Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing
aa r X i v : . [ m a t h . A P ] N ov INVARIANT MEASURES FOR NONLINEAR CONSERVATION LAWSDRIVEN BY STOCHASTIC FORCING
GUI-QIANG G. CHEN PETER H.C. PANG
Abstract.
Some recent developments in the analysis of long-time behaviors of stochasticsolutions of nonlinear conservation laws driven by stochastic forcing are surveyed. Theexistence and uniqueness of invariant measures are established for anisotropic degenerateparabolic-hyperbolic conservation laws of second-order driven by white noises. Somefurther developments, problems, and challenges in this direction are also discussed. Introduction
The analysis of long-time behaviors of global solutions is the second pillar in the theory ofpartial differential equations (PDEs), after the analysis of well-posedness. For the analysisof solution behaviors in the asymptotic regime, we seek to understand the global propertiesof the solution map, such as attracting or repelling sets, stable and unstable fixed points,limiting cycles, or chaotic behaviors that are properly determined by the entire systemrather than a given path.The introduction of noises usually serves to model dynamics phenomenologically – dy-namics too complicated to model from first principles, or dynamics only the statistics ofwhich are accurately known, or dynamics almost inherently random such as the decision ofmany conscious agents – or a combination of such behaviors. Mathematically, noises intro-duce behaviors that differ from deterministic dynamics, displaying much richer phenomenasuch as effects of dissipation, ergodicity, among others ( cf. [29, 37, 47, 53, 72, 100] and thereferences cited therein). These phenomena are of intrinsic interest.In this paper, we focus our analysis mainly on white-in-time noises. Indeed, they are themost commonly studied class of noises, though space-time white noises (such as in [26])and more general rough fluxes ( e.g. [78, 79, 92]) have also been considered. The reasonfor the prevalence of white noises as a basic model is not difficult to understand. First,Brownian motion occupies the unusual position of being simultaneously a martingale anda L´evy process. More importantly, with increments that are not only independent butalso normally distributed, it commands a level of universality by virtue of the central limittheorem. Some of the ideas, techniques, and approaches presented here can be applied toequations with more general or other forms of stochastic forcing.The organization of this paper is as follows: In §
2, the notion of invariant measures is firstintroduced, then the Krylov-Bogoliubov approach for the existence of invariant measures ispresented, and some methods for the uniqueness of invariant measures including the strong
Mathematics Subject Classification.
Primary: 35B40, 35K65, 37-02, 37A50, 37C40, 60H15. Sec-ondary: 35Q35, 58J70, 60G51, 60J65.
Key words and phrases.
Stochastic solutions, entropy solutions, invariant measures, existence, unique-ness, stochastic forcing, anisotropic degenerate, parabolic-hyperbolic equations, long-time behavior.
Feller property and the coupling method are discussed. In §
3, some recent developments inthe analysis of long-time behaviors of solutions of nonlinear stochastic PDEs are discussed.In §
4, we establish the existence of invariant measures for nonlinear anisotropic parabolic-hyperbolic equations driven by white noises. In §
5, we establish the uniqueness of invariantmeasures for the stochastic anisotropic parabolic-hyperbolic equations. In §
6, we presentsome further developments, problems, and challenges in this research direction.2.
Invariant Measures
In this section, we first introduce the notion of invariant measures for random dynamicsystems, and then present several approaches to establish the existence and uniqueness ofinvariant measures.2.1.
Notion of invariant measures.
The notion of invariant measures on a dynamicalsystem is quite straightforward. Let ( X , Σ , µ ) be a measure space, and let S : X → X be amap. System ( X , Σ , µ, S ) is a measure-preserving system if µ ( S − A ) = µ ( A ) for any A ∈ Σ.Then µ is called an invariant measure of map S .On a random dynamic system (RDS), there is an added layer of complexity. We followthe standard definitions in [1]; see also [23] for further references on RDSs and [50, 55] in aspecifically parabolic SPDE context.Let (Ω , F , P ) be a probability space, and let θ t : Ω → Ω be a collection of probability-preserving maps. A measurable RDS on a measurable space ( X , Σ) over quadruple (Ω , F , P , θ t )is a map: ϕ : R × Ω × X → X satisfying the following:(i) Measurability: ϕ is ( B ( R ) ⊗ F ⊗ Σ , Σ)–measurable;(ii) Cocycle property: ϕ ( t, ω ) = ϕ ( t, ω, · ) : X → X is a cocyle over θ : ϕ (0 , ω ) = id X ,ϕ ( t + s, ω ) = ϕ ( s, θ t ω ) ϕ ( t, ω ) , where B ( R ) denotes the collection of Borel sets in R .We think of Ω × X → Ω as a fibre bundle with fibre X . On the bundle, we have the skewproduct defined as Θ t = ( θ t , ϕ ). Then the invariant measures can be defined as follows: Definition 2.1 (Invariant measures) . An invariant measure on a RDS ϕ over θ t is aprobability measure µ on (Ω × X , F ⊗
Σ) satisfying(Θ t ) ∗ µ = µ, µ ( · , X ) = P , where (Θ t ) ∗ µ := µ ◦ (Θ t ) − is the pushforward measure.Any probability measure µ on Ω × X admits a disintegration: µ ( ω, u ) = ν ω ( u ) P ( ω ) . A measure ν ω is stationary if ϕ ( t, ω ) ∗ ν ω = ν θ t ω . Let { F t } t ≥ be a filtration associated with the RDS ϕ , i.e. an increasing sequence of σ –sub-algebras of F by which ϕ ( t, · , x ) is measurable (adapted). A Markov invariant
NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 3 measure is an invariant measure for which map: ω ν ω (Γ) is ( F , B ( R ))–measurable forany Γ ∈ B ( X ) [74, § cf. [24, 25]). In the context of dissipative PDEs perturbed by noises, it can beshown that the Hausdorff dimension of an attracting set is finite by the methods similar tothose used in the deterministic case (see [98] and the references cited therein) of linearizingthe flow and estimating the sums of global Lyapunov exponents ( cf . [25, 31, 32, 95]).2.2. Approaches for the existence of invariant measures.
There are several ap-proaches to establish the existence of invariant measures. One of the approaches is theKrylov-Bogoliubov approach, as we are going to discuss here. Another approach is viaKhasminskii’s theorem [30]. Both of them are based on the compactness property providedby the Prohorov theorem.We first recall that a sequence of probability measures { ν n } on a measure space X is tight if, for every ǫ >
0, there is a compact set K ǫ ⊆ X such that ν n ( X \ K ǫ ) ≤ ǫ uniformly in n. Lemma 2.1 (Prohorov theorem) . A tight sequence of probability measures ν n is weak*–compact in the space of probability measures ; that is, there exist a subsequence ( still de-noted ) ν n and a probability measure ν such that ν n ∗ ⇀ ν . Theorem 2.1 (Krylov-Bogoliubov theorem) . Let P t be a semigroup satisfying the F ellerproperty that φ ∈ C ( X ) implies P s φ ∈ C ( X ) for any s > , and let µ be a probabilitymeasure on X such that the measure sequence: ν T = 1 T ˆ T P ∗ t µ d t (2.1) is tight. Then there exists an invariant measure for P t . The key of its proof is that the invariant measure generated by the Krylov-Bogoliubovtheorem is the weak*–limit of ν T as T → ∞ . This can be seen as follows: By the Prohorovtheorem, the tight sequence has a weakly converging subsequence (still denoted as) { ν T } for T ranging over a unbounded subset of R , whose limit is ν ∗ . Then h P ∗ s ν ∗ , ϕ i X = h ν ∗ , P s ϕ i X = lim T →∞ T ˆ T h P ∗ t µ, P s ϕ i X d t = lim T →∞ T ˆ T h P ∗ t + s µ, ϕ i X d t = lim T →∞ T ˆ T h P ∗ t µ, ϕ i X d t + lim T →∞ T ˆ T + sT h P ∗ t µ, ϕ i X d t − lim T →∞ T ˆ s h P ∗ t µ, ϕ i X d t = lim T →∞ ˆ X ϕ ( u ) dν T . GUI-QIANG G. CHEN PETER H.C. PANG
In the above, we require the Feller property to execute the first equality, as P s ϕ has toremain continuous. With this, the second and third terms after the fourth equality abovetend to zero in the limit T → ∞ , as s is fixed.The following lemma provides two sufficient conditions for the tightness of { ν T } . Lemma 2.2.
A measure sequence { ν T } is tight if one of the following conditions holds :(i) { P ∗ t µ } is tight ;(ii) { P t } are compact for t > so that P t ( X ) ⊆ Y for almost all t > for a Banach space Y such that there is a compact embedding Y ֒ → X and thereexists C > independent of T so that T ˆ T k P t u k Y d t ≤ C for any u ∈ X ; (2.2) In addition, µ = δ u for some u ∈ X .Proof. For (i), we know that, for any ǫ >
0, there is a compact set K ǫ ⊆ X such that P ∗ t µ ( X \ K ǫ ) ≤ ǫ uniform in t > . Then ν T ( X \ K ǫ ) ≤ T ˆ T P ∗ t µ ( X \ K ǫ ) dt ≤ ǫ. For (ii), let K R = { u ∈ X : k u k Y ≤ R } . Since Y ֒ → X is compact, K R is compact in X .If u ∈ X \ K R , then k u k Y > R . Writing f ( · ) = k · k Y , then ν T ( X \ K R ) ≤ ν T ( { f ( u ) > R } ) . Applying the Markov inequality to f , we have ν T ( X \ K R ) ≤ R ˆ X f ( u ) d ν T ( u )= 1 RT ˆ T ˆ X ( P t f )( u ) d µ ( u ) d t. Since µ = δ u for some u ∈ X , then1 RT ˆ T ˆ X ( P t f )( u ) d µ ( u ) d t = 1 RT ˆ T ( P t f )( u ) d t = 1 RT ˆ T f ( u ( t )) d t. Therefore, if the temporal average (2.2) is bounded, then, for any ǫ >
0, we can choose
R > ǫ to conclude ν T ( X \ K R ) ≤ ν T ( { f ( u ) > R } ) < ǫ. In this way, a compact set K R has been found such that ν T ( X \ K R ) ≤ ǫ , which impliesthat { ν T } is tight. (cid:3) NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 5
This framework can be further refined. An example of such an extension can be foundin [21], in which the Feller property could not be proved in the context of the one-dimensional stochastic Navier-Stokes equations. Whilst the Feller condition is not available,the continuous dependence (without rates) can be shown. By using the continuous depen-dence, a class of functions,
G ⊇ C ( X ), is defined so that G is continuous on the elementsof the solution space with finite energy, though not necessarily the entire solution space.With these, it has been shown in [21] that P t is invariant under G . Then the existence ofinvariant measures is proved in two steps: First, an energy bound is employed to yield thetightness, so that the existence of a limiting measure is shown to exist; then the limitingmeasure is shown to be invariant (without invoking the Feller property) by using the con-tinuity condition imposed on G and following the arguments as in the proof of Theorem2.1.2.3. Approaches for the proof of the uniqueness of invariant measures.
