Small Time Asymptotics for SPDEs with Locally Monotone Coefficients

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.


Introduction
The small time LDP mainly studies the asymptotic behavior of the tails of a family of probability distributions at a given point in space when the time is very small. Specifically, we focus on the limiting behavior of the solution in time interval [0, t] as t goes to zero. The study of the small time asymptotics (large deviations) of finite dimensional diffusion processes was initiated by Varadhan in the influential work [55]. Due to its wide applications in extremal events arising in risk management, mathematical finance, statistical mechanics, quantum physics and many other areas, large deviation theory has become an important component of modern applied probability, see, e.g. [5,6,10,13,20,21,22,24,50,53,55] and references therein.
Another main point being that the small time behaviour of a diffusion process can be characterized in terms of an energy/distance function on a Riemannian manifold, whose metric is induced from the inverse of the diffusion coefficient, that is, such small time asymptotics will be useful to get the following Varadhan identity lim t→0 2t log P(X(0) ∈ A 1 , X(t) ∈ A 2 ) = −d 2 (A 1 , A 2 ), where d is an appropriate Riemann distance associated with the diffusion generated by X, see, e.g. [1,4,12,30,54,60] and references therein.
Apart from the above motivations, the small time asymptotic itself is also theoretically interesting, which has been studied a lot in the literatures. For instance, the small time asymptotics of infinite dimensional diffusion processes were studied in [2,3,19,31,60]. Subsequently, many authors have endeavored to derive the small time LDP for different types of SPDEs. An important development concerning small time LDP for stochastic 2D Navier-Stokes equation was established by Xu and Zhang [58]. In [48], Röckner and Zhang studied small time LDP for stochastic 3D tamed Navier-Stokes equation. Moreover, the second named author with Röckner and Zhu [42] also obtained the small time LDP for stochastic 2D quasi-geostrophic equations in the sub-critical case. The small time LDP of stochastic 3D primitive equations was investigated by Dong and Zhang [18]. Recently, the small time LDP of scalar stochastic conservation laws was also studied in [59]. The reader might refer to [11,12,32,35] and references therein for further results on this subject.
However, most papers in the literature investigated small time asymptotics (LDP) only for semilinear type SPDEs. On the other hand, some very interesting quasilinear SPDEs have been studied a lot recently, such as stochastic porous media equation and stochastic p-Laplace equation, see e.g. [25,26,27,36,37,40,41,44,47,56]) and references therein. We would like to know whether small time asymptotics (LDP) results also hold for those SPDE models. This is one of the main motivations for us to study the small time LDP for a class of nonlinear SPDEs, where the coefficients satisfy local monotonicity condition under the (generalized) variational framework.
The variational framework has been used intensively for studying SPDE where the coefficients satisfying the classical monotonicity and coercivity conditions. It was first investigated in the seminal works of Pardoux [46] and Krylov and Rozovskii [33], where they adapted the monotonicity tricks to prove the existence and uniqueness of solutions for a class of SPDE. Recently, this framework has been substantially extended by the second named author and Röckner in [38,39,40,41] for more general class of SPDE with coefficients satisfying the generalized coercivity and local monotonicity conditions. In recent years, various properties for SPDEs with monotone or locally monotone coefficients has been intensively investigated in the literature, such as small noise LDP [37,43,47,57], random attractors [25,26,27,28], Harnack inequality and applications [36], Wong-Zakai approximation and support theorem [44], ultra-exponential convergence [56], and existence of optimal controls [15].
The proof of the main result here mainly follows the idea in Zhang's work [60] by using exponential equivalence arguments, which is a very powerful method used by many scholars to study the small time LDP for SPDEs, see, e.g. [11,18,35,42,48,58,59]. More precisely, consider a zero drift stochastic differential equation with the same initial data (see (3.2) below), where the small noise and small time asymptotics problems are equivalent. It is easy to see that the small noise LDP for the solution Y ε of zero drift stochastic differential equation holds, thus our task is to show that the law of X ε and Y ε are exponentially equivalent (see (3.3) below). Comparing with some related works on small time LDP for SPDEs, to deal with the stochastic differential equation with zero drift, one usually assumes that there exists another Hilbert space H 1 which is densely embedded in state space H. Working in the space of continuous H 1 -valued trajectories, one is able to get the H 1 -norm estimates by applying Itô's formula to · 2 H 1 . However, in the variational framework, we work with the Gelfand triple V ⊂ H ⊂ V * , where V is a reflexive Banach space such that V ⊂ H is continuously and densely, and it is unavailable to get the Vnorm estimates by applying Itô's formula to · 2 V (e.g. in the quasilinear SPDE case). In order to overcome this difficulty, we use the concept of 2-smooth Banach space and get the V -norm estimates using the crucial BDG type inequality proved by Seidler [49] (cf. [63] for recent generalization) for stochastic integrals in the 2-smooth Banach space, where the sharp constant p 1/2 also plays an important role in our proof. This 2-smooth Banach space is introduced for establishing a theory of stochastic integration in Banach spaces and typical examples of such spaces are L p spaces with p ≥ 2 and Sobolev spaces W s,p 0 with p ≥ 2 and s ≥ 1. Thus, our main result is applicable to various types of SPDEs such as stochastic porous media equation, stochastic p-Laplace equation, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, by applying the abstract result to concrete models, our main result could cover the results in [58,35], where the small time LDP for stochastic 2D Navier-Stokes equation and stochastic Ladyzhenskaya model was studied respectively. Moreover, to the best of our knowledge, our small time LDP results for general quasilinear SPDEs with multiplicative noise (such as porous media equation and p-Laplace equation) seem to be new in the literature.
The rest of the paper is organized as follows. In Section 2, we introduce the variational framework and formulate our main result. Section 3 is devoted to proving our main result. In Section 4, we apply the main result to various SPDE models as applications. Throughout the paper, C and C p will denote positive constants which may change from line to line, here C p emphasize the dependence on parameter p.

