; 0 < ε ≤ 1} ⊂ A M with h ε (t) = k≥1 h ε k (t)e k , we consider the following equation and there exists a kinetic measurem ε ∈ M + 0 (T N × [0, T ] × R) such thatf ε := Iūε >ξ fulfills that for all ϕ ∈ where G 2 := k≥1 |g k | 2 and ν ε Due to Theorem 3.1 (the sufficient condition B) and Theorem 4.8, we only need to prove the following result to establish the main result.
Clearly, it holds that Moreover, it follows that , a.s..