It is wellknown that the invariant measures of a map form a convex set in the probability spaceon X . By the Krein-Milman theorem, a convex set is the closure of convex combinationsof its extreme points. These extreme points µ happen to be ergodic measures, which arecharacterized as the property that, for a measurable subset A ⊆ X , µ (( S − A )∆ A ) = 0 ⇐⇒ µ ( A ) = 0 or µ ( A ) = 1 , where A ∆ B := ( A \ B ) ∪ ( B \ A ).Ergodic measures heuristically carve up the solution space into essentially disjoint sub-sets, since any two ergodic measures of a process either coincide or are singular with respectto one another. This is a simple consequence of the property stated above.It also follows from the extremal property of ergodic measures that, if there are morethan one invariant measure, then there are at least two ergodic measures.There are several approaches to establish the uniqueness of invariant measures. The Strong Feller Property : This is one of the common conditions used to ensurethe uniqueness.
Definition 2.2 (Strong Feller property) . A Markov transition semigroup P t is of thestrong Feller property at time t if P t ϕ is continuous for every bounded measurable ϕ : X → R .The strong Feller property guarantees the uniqueness of invariant measures [42, 71]; seealso [30, Theorem 5.2.1], [86], and the references cited therein.The strong Feller property always holds for transition semigroups of processes associ-ated with nonlinear stochastic evolution equations with Lipschitz nonlinear coefficients andnondegenerate diffusion ( e.g. [91]). The Coupling Method : This method is a powerful tool in probability theory intro-duced in Doeblin-Fortet [39, 40], which can be used to show the uniqueness of invariantmeasures.The general argument proceeds as follows: Let X t be a Markov process with initialdistribution µ , and let Y t be an independent copy of the process with an initial distributionthat is an invariant measure µ . Then the first meeting time T is a stopping time, and the GUI-QIANG G. CHEN PETER H.C. PANG process defined by Z t = ( X t for t < T ,Y t for t ≥ T is also a copy of X t by the strong Markov property.Using the definition of Z t , we can write P ∗ t µ − µ = ( Z t ) ∗ P − ( Y t ) ∗ P = ( { t< T } Z t ) ∗ P + ( { t ≥T } Z t ) ∗ P − ( { t< T } Y t ) ∗ P − ( { t ≥T } Y t ) ∗ P = ( { t< T } Z t ) ∗ P − ( { t< T } Y t ) ∗ P . Then the total variation norm of P ∗ t µ − µ can be estimates as k P ∗ t µ − µ k T V ≤ ˆ ( { t< T } Z t ) ∗ d P ( u ) + ˆ ( { t< T } Y t ) ∗ d P ( u )= P ( { t < T } ) . Assume that T can be shown to be almost surely finite. Then, as t → ∞ , we see that P ∗ t µ → µ , and there is only one invariant measure.The coupling method has other applications in various different settings and can be im-plemented in qualitatively different ways; see also [77, 101] and the references cited therein.In our applications for the uniqueness of invariant measures for stochastic anisotropicparabolic-hyperbolic equations in § T will be slightly modified to be the time of entryinto a small ball. Moreover, instead of the use of independent copies, we take two solutionsstarting at different initial data, since our Markov processes are solutions of the equationswith pathwise uniqueness properties.First, we show in § u and v enter a given ball in finite time,almost surely. This is a stopping time. From this, by the strong Markov property, weconstruct a sequence of increasing, almost surely finite stopping times in (5.2), which arespaced at least T apart, for some T > § T >
0, if a solution starts within the samegiven ball, and the noise is uniformly small in W , ∞ x over a duration of length T , thenthe temporal average of k u ( t ) k L x over that temporal interval can be taken to be smallerthan some ǫ . Since the noise is σ ( x ) W , the uniform smallness in W , ∞ x over an interval[ T , T + T ] depends entirely on the size of W .We see that, for T >
0, the probability that the change in the noise remains smallbetween T and T + T is strictly positive. By the strong Markov property, we can replace T with any other stopping time ( e.g. the one in the sequence constructed) spaced at least T apart. Using the L –contraction, we show finally in § T , T + T ], with T inthe sequence of increasing stopping times, is bounded by the probability that the noise islarge in W , ∞ over all such sequences. This must be vanishingly small, as the probabilityis strictly less than one on each individual sequence. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 7 Nonlinear Conservation Laws driven by Stochastic Forcing
In this section, we discuss one strand of the recent developments in the analysis oflong-time behaviors of global solutions of nonlinear conservation laws driven by stochasticforcing.3.1.
The stochastic Burgers equation.
The Burgers equation is the archetypal nonlin-ear transport equation in many ways. The stochastic Burgers equation has also been usedin turbulence and interface dynamics modelling; see [28, 62, 66, 76] and the references citedtherein.The existence of a non-trivial invariant measure of the process associated to the one-dimensional Burgers equation driven by an additive spatially periodic white noise was firstderived in Sinai [96].The long-time behavior of global solutions of the Burgers equation in one spatial dimen-sion driven by space-time white noise has also been considered in the form: ∂ t u + ∂ x (cid:0) u (cid:1) = ∂ xx u + ∂ xt ˜ W , where ˜ W := ˜ W ( x, t ) is a zero-mean Gaussian process with a covariance function given by E [ ˜ W ( x, t ) ˜ W ( y, s )] = ( x ∧ y )( t ∧ s ) . Apart from the global well-posedness in L ( R ), it is known that an invariant measure forthe transition semigroup exists, for example, via an argument of [25,51] by using the ergodictheorem [28, 62]. Similar techniques have also been applied to study the two-dimensionalNavier-Stokes equations driven by space-time white noises ( e.g. [26]).Attention in the development of the stochastic Burgers equation with vanishing viscosityhas also been turned to the question of additive (spatial) noise in an equation of the form: ∂ t u + ∂ x (cid:0) u (cid:1) = ∞ X k =0 ∂ x F k ( x )d W k + ε∂ xx u. (3.1)The existence of invariant measures for equation (3.1) with ε = 0 is known ( e.g. [45]). Oneof the key points is that there is enough energy dissipation in the inviscid limiting solutionsas ε → § A [ y ( t )] = 12 ˆ t t ˙ y ( s ) d s + ˆ t t X k F k ( y ( s )) d W k ( s )are the curves that satisfy Newton’s equations for the characteristics. These minimizershave an existence and uniqueness property with probability one. Through this, a one force-one solution principle has been shown, in which the random attractor consists of a singletrajectory almost surely, which in turn leads to the proof of the existence of an invariantmeasure for (3.1). GUI-QIANG G. CHEN PETER H.C. PANG
Kinetic formulation.
The theory of kinetic formulation has been developed overthe last three decades ( cf.
Perthame [90] and the references cited therein). In particular,the compactness of entropy solutions of multidimensional scalar hyperbolic conservationlaws with a genuine nonlinearity was first established by Lions-Perthame-Tadmor in [80]via combining the kinetic formulation with corresponding velocity averaging. The velocityaveraging is a technique whereby a genuine nonlinearity condition ( i.e. a non-degeneracycondition on the nonlinearity) can be shown to imply the compactness (or even improvedfractional regularity under a stronger condition) of solutions via the kinetic formulation, asseen in subsequent sections, especially in condition (3.14).We discuss the kinetic formulation in the context of scalar hyperbolic conservation lawshere.One of the inspirations for a kinetic formulation originated from the kinetic theory ofgases. One starts with a simple step function as the kinetic function : χ r ( ξ ) := χ ( ξ, r ) = < ξ < r, − r < ξ < , . Then, for any η ∈ C , the following representation formula holds: ˆ R η ′ ( ξ ) χ u ( ξ ) d ξ = η ( u ) − η (0) . (3.2)A simple combination of kinetic functions yields | u − v | = ˆ (cid:0) | χ u | + | χ v | − χ u χ v (cid:1) d ξ. (3.3)This provides an approach to the derivation of the L –contraction between two solutions,by estimating the terms on the right.There are several variations on the form of the kinetic function. Since | u − v | = ( u − v ) + + ( v − u ) + , it suffices for a variation, or combinations, of the kinetic function to capture( u − v ) + , which is simpler than (3.3). This can be done by considering the following kineticfunction: ˜ χ u := ˜ χ ( ξ, u ) = 1 − H ( ξ − u ) = H ( u − ξ ) , where H = [0 , ∞ ) is the Heaviside step function. We then have the representation formula: η ( u ) = ˆ R η ′ ( ξ ) ˜ χ u ( ξ ) d ξ for η ∈ C with η ( −∞ ) = 0 . In particular, ( u − v ) + = ˆ ˜ χ u ( ξ ) (cid:0) − ˜ χ v ( ξ ) (cid:1) d ξ. (3.4)Such a kinetic function has been popularized by [80] and has been used, inter alia, in[12, 33–35], and even as far back as [61].The usefulness of the kinetic function can be seen in the kinetic formulation of scalarconservation laws, in which the kinetic variable takes the place of the solution in the non-linear coefficients so that a degree of linearity is restored for analysis. In this formulation,many powerful linear methods such as the Fourier transform become not only applicable,but also natural. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 9
Following Chen-Pang [12], we now derive the kinetic formulation of nonlinear anisotropicparabolic-hyperbolic equations of second order: ∂ t u + ∇ · F ( u ) = ∇ · ( A ( u ) ∇ u ) + σ ( u, x ) ∂ t W, (3.5)where F is a locally Lipschitz vector flux function of polynomial growth, A is a positivesemi-definite matrix function with continuous entries of polynomial growth, and ∇ = ∇ x :=( ∂ x , · · · , ∂ x d ).Consider the vanishing viscosity approximation to (3.5): ∂ t u ε + ∇ · F ( u ε ) = ∇ · (cid:0) ( A ( u ε ) + ε I) ∇ u ε (cid:1) + σ ( u ε , x ) ∂ t W, where I is the identity matrix. Let η ∈ C be an entropy with η (0) = 0. Using the Itoformula, we have ∂ t η ( u ε ) = − η ′ ( u ε ) ∇ · F ( u ε ) + η ′ ( u ε ) σ ( u ε , x ) ∂ t W + 12 η ′′ ( u ε ) σ ( u ε , x )+ ∇ · (cid:0) η ′ ( u ε ) A ( u ε ) ∇ u ε (cid:1) − η ′′ ( u ε ) A ( u ε ) : (cid:0) ∇ u ε ⊗ ∇ u ε (cid:1) + ε ∆ η ( u ε ) − εη ′′ ( u ε ) |∇ u ε | , where we have used the notation: A : B = P i,j a ij b ij for matrices A = ( a ij ) and B = ( b ij )of the same size.Applying the representation formula (3.2) yields ∂ t ˆ η ′ ( ξ ) χ u ε d ξ = − ∇ · (cid:16) ˆ η ′ ( ξ ) F ′ ( ξ ) χ u ε d ξ (cid:17) + h σ ( · , x ) ∂ t W ( t ) δ ( · − u ε ) , η ′ ( · ) i + ∇ : (cid:16) ˆ η ′ ( ξ ) A ( ξ ) χ u ε d ξ (cid:17) − h A ( · ) : ( ∇ u ε ⊗ ∇ u ε ) δ ( · − u ε ) , η ′′ ( · ) i− h ε |∇ u ε | δ ( · − u ε ) , η ′′ ( · ) i + 12 h σ ( · , x ) δ ( · − u ε ) , η ′′ ( · ) i + ε ∆ (cid:16) ˆ η ′ ( ξ ) χ u ε d ξ (cid:17) . Assume that u ε ( x, t ) → u ( x, t ) a.e. almost surely as ε →