Framework and main result
Let (H, ·, · ) be a real separable Hilbert space identified with its dual space H * by the Riesz isomorphism. Let (V, ·, · V ) be a reflexive Banach space which is continuously and densely embedded into H. Then we have the following Gelfand triple: where V * is the dual space of V . Let V * ·, · V denote the dualization between V and V * , then it follows that Let {W (t)} t≥0 be a cylindrical Wiener process in a separable Hilbert space (U, ·, · U ) on a complete filtered probability space (Ω, F , F t ; P). Let (L 2 (U; H), · 2 ) denote the space of all Hilbert-Schmidt operators from U to H.
In this paper, we consider the following stochastic evolution equation: Let us now state the precise conditions on the coefficients of (2.1).
(A2) (Local monotonicity) where ρ : V → [0, +∞) is a measurable function and locally bounded in V such that In order to study the small time LDP, we also need to estimate the stochastic integrals in the Banach space V . For a more specific example, consider the stochastic p-Laplace equation, it is common to take V = W 1,p 0 for p ≥ 2 as in [40] and therefore we need to ensure the existence of the stochastic integral in (2.1) as an W 1,p 0 -valued process. We recall that the Sobolev spaces W 1,p 0 with p ≥ 2 belong to the class of 2-smooth Banach spaces since they are isomorphic to L p (0, 1) according to [52,Remark 2 in Section 4.9] and hence they are well suited for the stochastic Itô integration (see e.g. Brzézniak et al. [7,8] for the precise construction of the stochastic integral).
In this work, we assume V is a 2-smooth Banach space. Let us denote by γ(U, V ) the space of the γ-radonifying operators from U to V . We recall that Υ ∈ γ(U, V ) if the series k≥1 γ k Υ(u k ) converges in L 2 (Ω, U) for any sequence (γ k ) k≥0 of independent Gaussian real-valued random variables on a probability space (Ω,F,P) and any orthonormal basis (u k ) k≥0 of U. Then, the space γ(U, V ) is endowed with the norm (which does not depend on (γ k ) k≥0 , nor on (u k ) k≥0 and is a Banach space). In the following, we shall write · γ instead of · γ(U,V ) for the simplicity of notations. We assume the following conditions on B.
Assumption 2.2. There exists constant C such that the following conditions hold for all Remark 2.2. By Assumption 2.1 and Assumption 2.2, the coercivity of A and B is easily obtained as Now, we recall the following definition.
is called a strong solution of (2.1), if for its dt ⊗ P-equivalence class we have and the following identity hold P-a.s.
The following well-posedness result is due to the second name author and Röckner [ For ε > 0, in this paper, we aim to study the probabilistic asymptotic behavior for small time process X(εt) as ε → 0.
Define a functional I(g) on C([0, T ]; H) by is absolutely continuous and such that Now we state the main result of this paper. (1) In section 4 below, we will apply Theorem 2.2 to concrete examples of SPDE models as applications. In particular, this covers the results in [58,35], where the small time LDP for stochastic 2D Navier-Stokes equation and stochastic Ladyzhenskaya model was studied respectively.
(2) Furthermore, Theorem 2.2 can also be applied to study the small time LDP for many other SPDEs, such as stochastic Burgers type equation, stochastic porous media equation, stochastic p-Laplace equation, stochastic 2D Boussinesq equations, stochastic 2D magneto-hydrodynamic equations, stochastic 2D magnetic Bénard problem, stochastic 3D Leray-α model, stochastic shell models of turbulence and stochastic power law fluid equation, which seem to not have been established in the literature before.