0. Then, taking η ′ ( ξ ) as a testfunction and letting ε →
0, we arrive heuristically at the formulation: ∂ t χ u + F ′ ( ξ ) · ∇ χ u = A ( ξ ) : ∇ χ u + σ ( ξ, x ) ∂ t W ( t ) δ ( ξ − u ) + ∂ ξ ( m u + n u − p u ) , (3.6)which holds in the distributional sense, where m u , n u , and p u are Radon measures that arethe limits of the following measure sequences: ε |∇ u ε | δ ( ξ − u ε ) ⇀ m u , A ( ξ ) : (cid:0) ∇ u ε ⊗ ∇ u ε (cid:1) δ ( ξ − u ε ) ⇀ n u , σ ( ξ, x ) δ ( ξ − u ε ) ⇀ p u . The Radon measure m u is the kinetic dissipation measure and n u is the parabolic defectmeasure , which capture the dissipation from the vanishing viscosity terms and the degen-erate parabolic terms, respectively. In addition, the Radon measure p u = 12 σ ( ξ, x ) δ ( ξ − u )arises from the Itˆo correction. As A is positive semi-definite, it is manifest that m u , n u ,and p u are all non-negative.More precisely, the parabolic defect measure n u ≥ ϕ ∈ C ( R × R d × R + ), n u ( ϕ ) = ˆ R + ˆ R d ϕ ( u ( x, t ) , x, t ) (cid:12)(cid:12) ∇ x · (cid:0) ˆ u α ( ζ ) dζ (cid:1)(cid:12)(cid:12) d x d t. (3.7)The kinetic dissipation measure m u ≥ B cR ⊂ R as the complement of the ball of radius R ,lim R →∞ E (cid:2) ( m u + n u )( B cR × T d × [0 , T ]) (cid:3) = 0; (3.8)(ii) For any ϕ ∈ C ( R × R d ), ˆ R × R d × [0 ,T ] ϕ ( ξ, x ) d( m u + n u )( ω ; ξ, x, t ) ∈ L (Ω) (3.9)admits a predictable representative (in the L –equivalence classes of functions).Then, following Chen-Pang [12], we introduce the notion of kinetic solutions: Definition 3.1 (Stochastic kinetic solutions) . A function u ∈ L p (Ω × [0 , T ]; L p ( R d )) ∩ L p (Ω; L ∞ ([0 , T ]; L p ( R d )))is called a kinetic solution of (3.5) with initial data: u | t =0 = u , provided that u satisfiesthe following:(i) ∇ · (cid:0) ´ u α ( ξ ) dξ (cid:1) ∈ L (Ω × R d × [0 , T ]);(ii) For any bounded ϕ ∈ C ( R ), the Chen-Perthame chain rule relation in [13] holds: ∇ · (cid:16) ˆ u ϕ ( ξ ) α ( ξ ) d ξ (cid:17) = ϕ ( u ) ∇ · (cid:16) ˆ u α ( ξ ) d ξ (cid:17) (3.10)in D ′ ( T d ) and almost everywhere in ( t, ω ).(iii) There is a kinetic measure m u ≥ P - a.e. such that, given the parabolic defectmeasure n u , the following holds almost surely: For any ϕ ∈ C ∞ c ( R , R d × [0 , T )), − ˆ T ¨ χ ( ξ, u ) ∂ t ϕ d ξ d x d t − ¨ χ ( ξ, u ) ϕ ( ξ, x,
0) d ξ d x = ˆ T ¨ χ ( ξ, u ) F ′ ( ξ ) · ∇ ϕ d ξ d x d t + ˆ T ¨ χ ( ξ, u ) A ( ξ ) : ∇ ϕ d ξ d x d t + ˆ T ¨ ϕ ξ d( m u + n u )( ξ, x, t ) − ˆ T ˆ ϕ u ( u, x, t ) σ ( u, x ) d x d t − ˆ T ˆ ϕ ( u, x, t ) σ ( u, x ) d x d W almost surely . (3.11) NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 11
Equation (3.11) is obtained by testing (3.6) with ϕ and using the chain rule (3.10).3.3. General scalar hyperbolic conservation laws driven by stochastic forcing.
In Feng-Nualart [49], the well-posedness was studied for the one-dimensional scalar conser-vation laws driven by white noise: ∂ t u + ∂ x F ( u ) = ˆ z ∈ Z σ ( u, x ; z ) d z W ( t, z ) , where Z is a metric space, and W is a space-time Gaussian noise martingale randommeasure with respect to a filtration { F t } t ≥ satisfying E (cid:2) W ( t, A ) ∩ W ( t, B ) (cid:3) = µ ( A ∩ B ) t for measurable sets A, B ⊂ Z , with a σ -finite Borel measure µ on Z . The well-posednesstheory was developed around the notion of strong stochastic entropy solutions introducedin Definition 2.6 in [49] when t ∈ [0 , T ) for any fixed T ∈ (0 , ∞ ). In addition to the usualdefinition of entropy solutions, the following further conditions on the solution, u = u ( x, t ),for t ∈ [0 , T ] are required:For any smooth approximation function β ( u ) of function u + on R and any ϕ ∈ C ∞ ( R d × R d ) with ϕ ≥
0, and for any F t –adapted function v satisfying sup ≤ t ≤ T E [ k v k pL px ] < ∞ ,there exists a deterministic function { A ( s, t ) : 0 ≤ s ≤ t } such that the functional f ( r, z, u, y ) := ˆ R d β ′ ( v ( x, r ) − u ) σ ( v ( x, r ) , x ; z ) ϕ ( x, y ) d x satisfies E (cid:2) ˆ ˆ ( s,t ] × Z f ( r, z, u ( y, t ) , y )d W ( r, z ) d y (cid:3) ≤ E (cid:2) ˆ ( s,t ] × Z ˆ ∂ v f ( r, z, v ( y, r ) , y ) σ ( y, u ( y, r ); z ) d y d r d µ ( z ) (cid:3) + A ( s, t ) , and that there is a sequence of partitions of [0 , T ] so thatlim max | t i +1 − t i |→ m X i =1 A ( t i , t i +1 ) = 0 . This notion of a solution addresses the problem that, in any direct adaptation of thedeterministic notion of entropy solutions, one encounters the question of adaptiveness ofthe Itˆo integral in the noise-noise interaction. With this notion, in [49], the L –contractionand comparison estimates of strong stochastic entropy solutions in any spatial dimensionwere established, while the existence of solutions is limited to the one-dimensional casebased on the compensated compactness argument in Chen-Lu [11].In Chen-Ding-Karlsen [9], the existence theory for strong stochastic entropy solutionswas established for any spatial dimension with the key observation that the following BV bound is a corollary from the L –contraction inequality: E (cid:2) k u ( t ) k BV (cid:3) ≤ E (cid:2) k u k BV (cid:3) , which provides the strong compactness required for the existence theory in any spatialdimension. More precisely, the following theorem holds: Theorem 3.1.
Consider the Cauchy problem of the equation : ∂ t u + ∇ · F ( u ) = σ ( u ) ∂ t W (3.12) with initial condition : u | t =0 = u , (3.13) satisfying E (cid:2) k u k pL p + k u k BV (cid:3) < ∞ for p > , where F is a locally Lipschitz function of polynomial growth and σ is a globally Lipschitzfunction. Then there exists a unique strong stochastic entropy solution of the Cauchyproblem (3.12) – (3.13) satisfying E (cid:2) k u ( t ) k BV (cid:3) ≤ E (cid:2) k u k BV (cid:3) . This existence theory in L p ∩ BV can also been extended to the second-order equations(3.5) as established in Chen-Pang [12], including the case with heterogeneous flux functions F = F ( u, x ) ( i.e. the space-translational variant case).A well-posedness theory can also be developed for kinetic solutions to the multidimen-sional scalar balance laws with stochastic force (3.12), by employing the Gy¨ongy-Krylovframework where the existence of a martingale solution with pathwise uniqueness guaran-tees the strong existence; see [34]. In particular, the existence of martingale solutions canbe proved via the notion of kinetic solutions. These results can be extended ( e.g. [33, 67])to encompass degenerate parabolic equations: ∂ t u + ∇ · F ( u ) − ∇ · ( A ( u ) ∇ u ) = σ ( u ) ∂ t W. A well-posedness theory has also been established based on the viscosity solutions (suchas in [3]). To achieve this, the difficulties caused by the noise-noise interaction that hasa non-zero correlation for the multiplicative noise case are avoided by directly comparingtwo entropy solutions to a viscosity solution.In Karlsen-Størrensen [70], these different viewpoints have been partially reconciled viaa Malliavin viewpoint, in which the constant in the Kruzhkov entropy is interpreted as aMalliavin differentiable variable.Long-time asymptotic results concerning the existence and uniqueness of invariant mea-sures have followed the well-posedness theory. Concerning the stochastic balance law: ∂ t u + ∇ · F ( u ) = Φ( x ) d B, with evolution on torus T d , where B = P k e k W k is a cylindrical Wiener process, { e k } is acomplete orthonormal basis of a Hilbert space, Φ is a Hilbert-Schmidt operator given byΦ( x ) = P k g k ( x ) e k , and g k ( x ) satisfies ˆ T d g k ( x ) d x = 0 , the existence and uniqueness of invariant measures were shown in [35]. In this case, thenoise is additive ; that is, it depends only on the spatial variable, but is independent of thesolution – a point to which we will return.These results can be summarized as follows: NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 13
Theorem 3.2.
Let F satisfy the non-degeneracy condition : For some b < and a constant C > , δ ( ε ) := ˆ ∞ e − t sup τ ∈ R , | ˆ k | =1 L ( { ξ : | F ′ ( ξ ) · ˆ k + τ | ≤ εt } ) d t ≤ Cε b (3.14) for the Lebesgue measure L on R , in addition to the condition that | F ′′ ( ξ ) | . | ξ | + 1 . Thenthere exits an invariant measure to the process. Furthermore, if | F ′′ ( ξ ) | . is bounded,then the invariant measure is unique. The bounds for the spaces on which the invariant measures are supported have alsobeen derived. This result has been obtained by employing the velocity averaging. It hasbeen built also on the related ideas of kinetic solutions, which is first applied to the velocityaveraging in the deterministic context. They avoided the question of the Fourier transformsof the Wiener process by introducing regularizing operators.Similar results were also derived for ∂ t u + ∇ · ( F ( u ) ◦ d W ) = 0 , by further employing the conservative form as considered in Lions-Perthame-Souganidis[78, 79]; see Gess-Souganidis [59]. A generalization of this with a degenerate parabolicterm ∇ · ( A ( u ) ∇ u ) has also been considered in [48, 60]. In particular, Fehrman-Gess [48]investigated the well-posedness and continuous dependence of the stochastic degenerateparabolic equations of porous medium type, including the cases with fast diffusion andheterogeneous fluxes.By using the methods developed in [57, 64, 65, 74] and developing the probabilistic Gron-wall inequality based on delicate reasoning about a stopping time, such MHD equationsdriven by additive noise of zero spatial average in the vanishing Rossby number and vanish-ing magnetic Reynold’s number limit were also shown to have a unique invariant measure(that is necessarily ergodic) in [56].4. Stochastic Anisotropic Parabolic-Hyperbolic EquationsI: Existence of Invariant Measures
In this section, we present an approach for establishing the existence of invariant mea-sures for nonlinear anisotropic parabolic-hyperbolic equations driven by stochastic forcing: ∂ t u + ∇ · F ( u ) = ∇ · ( A ( u ) ∇ u ) + σ ( x ) ∂ t W, (4.1)where A is positive semi-definite, and σ has zero average over T d . The main focus ofthis section is on the presentation of the approach, so we do not seek the optimality of theresults, while the results presented below can be further improved by refining the argumentsand technical estimates required for the approach which is out of the scope of this section.More precisely, we establish the following theorem: Theorem 4.1.