Proof of main result
In this section, we will give the proof of Theorem 2.2, which is mainly based on the exponential equivalence arguments.
More precisely, for ε > 0, by the scaling property of the Wiener process, it is easy to see that the small time process X(ε·) coincides in law with the solution of the following stochastic evolution equation: Now, let Y ε (·) be the solution of the following stochastic differential equation: and ν ε be the law of Y ε (·) on the C([0, T ]; H). Then, applying the weak convergence approach developed by Budhiraja and Dupuis [9], it easy to get that ν ε satisfies the LDP with rate function I(·) given by (2.2) (see, e.g. [37,43]). Therefore, our task now is to show that the two families of probability measures µ ε and ν ε are exponentially equivalent, that is, for any δ > 0, We begin the proof with the following lemma which provides an estimate of the probability that the solution of (3.1) leaves an energy ball. Set Then, we claim that Proof. According to Itô's formula (cf. [40,Theorem 4.2.5]) and Remark 2.2, we have Then it is easy to get which implies that for any q ≥ 2 To estimate the stochastic integral term, we will use the following martingale inequality from [16] that there exists a universal constant C such that, for any q ≥ 2 and for any continuous martingale M t with M 0 = 0, one has Therefore, combining the above estimates yields where the second inequality is due to Minkowski's inequality. Applying Gronwall's lemma we obtain that Using Chebyshev's inequality, for any M > 0, we have Taking q = 2/ε, we get Then, it is easy to see Let M → ∞ on both sides of (3.6), we complete the proof.
Let X ε n be the solution of (2.1) with the initial value x n . From the proof of Lemma 3.1, it follows that Let Y ε n be the solution of (3.2) with the initial value x n , i.e.
Then we can get the following estimate.
where in the last inequality we use Minkowski's inequality and the constant C is independent of q and ε. Then, it is easy to get Applying Gronwall's Lemma yields Fixing M and taking q = 1/ε, we have Let M → ∞ on both sides of (3.9), we complete the proof. Now, we establish the exponential convergence of X ε n − X ε . Lemma 3.3. For any δ > 0, we have Proof. For M > 0, define stopping time Using Itô's formula, the local monotone condition (A2), we can get Then by the martingale inequality (3.5), we obtain Applying Gronwall's lemma yields .
Fixing M and taking q = 2/ε we get By Lemma 3.1, for any R > 0, there exists a constant M such that for every ε ∈ (0, 1] the following inequality holds: Combining (3.10) and (3.12), we conclude that there exists a positive integer N such that for any n ≥ N and ε ∈ (0, 1], Since R is arbitrary, the assertion of the lemma follows.
We also need to establish the exponential convergence of Y ε n − Y ε .
Lemma 3.4. For any δ > 0, we have Proof. From (3.2) and (3.8), it is easy to see Then by the Assumption (2.2) and martingale inequality (3.5), we obtain where the constant C is independent of q and ε. Utilizing Gronwall's lemma, we get Applying the same argument as the proof of (3.10) in Lemma 3.3, we complete the proof.
By condition (A3) and Young's inequality, we have where θ < η is a small positive constant. Then by condition (A2), we obtain that Then applying Gronwall's lemma yields that Using (3.5), we obtain by the definition of the stopping time τ n M,ε that Applying Gronwall's lemma again, we obtain that Fixing M and taking q = 2/ε we have By (3.7) and Lemma 3.2, for any R > 0, there exists a constant M such that the following inequalities hold: For such M, by (3.14) and the definition of stoping time τ n M,ε , there exists ε 0 , such that for every ε satisfying 0 < ε ≤ ε 0 , Combining (3.15), (3.16) and (3.17), we conclude that there exists ε 0 , such that for every ε satisfying 0 < ε ≤ ε 0 , Since R is arbitrary, the assertion of the lemma follows.
We can now complete the proof of our main result.

Application to examples
The main result of this paper is applicable to a large class of SPDE with local monotone coefficients, and we illustrate the applicability of our main result to the following concrete examples of SPDE models. In this section we use Λ ⊆ R d to denote an open bounded domain with a smooth boundary and C ∞ 0 (Λ, R d ) denote the set of all smooth functions from Λ to R d with compact support. For p ≥ 1, let L p (Λ, R d ), · L p be the vector valued L p -space. For any integer m > 0, let W m,p 0 (Λ, R d ) denote the standard Sobolev space on Λ with values in R d , i.e. the closure of C ∞ 0 (Λ, R d ) with respect to the following norm: For the reader's convenience, we recall the following Gagliardo-Nirenberg interpolation inequality (cf. e.g. [51, Theorem 2.1.5]). If m, n ∈ N and q ∈ [1, ∞] such that then there exists a constant C > 0 such that
Remark 4.1. If d = 1, f (x) = x and g = 0, Theorem 4.1 can be applied to the classical stochastic Burgers equation. Here, we also allow a polynomial perturbation term g in the drift of (4.2). For example, one can take g(x) = −x 3 + c 1 x 2 + c 2 x (c 1 , c 2 ∈ R) and show that (4.3)-(4.4) hold. Hence (4.2) also covers some stochastic reaction-diffusion type equations.
Besides from the example of semilinear SPDE above, we can also apply the main result to the following quasilinear SPDEs such as stochastic p-Laplace equation and stochastic porous media equation, which have been studied a lot in recent years see e.g. [25,26,27,36,37,40,41,44,47,56]) and references therein.

Stochastic p-Laplace equation
We consider the triple V := W 1,p 0 ⊂ H := L 2 (Λ) ⊂ (W 1,p 0 ) * = V * and the following stochastic p-Laplace equation where 2 ≤ p ≤ ∞, 1 ≤p ≤ p, c is positive constant and W (t) is a cylindrical Wiener process in U defined on a probability space (Ω, F , F t , P).
Consider the small time process X(εt) and let µ ε be the law of X(ε·) on C([0, T ], H), by applying our main result, we formulate the small time LDP for Eq. (4.5).

Stochastic porous media equation
The main result in this work can also be applied to stochastic porous media equation. Let where Ψ, Φ : [0, T ] × R → R are measurable and continuous in the second variable, W (t) is a cylindrical Wiener process in U defined on a probability space (Ω, F , F t , P). Suppose that there exist two constants δ > 0 and K such that It is easy to see that the drift part of Eq. (4.6) satisfies the conditions (A1)-(A3) (cf. [37,Example 5.3]). Let µ ε be the law of X(ε·) on C([0, T ], H), by applying our main result, we formulate the small time LDP for Eq. (4.6). Theorem 4.3. (stochastic porous media equation) Assume that Ψ, Φ satisfy the above conditions (4.7)-(4.8) and B satisfies the assumption 2.2, then (4.6) has a unique solution X(t) and µ ε satisfies the LDP with the rate function I(·) given by (2.2).
Remark 4.2. If we take L = ∆, the Laplace operator on a smooth bounded domain in a complete Riemannian manifold with Dirichlet boundary condition. A simple example for Ψ and Ψ satisfy the above conditions (4.7)-(4.8) is given by for some strictly positive continuous function f and bounded function g on [0,T].
In the following, we will show that the main result is also applicable to many stochastic hydrodynamical systems. We define the stokes operator A by