Let F and A satisfy the nonlinearity-diffusivity condition : There exist β ∈ (1 , , κ ∈ (0 , , and C > , independent of λ , such that sup τ ∈ R , | ˆ k | =1 ˆ λ ( A ( ξ ) : ˆ k ⊗ ˆ k + λ )( A ( ξ ) : ˆ k ⊗ ˆ k + λ ) + λ β | F ′ ( ξ ) · ˆ k + τ | d ξ =: η ( λ ) ≤ Cλ κ → as λ → . In addition, let F and A satisfy the condition : | F ′′ ( ξ ) | . | ξ | + 1 , | A ′ ( ξ ) | . | ξ | + 1 . (4.3) Then there exists an invariant measure to the process associated with the solutions to (4.1) . The approach is motivated by Debussche-Vovelle [35] by extending the case from first-order scalar balance laws to the second-order degenerate parabolic-hyperbolic equation(4.1). The first-order case is handled in [35], based on the velocity averaging and builton Lemma 2.4 of Bouchut-Desvillettes [6]. In our approach, we require a modified versionof this lemma, which is incorporated into the calculation that allows us to exploit thecancellations in an oscillatory integral in this more general case than the first-order case.We now proceed to prove the theorem as follows:(i) First we incorporate regularizing operators into the equation in order to exploit thebounds that can be provided in the Duhamel representation of the solution.(ii) We separate the Duhamel representation of the solution into four different sum-mands, the W s,q norms of which we estimate.(iii) Adding these estimates together by the triangle inequality and using the compactinclusion of W s,q into a suitable L q norm allow us to invoke the Krylov-Bogoliubovmachinery described in § ∂ t χ u + (cid:0) F ′ ( ξ ) · ∇ − A ( ξ ) : ∇ ⊗ ∇ (cid:1) χ u = ∂ ξ ( m u + n u − p u ) + σ ( x ) δ ( ξ − u ) ∂ t W. (4.4)In order to handle the two measures: m u + n u − p u and σ ( x ) δ ( ξ − u ), we need to regularizethe operators as in [35], by adding γ ( − ∆) α + θ I to each side: ∂ t χ u + (cid:0) F ′ ( ξ ) · ∇ − A ( ξ ) : ∇ ⊗ ∇ + γ ( − ∆) α + θ I (cid:1) χ u = (cid:0) γ ( − ∆) α + θI (cid:1) χ u + ∂ ξ ( m u + n u − p u ) + σ ( x ) δ ( ξ − u ) ∂ t W (4.5)for α = β − β ∈ (0 , ) for some β ∈ (1 ,
2) required in the nonlinearity-diffusivity condition(4.2).We adapt the semigroup approach. There are specific reasons to include these regular-izing operators: In order to estimate the measure, σ ( x ) δ ( ξ − u ) ∂ t W , we require a spatialregularization provided by ( − ∆) α and temporal decay provided by θI .More specifically, let S ( t ) be the semigroup of operator ∂ t + (cid:0) F ′ ( ξ ) · ∇ − A ( ξ ) : ∇ ⊗ ∇ + γ ( − ∆) α + θ I (cid:1) : S ( t ) f ( x ) = e − ( F ′ ( ξ ) ·∇− A ( ξ ): ∇⊗∇ + γ ( − ∆) α + θI ) t f = e − θt (cid:0) e t A ( ξ ): ∇⊗∇− tγ ( − ∆) α f (cid:1) ( x − F ′ ( ξ ) t ) for any f = f ( x ) . (4.6)Then we can express the solution, χ u , to the kinetic formulation in the mild formulation: χ u = S ( t ) χ u ( ξ, x,
0) + ˆ t S ( s )( γ ( − ∆) α − θI ) χ u ( ξ, x, t − s ) d s + ˆ t S ( t − s ) ∂ ξ ( m u + n u − p u )( ξ, x, s ) d s + ˆ t S ( t − s ) σ ( x ) δ ( ξ − u ( x, s )) d W s . NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 15
This leads to the decomposition: u = u + u ♭ + M + M , (4.7)where u ( x, t ) = ˆ S ( t ) χ u ( ξ, x,
0) d ξ, (4.8) u ♭ ( x, t ) = ˆ ˆ t S ( s )( γ ( − ∆) α − θI ) χ u ( ξ, x, t − s ) d s d ξ, (4.9) h M , ϕ i = ˆ t ˆ h ∂ ξ ( m u + n u − p u )( · , x, t − s ) , S ∗ ( s ) ϕ i d x d s, (4.10) h M , ϕ i = ˆ t ˆ h δ ( · − u ( x, s )) , S ∗ ( t − s ) ϕ i σ ( x ) d x d W s , (4.11)where S ∗ ( t ) is the dual operator of the semigroup operator S ( t ), and ϕ ∈ C ( T d ).We now estimate each of the four terms separately in each subsection: The first two inte-grals are essentially “deterministic” parts and estimated by the velocity averaging method,and the final two integrals incorporate stochastic elements and are treated by a kernelestimate on semigroup S ( t ).4.1. Analysis of u . Notice that the local Fourier transform in x ∈ T d for any periodicfunction g ( x, · ) in x with period P = ( P , · · · , P d ) is:ˆ g ( k, · ) = 1 | T d | ˆ T d g ( x, · ) e − ix · k d x, where frequencies k = ( k , · · · , k d ) are discrete: k i = 2 πP i n i , n i = 0 , ± , ± , · · · , i = 1 , · · · , d. Taking the Fourier transform in x and integrating in ξ , we have c u ( k, t ) = ˆ ˆ S ( t ) c χ u ( ξ, k,
0) d ξ = ˆ e − ( iF ′ ( ξ ) · k + A ( ξ ):( k ⊗ k )+ ω k | k | ) t c χ u ( ξ, k,
0) d ξ, where ω k = γ | k | α − + θ | k | − .For simplicity, we denote ˆ k = k | k | and A = A ( ξ, ˆ k ) = A ( ξ ) : ˆ k ⊗ ˆ k . Then we square theabove and integrate in t from 0 to T to obtain ˆ T | c u ( k, t ) | d t = ˆ T (cid:12)(cid:12)(cid:12)(cid:12) ˆ e − ( iF ′ ( ξ ) · ˆ k + A ( ξ, ˆ k ) | k | + ω k ) | k | t c χ u ( ξ, k,
0) d ξ (cid:12)(cid:12)(cid:12)(cid:12) d t ≤ | k | ˆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ { s> } e − ( iF ′ ( ξ ) · ˆ k + A ( ξ, ˆ k ) | k | + ω k ) s c χ u ( ξ, k,
0) d ξ (cid:12)(cid:12)(cid:12)(cid:12) d s. (4.12)Notice that it is impossible to extract the entire non-oscillatory part of the exponentialfrom the integral in ξ , as was done with the lemma of Bouchut-Desvillettes [6]. However, by extending the range of integration over all R to make the function in s smoother so thatits transform has better decay properties, we can partially exploit the cancellations later: ˆ T | c u ( k, t ) | d t ≤ | k | ˆ ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ e iF ′ ( ξ ) · ˆ ks e − ( ω k + A| k | ) | s | c χ u ( ξ, k,
0) d ξ (cid:12)(cid:12)(cid:12)(cid:12) d s. (4.13)We can evaluate the temporal Fourier transform of the integrand explicitly: F − n e iF ′ ( ξ ) · ˆ ks e − ( ω k + A| k | ) | s | o ( τ ) = − A| k | + ω k )( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | . Next, using the Parseval identity in the temporal variable and the Cauchy-Schwarzinequality, we have ˆ T (cid:12)(cid:12)c u ( k, t ) (cid:12)(cid:12) d t ≤ | k | ˆ ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ e iF ′ ( ξ ) · ˆ ks e − ( ω k + A| k | ) | s | c χ u ( ξ, k,
0) d ξ (cid:12)(cid:12)(cid:12)(cid:12) d s = 1 | k | ˆ ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F − (cid:26) ˆ e iF ′ ( ξ ) · ˆ ks e − ( ω k + A| k | ) | s | c χ u ( ξ, k,
0) d ξ (cid:27) ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) d τ = 4 | k | ˆ ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ A| k | + ω k ( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | c χ u ( ξ, k,
0) d ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d τ ≤ | k | ˆ ∞−∞ ˆ (cid:12)(cid:12)c χ u ( ξ, k, (cid:12)(cid:12) A| k | + ω k ( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | d ξ ! × ˆ A| k | + ω k ( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | d ξ ! d τ ≤ | k | ω k ˆ (cid:12)(cid:12)c χ u ( ξ, k, (cid:12)(cid:12) ˆ A| k | + ω k ( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | d τ ! d ξ × sup τ ˆ ω k ( A| k | + ω k )( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | d ξ. Notice that the integral ˆ A| k | + ω k ( A| k | + ω k ) + | F ′ ( ξ ) · ˆ k + τ | d τ is a constant for fixed ξ by the translation invariance of d τ .Now invoking (4.2) and setting λ = ω k | k | , we have ˆ T | c u ( k, t ) | d t ≤ C | k | ω k η ( ω k | k | ) ˆ | c χ u ( ξ, k, | d ξ for some constant C depending on γ and θ . That is, ˆ T | k | κ ω − κk | c u ( k, t ) | d t ≤ C ˆ | c χ u ( ξ, k, | d ξ. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 17
Since u has null average over T d , c u (0 , t ) = ˆ T d u ( x, t ) d x = ˆ c χ u ( ξ, ,
0) d ξ = ¨ χ u ( ξ, x,
0) d ξ d x = ˆ T d u ( x ) d x = 0 . (4.14)Then, summing over all the discrete frequencies k with | k | 6 = 0, using the Planchereltheorem again — in space this time — and noting that ω k ≥ γ | k | α − , we have the estimate ˆ T k u k H (1 − α ) κ + αx d t ≤ C k u k L x . (4.15)4.2. Analysis of u ♭ . The calculation is similar: ˆ T | b u ♭ ( k, t ) | d t = ˆ T (cid:12)(cid:12)(cid:12)(cid:12) ˆ ˆ t ˆ S ( s )( γ | k | α + θI ) c χ u ( ξ, k, t − s ) d s d ξ (cid:12)(cid:12)(cid:12)(cid:12) d t = ˆ T (cid:12)(cid:12)(cid:12)(cid:12) ˆ T { t − s ≥ } ˆ e − ( iF ′ ( ξ ) · k + ω k | k | + A ( ξ ): k ⊗ k ) s ω k | k | c χ u ( ξ, k, t − s ) d ξ d s (cid:12)(cid:12)(cid:12)(cid:12) d t ≤ (cid:16) ˆ ∞ ω k | k | e − ω k | k | s d s (cid:17) × ˆ T (cid:16) ˆ T (cid:12)(cid:12)(cid:12) ˆ e − ( iF ′ ( ξ ) · k + ω k | k | / A ( ξ ): k ⊗ k ) s p ω k | k | c χ u ( ξ, k, t ) d ξ (cid:12)(cid:12)(cid:12) d s (cid:17) d t, where we have used the Cauchy-Schwarz inequality and extended the domain of the innertemporal integration to [0 , ∞ ).This leaves us in the exact position of Eq. (4.12) with an additional temporal integralin t (applied only to the kinetic function c χ u ) and an additional factor of | k | ω k . Therefore,we can conclude as in Eq. (4.15) by using the zero-spatial average property (4.14) and ω k ≤ ( γ + θ ) | k | α − that ˆ T k u ♭ ( t ) k H (1 − α ) κx d t ≤ C ˆ T k u ( t ) k L x d t. (4.16) Remark . Condition (4.2) is reminiscent of the nonlinearity condition given in the de-terministic setting by Chen-Perthame [14]. If we discard the regularising operator ( − ∆) α in (4.5), i.e. by setting α = 0, then β = 1 in (4.2). On the other hand, we can choose β sufficiently close to 2 so that (4.2) holds, by selecting α close to . For both cases, weare able to conclude that the u ♭ -part of the solution operator is compact. However, as wewill see below, the regularizing effect of ( − ∆) α is crucial in estimating (4.10)–(4.11) inthe way as we do, via (4.19), in the next subsections. As the two terms (4.10) and (4.11)arise from the martingale and the Itˆo approximation, respectively, this decay requirementbeyond o (1) does not appear in the deterministic setting. Analysis of M . Next we turn to the analysis of the two measures M and M .For this, we follow [35] closely, since the only difference is the parabolic defect measure,which has the same sign as the kinetic dissipation measure, and the magnitude of thekinetic dissipation measure is never invoked in [35]. In this and the following sections, werepeatedly apply bound (4.19) in order to pursue the compactness estimates.From (4.6), we see ∂ ξ (cid:0) S ∗ ( t − s ) h ( ξ, x ) (cid:1) = ( t − s ) F ′′ ( ξ ) · ∇ ( S ∗ ( t − s ) h ) + A ′ ( ξ ) : ∇ ( S ∗ ( t − s ) h ) + S ∗ ( t − s ) ∂ ξ h. (4.17)Then we have h M , ϕ i = − ˆ t ¨ ∂ ξ ( S ∗ ( s ) ϕ ) d( m u + n u − p u )( ξ, x, t − s )= ˆ t ¨ ( t − s ) F ′′ ( ξ ) · ∇ ( S ∗ ( t − s ) ϕ ) d( m u + n u − p u )( ξ, x, t − s )+ ˆ t ¨ A ′ ( ξ ) : ∇ ( S ∗ ( t − s ) ϕ ) d( m u + n u − p u )( ξ, x, t − s ) . (4.18)Now we show the following total variation estimate. Lemma 4.1.
Let u : T d × [0 , T ] × Ω be a solution with initial data u . Let ψ ∈ C c ( R ) beany nonnegative and compactly supported continuous function, and Ψ = ´ s ´ r ψ ( t ) d t d r .Then E (cid:2) ˆ T d × [0 ,T ] × R ψ ( ξ ) d | m u + n u − p u | ( ξ, x, t ) (cid:3) ≤ D E (cid:2) k ψ ( u ) k L x,t (cid:3) + E (cid:2) k Ψ( u ) k L x (cid:3) , where D := k σ k L ∞ ( T ) .Proof. The proof is the same as that in [35] and involves bounding | m u + n u − p u | ≤ m u + n u + p u , so that E (cid:2) ˆ T ¨ ψ ( ξ ) d | m u + n u − p u | ( ξ, x, t ) (cid:3) ≤ E (cid:2) ˆ T ¨ ψ ( ξ ) d( m u + n u − p u )( ξ, x, t ) (cid:3) + 2 E (cid:2) ˆ T ¨ ψ ( ξ ) d p u ( ξ, x, t ) (cid:3) = E (cid:2) − ˆ Ψ( u ) d x (cid:12)(cid:12)(cid:12)(cid:12) T (cid:3) + E (cid:2) ˆ T ˆ σ ( x ) ψ ( u ) d x d t (cid:3) , by using the kinetic equation in the sense of (3.11). Now, using the non-negativity of ψ ,we have E (cid:2) ˆ T ¨ ψ ( ξ ) d | m u + n u − p u | ( ξ, x, t ) (cid:3) ≤ E (cid:2) ˆ Ψ( u ) d x (cid:3) + D E (cid:2) ˆ T ˆ ψ ( u ) d x d t (cid:3) . (cid:3) This estimate is quite crude, as one does not take the cancellation between measures m u + n u and p u , both non-negative, into account. Since there is no available way toquantify m u + n u , this is the best possible at the moment. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 19
In addition to a total variation estimate, we also require the kernel estimate: (cid:13)(cid:13)(cid:13) ( − ∆) ˆ β e ( A : ∇⊗∇− γ ( − ∆) α ) t (cid:13)(cid:13)(cid:13) L px,ξ → L qx,ξ ≤ C ( γt ) − d α (cid:16) p − q (cid:17) − ˆ β α . (4.19)The reason for the no additional improvement over the estimate for operator e t A : ∇⊗∇ isthat we have not specified how degenerate A is — it may well be simply the zero matrix.It is the use of this kernel estimate that necessitates the inclusion of the regularizations γ ( − ∆) α + θI .By the kernel estimate (4.19), we have k ( − ∆) ˆ β ∇ ( S ∗ ( t ) ϕ ) k L ∞ x,ξ ≤ C ( γt ) − µ e − θ t k ϕ k L px , k ( − ∆) ˆ β ∇ ( S ∗ ( t ) ϕ ) k L ∞ x,ξ ≤ C ( γt ) − µ − α e − θ t k ϕ k L px , where µ := ˆ β +12 α + d ( α − αp ′ ) for p ′ >
1, and the universal constant C is independent of γ and θ .Inserting these estimates into (4.18), we have the estimate: E (cid:2) ˆ T h ( − ∆) ˆ β M , ϕ i d t (cid:3) = E (cid:2) ˆ T ˆ t ¨ ( − ∆) ˆ β F ′′ ( ξ ) · ∇ ( S ∗ ( t − s ) ϕ ) d( m u + n u − p u )( ξ, x, t − s ) d t + ˆ T ˆ t ¨ ( − ∆) ˆ β A ′ ( ξ ) : ∇ ( S ∗ ( t − s ) ϕ ) d( m u + n u − p u )( ξ, x, t − s ) d t (cid:3) ≤ E (cid:2) ˆ T ˆ k ( − ∆) ˆ β ∇ ( S ∗ ( t − s ) ϕ ) k ∞ | F ′′ ( ξ ) | ( t − s ) d | m u + n u − p u | ( ξ, x, s ) d t (cid:3) + E (cid:2) ˆ T ˆ k ( − ∆) ˆ β ∇ ( S ∗ ( t − s ) ϕ ) k ∞ | A ′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) d t (cid:3) . By the presence of factor e − θ ( t − s ) , we can also bound the outer temporal integral byusing the definition of the Gamma function:Γ( z ) = ˆ ∞ x z − e − x d x so that, taking h· , ·i as the L p ′ ( T d )– L p ( T d ) pairing, E (cid:2) ˆ T h ( − ∆) ˆ β M , ϕ i d t (cid:3) ≤ ˆ T ( γτ ) − µ e − θτ d τ E (cid:2) ˆ R × T d × [0 ,T ] k ϕ k L px | F ′′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) + ˆ T ( γτ ) − µ − α e − θτ d τ E (cid:2) ˆ R × T d × [0 ,T ] k ϕ k L px | A ′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) ≤ Cθ µ +1 γ − µ | Γ( − µ + 1) | E (cid:2) ˆ R × T d × [0 ,T ] k ϕ k L px | F ′′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) + Cθ µ − − α γ − µ − α (cid:12)(cid:12) Γ( − µ + 1 − α ) (cid:12)(cid:12) × E (cid:2) ˆ R × T d × [0 ,T ] k ϕ k L px | A ′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) . By duality, the total variation estimate, and the sublinearity of F ′′ and A ′ , we have E (cid:2) k M k L t W ˆ β,p ′ x (cid:3) ≤ Cθ µ +1 γ − µ | Γ( − µ + 1) | E (cid:2) ˆ R × T d × [0 ,T ] | F ′′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) + Cθ µ − − α γ − µ − α (cid:12)(cid:12) Γ( − µ + 1 − α ) (cid:12)(cid:12) E (cid:2) ˆ R × T d × [0 ,T ] | A ′ ( ξ ) | d | m u + n u − p u | ( ξ, x, s ) (cid:3) ≤ C (cid:16) θ µ +1 γ − µ | Γ( − µ + 1) | + θ µ − − α γ − µ − α (cid:12)(cid:12) Γ( − µ + 1 − α ) (cid:12)(cid:12)(cid:17) × (cid:16) ˆ T E (cid:2) k u ( t ) k L x (cid:3) d t + E (cid:2) k u k L x (cid:3)(cid:17) , (4.20)where we have chosen γ and θ such that C (cid:16) θ µ +1 γ − µ | Γ( − µ + 1) | + θ µ − − α γ − µ − α (cid:12)(cid:12) Γ( − µ + 1 − α ) (cid:12)(cid:12)(cid:17) ≤ ǫ , for sufficiently small ǫ to be determined later.4.4. Analysis of M . h M , ϕ i = ˆ t ˆ h δ ( · − u ( x, s )) , ϕ ( S ( t − s ) σ ( x )) i d x d W s . We again invoke the kernel estimate. In fact, it is here that the kernel estimate becomesindispensable. In the stochastic setting, with a forcing term given by σ ( x ) δ ( ξ − u ( x, t )) ∂ t W ,which does not easily lend itself to the space-time Fourier transform, one may not simplytake the Fourier transform on both sides so that the factor, i ( τ + F ′ ( ξ ) · k ) + A ( ξ ) : ( k ⊗ k ),on the left side can simply be divided out, with a certain genuine nonlinearity ( i.e. thenon-degeneracy condition; cf. [10, 80, 97]). Thus, we have to find a different way to handlethe forcing term. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 21
Expanding the effect of the semigroup, we have h M , ϕ i = ˆ t ˆ T d e − θ ( t − s ) ϕe − ( A ( ξ ): ∇⊗∇ + γ ( − ∆) α )( t − s ) σ ( x − F ′ ( u ( x, s ))( t − s )) d x d W s . Since σ is bounded in T d , we see that σ ( · − F ′ ( u ( · , s ))( t − s )) is bounded in x .The kernel estimate then gives k e − θ ( t − s ) ϕe − ( A ( u ): ∇⊗∇ + γ ( − ∆) α )( t − s ) σ ( · − F ′ ( u ( · , s ))( t − s )) k H ˆ βx ≤ C (cid:0) γ ( t − s ) (cid:1) − ˆ β α k σ k L x , just as in [35]. In the same way, we have E (cid:2)(cid:13)(cid:13) ˆ t h δ ( · − u ( x, s )) , S ( t − s ) σ ( x ) i d W s (cid:13)(cid:13) H ˆ βx (cid:3) ≤ Cγ − ˆ βα θ ˆ βα − (cid:12)(cid:12) Γ(1 − ˆ βα ) (cid:12)(cid:12) . Then we have E (cid:2) k M k H ˆ βx (cid:3) ≤ Cγ − ˆ βα θ ˆ βα − (cid:12)(cid:12) Γ(1 − ˆ βα ) (cid:12)(cid:12) . Completion of the existence proof.