Stochastic 2D Navier-Stokes equation
where P H (Helmholtz-Leray projection) is the projection operator from L 2 (Λ, R 2 ) to H, and the nonlinear operator Then (4.9) can then be written in form: Now, we study the following stochastic 2D Navier-Stokes equation where W (t) is a cylindrical Wiener process in U defined on a probability space (Ω, F , F t , P).
(2) Beside the stochastic 2D Navier-Stokes equation, many other hydrodynamical systems also satisfy the local monotonicity condition (A2) and growth condition (A3). For example, Chueshov and Millet [14] have studied the well-posedness and small noise LDP for an abstract stochastic evolution equations, covering a wide class of fluid dynamical models such as stochastic 2D Boussinesq equations, stochastic 2D magneto-hydrodynamic equations, stochastic 2D magnetic Bénard problem, stochastic 3D Leray-α model and also shell models of turbulence. We refer the reader to [14] (and the references therein) for the details of these models. Note that the assumptions in [14] imply the conditions (A1)-(A3) (cf. [44, section 3.1] for a detail proof).
(3) Furthermore, below we will show that the main result in this work is also applicable to stochastic power law fluid equation and stochastic Ladyzhenskaya model.

Stochastic power law fluid equation
As one of the important models in hydrodynamical, stochastic power law fluid equation can be used to characterize the dynamic properties of various incompressible non-Newtonian fluids. We can refer to [23,45] for the study of this type of equation.
Let Λ be the open bounded domain with smooth boundary on R d (d ≥ 2), u : Λ → R d be a vector field. Define where ν > 0 is the viscosity coefficient of the fluid, p > 1 is a constant. Now we study a hydrodynamic equation with a power law property: denote the velocity field of the fluid, p is pressure, f denote the external force of the fluid, The power law fluid equation defined above is the classical Navier-Stokes equation if p = 2. Now we consider the Gelfand triple: Let P H be the projection operator on L 2 (Λ; R d ) → H. Then we can extend the operator to the map(see [39]): In particular, we have Then the power law fluid equation defined above can be written in variational form: Now study the following stochastic power law fluid equation (4.11) dX(t) = (νAX(t) + F (X(t)) + f (x))dt + B(X(t))dW (t), where W (t) is a cylindrical Wiener process in U defined on a probability space (Ω, F , F t , P). Let µ ε be the law of X(ε·) on C([0, T ], H). We will show the small time LDP and its proof by applying our main result. 2 and B satisfy the assumption 2.2. Then (4.11) has a unique solution X(t) and µ ε satisfies the LDP with the rate function I(·) given by (2.2).
According to the inequality above, for any u, v ∈ V , we have the condition (A2) holds with ρ(v) = C ε v 4q 4q−d V and α = p. Note that Then we have Let q = dp d−p , γ = d (d+2)p−2d , by the Gagliardo-Nirenberg interpolation inequality (4.1) we have v 2 , the condition (A3) holds. Therefore, the assertion follows by Lemma 2.1 and Theorem 2.2.

Stochastic Ladyzhenskaya model
The Ladyzhenskaya model is a higher order variant of the power law fluid where the stress tensor has the form τ (u) : Λ → R d ⊗ R d ,τ (u) = 2µ 0 (1 + |e(u)| 2 ) p−2 2 e(u) − 2µ 1 ∆e(u). This model was pioneered by Ladyzhenskaya [34] and further analyzed by various authors (see [62] and the references therein). Compared to the power law fluids considered above, there is an additional fourth order term div(−2µ 1 ∆e(u)) present in the equation. The fluids are shear thinning when 1 < p < 2 and shear thickening when p > 2.
Martingale and stationary solutions for this model was established by Guo et al. in [29]. Moreover, the existence of random attractors for this model has been proved for p ∈ (1, 2), i.e. shear-thinning fluids, by Duan and Zhao in [62]. Recently, the small time LDP for this model has been studied for d = 2, p ∈ (1, 5 2 ] by Lin and Sun in [35]. Consider the Gelfand triple V ⊂ H ⊂ V * , where V = u ∈ W 2,2 0 (Λ; R d ) : div(u) = 0 in Λ ; H = u ∈ L 2 (Λ; R d ) : div(u) = 0 in Λ, u · n = 0 on ∂Λ . can be extended to the well defined operators: With these preparations, we can write our model in the abstract form (4.12) dX(t) = (Ã(X(t)) + F (X(t)) + f (x))dt + B(X(t))dW (t), where W (t) is a cylindrical Wiener process in U defined on a probability space (Ω, F , F t , P). Let µ ε be the law of X(ε·) on C([0, T ], H). We then have the small time LDP by applying our main result, which covers the result in [35]. Since the proof is similar with power law fluid in Theorem 4.5, we omit it here, the reader might refer to [35] for some further detailed calculations.