From (4.15)–(4.16), we have E (cid:2) k u + u ♭ + M k L t W s,qx (cid:3) ≤ E (cid:2) k u (0) k L x (cid:3) + E (cid:2) k u k L ([0 ,T ] ,L x ) (cid:3) + CT, where q > s > T E (cid:2) k u + u ♭ + M k L t W s,qx (cid:3) ≤ E (cid:2) k u + u ♭ + M k L t W s,qx (cid:3) , so that E (cid:2) k u + u ♭ + M k L t W s,qx (cid:3) ≤ CT (cid:16) E (cid:2) k u (0) k L x (cid:3) + E (cid:2) k u k L ([0 ,T ] ,L x ) (cid:3) + T (cid:17) . Then we have E (cid:2) k u + u ♭ + M k L t W s,qx (cid:3) ≤ C (cid:16) E (cid:2) k u (0) k L x (cid:3) + T (cid:17) + ǫ E (cid:2) k u k L ([0 ,T ] ,L x ) (cid:3) . From (4.20), we further have E (cid:2) k M k L ([0 ,T ] ,W ˆ β,p ′ x ) (cid:3) ≤ ǫ (cid:16) E (cid:2) k u k L ([0 ,T ] ,L x ) (cid:3) + E (cid:2) k u k L x (cid:3)(cid:17) . By the continuous embedding W s,qx ֒ → L x , E (cid:2) k u k L ([0 ,T ] ,W s,qx ) (cid:3) ≤ C ( α, ˆ β, γ, θ ) (cid:16) E (cid:2) k u (0) k L x (cid:3) + T (cid:17) . (4.21)Since W s,q is compactly embedded in L for q ≥
1, the Krylov-Bogoliubov mechanism( § Stochastic Anisotropic Parabolic-Hyperbolic EquationsII: Uniqueness of Invariant Measures
In this section, we prove the uniqueness of invariant measures for the second-order non-linear stochastic equations (4.1).
Theorem 5.1.
Let F and A satisfy the non-degeneracy condition (4.2) and the bounded-ness condition : | F ′′ ( ξ ) | . , | A ′ ( ξ ) | . . (5.1) Then the invariant measure established in Theorem is unique.
To show the uniqueness, we first show that the solutions enter a certain ball in L x infinite time almost surely. Then we show that the solutions, starting on a fixed ball, enterarbitrarily small balls almost surely, if the noise is sufficiently small in W , ∞ . This allowsus to conclude that any pair of balls enters an arbitrarily small ball of one another, sincethe noise is sufficiently small for any given duration with positive probability. This is theproperty of recurrence discussed in the coupling method in §
2, which implies the uniquenessof invariant measures. In showing the recurrence, we follow § Uniqueness I: Finite time to enter a ball.
The following lemma is proved in thesame way as in [35], via a Borel-Cantelli argument.
Lemma 5.1.
There are both a radius ˆ κ ( depending on the initial conditions ) and an almostsurely finite stopping time T such that a solution enters B ˆ κ (0) ⊆ L ( T d ) in time T . The proof uses the coupling method, where v is another solution to the same equationwith initial condition v (0) = v . It furnishes us with the recursively defined sequence ofstopping times, with T = 0 and T l = inf { t ≥ T l − + T : k u ( t ) k L x + k v ( t ) k L x ≤ κ } , (5.2)which are also almost surely finite.5.2. Uniqueness II: Bounds with small noise.
We now prove the following key lemmafor the pathwise solutions:
Lemma 5.2.
For any ǫ > , there are T > and ˜ κ > such that, for the initial conditions u satisfying k u k L x ≤ κ, and the noise satisfying sup t ∈ [0 ,T ] k σW k W , ∞ x ≤ ˜ κ, then T k u ( t ) k L x d t ≤ ǫ, where we have used the symbol ffl to denote the averaged integral. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 23
Proof.
One of the differences in our estimates from [35] is that a kernel estimate is used on v ♯F + v ♯A , instead of velocity averaging techniques, since the extra derivatives are requiredto be handled here. Of course, this method can also be applied to the first-order case sothat the need to estimate the average term ffl v ♯ d x in [35] can be eliminated. We dividethe proof into nine steps.1. Let u be a solution of ∂ t u + ∇ · F ( u ) + ∇ · ( A ( u ) ∇ u ) = σ ( x ) ∂ t W with initial condition u (0) = u , and let ˜ u be the solution to the same equation with initialcondition ˜ u satisfying k u − ˜ u k L x ≤ ǫ , k ˜ u k L x ≤ C ˆ κǫ − d , which can be found by convolving u with a mollifying kernel, where ˆ κ is the radius constantof Lemma 5.1.2. Consider the difference between solution ˜ u and noise σ ( x ) W : v = ˜ u − σ ( x ) W , whichis a kinetic solution to ∂ t v = −∇ · F ( v + σ ( x ) W ) + ∇ · (cid:0) A ( v + σ ( x ) W ) ∇ ( v + σ ( x ) W ) (cid:1) . The kinetic formulation for this equation can be derived as in (3.6): ∂ t χ v + F ′ ( ξ ) · ∇ χ v − A ( ξ ) : ∇ χ v = (cid:0) F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) (cid:1) · ∇ χ v − ∇ · (cid:0) ( A ( ξ ) − A ( ξ + σ ( x ) W )) ∇ χ v (cid:1) − F ′ ( ξ + σ ( x ) W ) δ ( ξ − v ) · ∇ ( σ ( x ) W ) + ∇ · (cid:0) A ( ξ + σ ( x ) W ) δ ( ξ − v ) ∇ ( σ ( x ) W ) (cid:1) − ∂ ξ (cid:0) δ ( ξ − v ) A ( ξ + σ ( x ) W ) : (cid:0) ∇ ( σ ( x ) W ) ⊗ ∇ ( σ ( x ) W ) (cid:1)(cid:1) (5.3)+ ∂ ξ ( m v + N v ) . A notable difference here is that the parabolic defect measure N v is not the limit of δ ( ξ − ( v ε + σ ( x ) W )) A ( ξ ) : (cid:0) ∇ ( v ε + σW ) ⊗ ∇ ( v ε + σ ( x ) W ) (cid:1) , but rather the limit of N uε = δ ( ξ − v ε ) A ( ξ + σ ( x ) W ) : (cid:0) ∇ v ε ⊗ ∇ v ε (cid:1) + δ ( ξ − v ε ) A ( ξ + σ ( x ) W ) : (cid:0) ∇ ( σ ( x ) W ) ⊗ ∇ ( σ ( x ) W ) (cid:1) + δ ( ξ − v ε ) A ( ξ + σ ( x ) W ) : (cid:0) ∇ v ε ⊗ ∇ ( σ ( x ) W ) (cid:1) . (5.4)The asymmetry in the cross term in failing to contain both ∇ v ⊗∇ ( σ ( x ) W ) and ∇ ( σ ( x ) W ) ⊗∇ v arises from the fact that the convex entropy used is Φ( v ), instead of Φ( v + σW ). Oneof the key insights in [13] is that, using the symmetry and nonnegativity of A , A can bewritten as the square of another symmetric, positive semi-definite matrix so that (5.4) isnon-negative. The limit of N uε is the non-negative parabolic defect measure N u .3. As before, we insert the regularizing operators: γ ( − ∆) α + θI (with fixed γ and θ in thiscase) on both sides. Again, we can decompose the solution into the following components: h v ( t ) , ϕ i = h v + v ♭ + v ♯F + v ♯A + M F + M A + M + M , ϕ i , with v ( x, t ) = ˆ S ( t ) χ v ( ξ, x,
0) d ξ,v ♭ ( x, t ) = ˆ ˆ t S ( s )( γ ( − ∆) α + θ I) χ v ( ξ, x, t − s ) d s d ξ,v ♯F ( x, t ) = ˆ ˆ t S ( t − s ) (cid:0) F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) (cid:1) · ∇ χ v ( ξ, x, s ) d s d ξ,v ♯A ( x, t ) = − ˆ ˆ t S ( t − s ) ∇ · (cid:0) ( A ( ξ ) − A ( ξ + σ ( x ) W )) ∇ χ v ( ξ, x, s ) (cid:1) d s d ξ, h M F , ϕ i = − ˆ ˆ t F ′ ( v + σ ( x ) W ) · ∇ ( σ ( x ) W ) ( S ∗ ( t − s ) ϕ )( x, v ( x, s )) d s d x, h M A , ϕ i = − ˆ ˆ t A ( v + σ ( x ) W ) : (cid:0) ∇ ( σ ( x ) W ) ⊗ ∇ ( S ∗ ( t − s ) ϕ )( v ( x, s ) , x ) (cid:1) d s d x, h M , ϕ i = − ¨ ˆ t ∂ ξ ( S ∗ ( t − s ) ϕ ) d( m v + N v )( ξ, x, s ) , h M , ϕ i = ˆ ˆ t ∂ ξ ( S ∗ ( t − s ) ϕ )( v ( x, s ) , x ) A ( v + σ ( x ) W ) : (cid:0) ∇ ( σ ( x ) W ) ⊗ ∇ ( σ ( x ) W ) (cid:1) d s d x. Now we estimate each of these integrals, with some variations from [35] especially forthe terms involving A . For this, C > ǫ, ˜ κ , and T .4. We first have the familiar estimates: ˆ T k v ( t ) k H αx d t ≤ Cγ r k u k L x , and ˆ T k v ♭ ( t ) k L x d t ≤ Cγ r +1 ˆ T k v ( t ) k L x d t from the velocity averaging arguments, where | r | < γ in the second estimate from those arguments, no matter what r might be).These imply T k v k L x d t ≤ CT − γ r k u k L x , (5.5) T k v ♭ ( t ) k L x d t ≤ Cγ r +12 (cid:16) T k v ( t ) k L x d t (cid:17) . (5.6)5. For v ♯F and v ♯A , we use the fact that (cid:0) F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) (cid:1) · ∇ χ v ( ξ, x, s )= ∇ · (cid:0) ( F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W )) χ v ( ξ, x, s ) (cid:1) − (cid:0) F ′′ ( ξ + σ ( x ) W ) · ∇ σ ( x ) W (cid:1) χ v ( ξ, x, s ) , (cid:0) A ( ξ ) − A ( ξ + σ ( x ) W ) (cid:1) ∇ χ v ( ξ, x, s )= ∇ · (cid:0) ( A ( ξ ) − A ( ξ + σ ( x ) W )) χ v ( ξ, x, s ) (cid:1) − (cid:0) A ′ ( ξ + σ ( x ) W ) ∇ σ ( x ) W (cid:1) χ v ( ξ, x, s ) . NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 25
Now we apply the kernel estimates. Let ϕ ∈ L be any test function, and let h· , ·i be thepairing in L . Then h v ♯F ( t ) , ϕ i = ¨ ˆ t ϕ S ( t − s ) ∇ · (cid:0) ( F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W )) χ v ( ξ, x, s ) (cid:1) d s d ξ d x − ¨ ˆ t ϕ S ( t − s ) (cid:0) ( F ′′ ( ξ + σ ( x ) W ) · ∇ σ ( x ) W ) χ v ( ξ, x, s ) (cid:1) d s d ξ d x = ¨ ˆ t ∇ ( S ∗ ( t − s ) ϕ ) · (cid:0) F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) (cid:1) χ v ( ξ, x, s ) d s d ξ d x − ¨ ˆ t S ∗ ( t − s ) ϕ (cid:0) F ′′ ( ξ + σ ( x ) W ) · ∇ σ ( x ) W (cid:1) χ v ( ξ, x, s ) d s d ξ d x. (5.7)Similarly, we have h v ♯A ( t ) , ϕ i = ¨ ˆ t ϕ S ( t − s ) ∇ : (cid:0) ( A ( ξ ) − A ( ξ + σ ( x ) W )) χ v ( ξ, x, s ) (cid:1) d s d ξ d x − ¨ ˆ t ϕ S ( t − s ) ∇ · (cid:0) A ′ ( ξ + σ ( x ) W ) ∇ ( σ ( x ) W ) χ v ( ξ, x, s ) (cid:1) d s d ξ d x = ¨ ˆ t ∇ ( S ∗ ( t − s ) ϕ ) : (cid:0) A ( ξ ) − A ( ξ + σ ( x ) W ) (cid:1) χ v ( ξ, x, s ) d s d ξ d x − ¨ ˆ t (cid:0) ∇ ( S ∗ ( t − s ) ϕ ) ⊗ ∇ σ ( x ) W (cid:1) : A ′ ( ξ + σ ( x ) W ) χ v ( ξ, x, s ) d s d ξ d x. (5.8)Notice that ˆ T ˆ t k∇ ( S ∗ ( t − s ) ϕ ) · (cid:0) F ′ ( · ) − F ′ ( · + σ ( · ) W ) (cid:1) χ v ( · , · , s ) k L x,ξ d s d t ≤ ˆ T ˆ t k∇ ( S ∗ ( t − s ) ϕ ) k L ∞ x,ξ k F ′ ( · ) − F ′ ( · + σ ( · ) W ) k L ∞ x,ξ k χ v ( · , · , s ) k L x,ξ d s d t ≤ ˆ T ˆ t k∇S ∗ ( t − s ) k L → L ∞ k ϕ k L x k F ′ ( · ) − F ′ ( · + σ ( · ) W ) k L ∞ x,ξ k χ v ( · , · , s ) k L x,ξ d s d t ≤ C ˜ κ k ϕ k L x sup s ∈ [0 ,T ] (cid:16) ˆ T e θ ( t − s ) ( γt ) − d +24 α d t (cid:17) ˆ T k v ( s ) k L x d s ≤ C ˜ κ k ϕ k L x γ − d +24 α θ d +24 α − ˆ ∞ e − t t − d +24 α d t ˆ T k v ( s ) k L x d s = C ˜ κ k ϕ k L x γ − d +24 α θ d +24 α − (cid:12)(cid:12) Γ(1 − d + 24 α ) (cid:12)(cid:12) ˆ T k v ( s ) k L x d s ; (5.9) ˆ T ˆ t k (cid:0) S ∗ ( t − s ) ϕ (cid:1) ∇ ( σ ( x ) W ) · (cid:0) F ′′ ( · + σ ( · ) W ) χ v ( · , · , s ) (cid:1) k L x,ξ d s d t ≤ ˆ T ˆ t kS ∗ ( t − s ) ϕ k L ∞ x,ξ k F ′′ ( · + σ ( · ) W ) k L ∞ x,ξ k σW k W , ∞ x k χ v ( · , · , s ) k L x,ξ d s d t ≤ C ˜ κ k ϕ k L x γ − d α θ d α − (cid:12)(cid:12) Γ(1 − d α ) (cid:12)(cid:12) ˆ T k v ( s ) k L x d s ; (5.10) ˆ T ˆ t k∇ ( S ∗ ( t − s ) ϕ ) : (cid:0) A ( · ) − A ( · + σ ( · ) W ) (cid:1) χ v ( · , · , s ) k L x,ξ d s d t ≤ ˆ T ˆ t k∇ ( S ∗ ( t − s ) ϕ ) k L ∞ x,ξ k A ( · ) − A ( · + σ ( · ) W ) k L ∞ x,ξ k χ v ( · , · , s ) k L x,ξ d s d t ≤ C ˜ κ k ϕ k L x γ − d +44 α θ d +44 α − (cid:12)(cid:12) Γ(1 − d + 44 α ) (cid:12)(cid:12) ˆ T k v ( s ) k L x d s ; (5.11) ˆ T ˆ t k (cid:0) ∇ ( S ∗ ( t − s ) ϕ ) ⊗ ∇ ( σ ( · ) W ) (cid:1) : A ′ ( · + σ ( · ) W ) χ v ( · , · , s ) k L x,ξ d s d t ≤ ˆ T ˆ t k∇ ( S ∗ ( t − s ) ϕ ) k L ∞ x,ξ k σW k W , ∞ x k A ′ ( · + σ ( · ) W ) k L ∞ x,ξ k χ v ( · , · , s ) k L x,ξ d s d t ≤ C ˜ κ k ϕ k L x γ − d +24 α θ d +24 α − (cid:12)(cid:12) Γ(1 − d + 24 α ) (cid:12)(cid:12) ˆ T k v ( s ) k L x d s. (5.12)Now, by (5.1), we have assumed that | F ′′ ( ξ ) | . , | A ′ ( ξ ) | . , and k σ ( x ) W k W , ∞ ≤ ˜ κ , so that we can use the estimates (the second from the first by thePoincar´e-Wirtinger inequality, since ´ T d σ ( x ) d x = 0): | F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) | + | F ′′ ( ξ + σ ( x ) W ) · ∇ ( σW ) | ≤ C ˜ κ, | A ( ξ ) − A ( ξ + σ ( x ) W ) | + | A ′ ( ξ + σ ( x ) W ) ∇ ( σW ) | ≤ C ˜ κ. Putting these estimate (5.7)–(5.12) back into the bound: k v ♯ · ( t ) k L x = sup k ϕ k L x =1 h v ♯ · ( t ) , ϕ i ,we have ˆ T k v ♯A ( t ) + v ♯F ( t ) k L x d t ≤ C ˜ κ (cid:16) γ − d +24 α θ d +24 α − | Γ(1 − d + 24 α ) | + γ − d α θ d α − | Γ(1 − d α ) | + γ − d +44 α θ d +44 α − | Γ(1 − d + 44 α ) | (cid:17) ˆ T k v ( t ) k L x d t. (5.13) NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 27
6. For M F and M A , we employ the kernel estimate and k σW k W , ∞ x ≤ ˜ κ to obtain |h M F , ϕ i| ≤ ˆ t k F ′ ( v + σW ) k L x kS ϕ k L ∞ x k∇ σW k L ∞ x d s, |h M A , ϕ i| ≤ ˆ t k A ( v + σW ) k L x k∇ ( S ϕ ) k L ∞ x k∇ σW k L ∞ x d s. Now, by (4.3), we have k F ′ ( v + σW ) k L x + k A ( v + σW ) k L x ≤ C (cid:0) k v ( t ) k L x + k σ k L x | W | (cid:1) . These give ˆ T k M F ( t ) + M A ( t ) k L x d t ≤ C ˜ κ ˆ T ˆ t (cid:0) k v ( s ) k L x (cid:1) e − θ ( t − s ) (cid:0) γ ( t − s )) − α (cid:1) d s d t ≤ C ˜ κ (cid:16) θ − + γ − α θ α − | Γ(1 − α ) | (cid:17) ˆ T (cid:0) k v ( s ) k L x (cid:1) d s. (5.14)7. For M , we have h M , ϕ i = ˆ ˆ t ∂ ξ ( S ∗ ( t − s ) ϕ )( v ( x, s ) , x ) A ( v + σ ( x ) W ) : (cid:0) ∇ ( σ ( x ) W ) ⊗ ∇ ( σ ( x ) W ) (cid:1) d s d x. We notice that ∂ ξ ( S ( t − s ) ϕ )( v ( x, s ) , x ) = ( t − s ) F ′′ ( v ( x, s )) · ∇ ( S ∗ ( t − s ) ϕ ) + A ′ ( v ( x, s )) : ∇ ( S ∗ ( t − s ) ϕ ) , as explained in (4.17). By (5.1), we have assumed that | F ′′ ( ξ ) | . , | A ′ ( ξ ) | . . Again we have k A ( v + σ ( x ) W ) k L x ≤ C (cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) . Finally, using the kernel estimate yields |h M , ϕ i|≤ C ˆ t ( t − s ) k F ′′ k L ∞ k∇ S ∗ ( t − s ) ϕ k L ∞ x,ξ k∇ σW k L ∞ x (cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) d s + C ˆ t ( t − s ) k A ′ k L ∞ k∇ S ∗ ( t − s ) ϕ k L ∞ x,ξ k∇ σW k L ∞ x (cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) d s ≤ C ˜ κ ˆ t ( t − s ) (cid:0) k∇ S ∗ ( t − s ) k + k∇ S ∗ ( t − s ) k (cid:1) k ϕ k L ∞ x (cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) d s. Therefore, we have ˆ T k M ( t ) k L x d t ≤ C ˜ κ ˆ T ˆ t ( t − s ) (cid:0) k∇ S ∗ ( t − s ) k + k∇ S ∗ ( t − s ) k (cid:1)(cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) d s d t ≤ C ˜ κ ˆ T ˆ t ( t − s ) (cid:0) ( γ ( t − s )) − α + ( γ ( t − s )) − α (cid:1) e − θ ( t − s ) (cid:0) k v ( s ) k L x + k σ k L x | W | (cid:1) d s d t, and ˆ T k M ( t ) k L x d t (5.15) ≤ C ˜ κ (cid:16) γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12) + γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12)(cid:17) ˆ T (cid:0) k v ( s ) k L x (cid:1) d s.
8. For the kinetic measure M , we use the total variation estimate again. First, with ϕ ∈ L ∞ x , |h M , ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12) ¨ ˆ t ∂ ξ ( S ∗ ( t − s ) ϕ ) d( m v + N v )( x, ξ, s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ¨ ˆ t (cid:0) ( t − s ) F ′′ ( ξ ) · ∇ ( S ∗ ( t − s ) ϕ ) + A ′ ( ξ ) : ∇ ( S ∗ ( t − s ) ϕ ) (cid:1) d( m v + N v )( x, ξ, s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ϕ k L ∞ x ¨ ˆ t (cid:0) γ − α ( t − s ) − α + γ − α ( t − s ) − α (cid:1) e − θ ( t − s ) d | m v + N v | ( x, ξ, s ) , so that ˆ T k M ( t ) k L x d t ≤ C (cid:16) γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12) + γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12)(cid:17) | m v + N v | ( R × T d × [0 , T ]) . NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 29
As in Lemma 4.1, we test equation (5.3) against ξ to find12 k v ( t ) k L x + | m v + N v | ( R × T d × [0 , t ]) ≤ k ˜ u k L x + (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ¨ ξ (cid:0) F ′ ( ξ ) − F ′ ( ξ + σ ( x ) W ) (cid:1) · ∇ χ v d ξ d x d s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ˆ vF ′ ( v + σ ( x ) W ) · ∇ ( σ ( x ) W ) d x d s (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ˆ A ( v + σ ( x ) W ) : (cid:0) ∇ ( σW ) ⊗ ∇ ( σW ) (cid:1) d x d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˜ u k L x + (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ¨ ξF ′′ ( ξ + σ ( x ) W )) · ∇ ( σ ( x ) W ) χ v d ξ d x d s (cid:12)(cid:12)(cid:12)(cid:12) + C ˜ κ (1 + ˜ κ ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ t ˆ v (cid:0) | v | + | σ ( x ) W | (cid:1) d x d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˜ u k L x + C ˜ κ ˆ t (cid:0) k v ( s ) k L x (cid:1) d s. Then Gronwall’s inequality implies | m v + N v | ≤ Ce C ˜ κt (cid:0) k ˜ u k L x + 1 (cid:1) ≤ Ce C ˜ κt (cid:0) ˆ κ ǫ − d + 1 (cid:1) . Therefore, we have ˆ T k M ( t ) k L x d t (5.16) ≤ C (cid:16) γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12) + γ − α θ α − (cid:12)(cid:12) Γ(2 − α ) (cid:12)(cid:12)(cid:17) e C ˜ κT (cid:0) ˆ κ ǫ − d + 1 (cid:1) . Completion of the estimates . First, we set α ≤ so that the instances of | Γ | are neverevaluated at a negative integer, where it is infinite. With the finite bound of all the valuesof | Γ | and finitely many instances of Γ in estimates (5.5)–(5.16) above, we can write thoseestimates as T k v ( t ) k L x d t ≤ CT − γ r k u k L x , T k v ♭ ( t ) k L x d t ≤ Cγ r +12 (cid:16) T k v k L x d t (cid:17) ≤ Cγ r +12 T (cid:0) k v k L x (cid:1) d t, T k v ♯F + v ♯A k L x d t ≤ C ˜ κ (cid:16) γ − d +24 α θ d +24 α − + γ − d α θ d α − + γ − d +44 α θ d +44 α − (cid:17) T k v ( t ) k L x d t, T k M F ( t ) + M A ( t ) k L x d t ≤ C ˜ κ (cid:16) θ − + γ − α θ α − (cid:17) T (cid:0) k v ( t ) k L x (cid:1) d t, T k M ( t ) k L x d t ≤ C ˜ κ (cid:16) γ − α θ α − + γ − α θ α − (cid:17) T (cid:0) k v ( t ) k L x (cid:1) d t, T k M k L x d t ≤ CT (cid:16) γ − α θ α − + γ − α θ α − (cid:17) e C ˜ κT (cid:0) ˆ κ ǫ − d + 1 (cid:1) . Combining all the estimates together yields T k v ( t ) k L x d t ≤ C T − γ r k u k L x + (cid:16) C γ r +12 + C ( γ, θ )(˜ κ + ˜ κ ) (cid:17)(cid:16) T k v k L x d s (cid:17) + C ( γ, θ ) T − e c ˜ κT (ˆ κ ǫ − d + 1) . We can choose γ , θ , T , and ˜ κ in that order so that, for some q to be determined, C γ r +12 ≤ qǫ. For α < , we see that every θ has positive power above, except in C ( γ, θ ) for the estimateof k M F + M A k L x,t , so that C ( ρ, θ ) involves θ with positive power. Therefore, we choose θ such that C ( γ, θ ) < , so that we can choose T sufficiently large such that C T − k u k L x + C ( γ, θ ) T − (cid:0) ˆ κ ǫ − d + 1 (cid:1) ≤ qǫ. Finally, we choose ˜ κ such that C ( γ, θ )˜ κ (1 + ˜ κ ) ≤ qǫ, and C ˜ κT ≤ qǫ. By taking q sufficiently small, we have T k v ( t ) k L x d t ≤ ǫ , T k ˜ u ( t ) k L x d t ≤ ǫ , which leads to T k u ( t ) k L x d t ≤ ǫ , by the following L -contraction property of the pathwise solutions: Lemma 5.3.
Let A be symmetric positive-semi-definite, and let both A ( ξ ) and F ( ξ ) beH¨older continuous and of polynomial growth. Then, for each initial data function u , thereexists a unique measurable u : T d × [0 , T ] × Ω → R solving (3.5) in the sense of Definition . Moreover, for u , ˜ u ∈ L ( T d ) , k u ( t ) − ˜ u ( t ) k L x ≤ k u − ˜ u k L x almost surely . For our case, k u ( t ) − ˜ u ( t ) k L x ≤ k u − ˜ u k L x ≤ ǫ . This completes the proof. (cid:3)
Remark . This almost sure L –contraction property is not available in the multiplica-tive case. In fact, it is not available in many other situations, such as in systems or fornon-conservative equations where the L -contraction is not ready to provide a stabilitycondition. It is of interest to study the uniqueness and ergodicity properties of invariantmeasures for equations without this property. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 31
Uniqueness III: Conclusion.
Let u and u be in L x . For a given ǫ >
0, let ˜ u and˜ u be in L x such that k u i − ˜ u i k L x ≤ ǫ . Denote their corresponding solutions u , u , ˜ u , and ˜ u , respectively. Let us now put ˜ u and ˜ u , in place of u and v in § T l = inf { t ≥ T l − + T : k ˜ u ( t ) k L x + k ˜ u ( t ) k L x ≤ κ } . As in [35], choosing T and ˜ κ as above, we obtain by the L –contraction (for the additivenoise, there is the L -contraction almost sure): P (cid:8) T l + T T l k u ( s ) − u ( s ) k L x d s ≤ ǫ (cid:12)(cid:12) F T l (cid:9) ≥ P (cid:8) T l + T T l k ˜ u ( s ) − ˜ u ( s ) k L x d s ≤ ǫ (cid:12)(cid:12) F T l (cid:9) ≥ P (cid:8) sup t ∈ [ T l , T l + T ] k σW ( t ) − σW ( T l ) k W , ∞ x ≤ ˜ κ (cid:12)(cid:12) F T l (cid:9) . Since ˜ κ >
0, and σ is Lipschitz, we can denote the positive probability of the event as λ . By the strong Markov property, we know that it does not change with l .This allows us to write P (cid:8) T l + T T l k u ( s ) − u ( s ) k L x d s ≥ ǫ for l = l , l + 1 , . . . , l + k (cid:9) ≤ (1 − λ ) k , so that P (cid:8) lim l →∞ T l + T T l k u ( s ) − u ( s ) k L x d s ≥ ǫ (cid:9) = P (cid:8) ∃ l ∀ l ≥ l : T l + T T l k u ( s ) − u ( s ) k L x d s ≥ ǫ (cid:9) = 0 . This limit exists as s
7→ k u ( s ) − u ( s ) k L x is non-increasing, by the L –contraction property.Then, by the same property, P (cid:8) lim t →∞ k u ( t ) − u ( t ) k L x ≥ ǫ (cid:9) = 0 . Therefore, almost surely, lim t →∞ k u ( t ) − u ( t ) k L x = 0 , which implies the uniqueness of the invariant measure.6. Further Developments, Problems, and Challenges
In this section, we discuss some further developments, problems, and challenges in thisdirection.
Further problems.
There are several natural problems that follow from the analysisdiscussed above. We restrict ourselves again to nonlinear conservation laws driven bystochastic forcing.One of the problems is the long-time behavior problem for solutions of nonlinear conser-vation laws driven by multiplicative noises. The noises of form ∇ · ( F ( u ) ◦ d W ) have beenconsidered in [59, 60], in which the dynamics remains in the zero-spatial-average subspaceof L ( T d ).The well-posedness for nonlinear conservation laws driven by multiplicative noises isquite well understood from several different perspectives – the strong entropy stochasticsolutions of Feng-Nualart [49] and of Chen-Ding-Karlsen [9], the viscosity solution methodsof Bauzet-Vallet-Wittbold [3], and the kinetic approach of Debussche-Hofmanov`a-Vovelle[33, 34], as we have mentioned above. Nevertheless, the long-time behavior problem forsolutions is wide open, since there is no effective way to control k u ( t ) k L x .We remark on two aspects of the noises that can affect qualitative long-time behaviorsof solutions:(i) The question seems to depend heavily on the roots and growth of the noise coef-ficient function σ ( u ) – If the noise is degenerate (not cylindrical), say σ ( u ) = 0for certain u = r ∈ R , then u ≡ r is a fixed point of the evolution. By the L –contraction, it is possible to prove certain long-time behavior results for the so-lutions for the unbounded noise coefficient function with one root. Both the growthof σ and how many roots it possesses affect the long-time behaviors of solutions,as is evident also in the analysis of other equations such as the KPP equation (dis-cussed below). In the case that σ has no roots, there are no fixed points. It ispossible that the nonlinear conservation laws driven by bounded noises with noroots have non-trivial invariant measures.(ii) If the noise is σ ( u )d B = P k g k ( u )d W k , where B = W k e k is a cylindrical Wienerprocess, the behaviors are expected to be very different from the case that the noiseis simply σ ( u )d W .Another natural direction to consider is the case of nonlinear systems of balance laws. Forthis case, such as for the isentropic Euler system, the kinetic formulation is not “pure” – itcontains the instances of the solution mixed with the kinetic operator ( cf. [81]). At present,it seems that the methods discussed above are not directly applicable to the systems.6.2. The Navier-Stokes equations.
The two-dimensional incompressible Navier-Stokesequations (INSEs) driven by stochastic forcing has been a subject of intense interest. Wefocus on the analysis of asymptotic behaviors of solutions to keep ourselves from gettingsidetracked.The existence of invariant measures for the 2-D INSEs on a regular bounded domain witha general noise that is a Gaussian random field and white-in-time has been known at leastsince [51]. The uniqueness and ergodicity for the 2-D INSEs have also been established;see [54] and the references therein for such results and further existence results of invariantmeasures under different conditions. These results have subsequently been improved, in-cluding for the noises that are localized in time and Gaussian in space, in [7, 8, 87, 88], andin some references cited in this paper.We remark particularly that the corresponding existence questions for the 2-D INSEswith multiplicative noises have been established, for example in [52], via the Skorohod
NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 33 embedding and a Faedo-Galerkin procedure, which have shown the existence of martin-gale solutions and stationary martingale solutions, from which in turn the existence of aninvariant measure can be derived.The asymptotic behaviors of 2-D INSEs driven by white-in-time noises or Poisson dis-tributed unbounded kick noises have been explored, and the existence and uniqueness ofinvariant measures for these systems are known. See also [73, 74, 89] for the related refer-ences.There are also more recent results on INSEs driven by space-time white noises in 2-D or3-D; see [26,27,102] and the references therein. For example, it is known that the transitionsemigroup of the Kolmogorov equation associated to the 3-D stochastic INSEs driven by acylindrical white noise has a unique (and hence ergodic) invariant measure.The existence of invariant measures for the compressible Navier-Stokes equations, evenin the 2-D case, is wide open.See [46, 83], and discussions in [82, Chp. 3] as well as references contained there forfurther treatments on ergodicity results; also see [84].6.3.
The asymptotic strong Feller property.
Using the 2-D INSEs as a springboard,the notion of the asymptotic strong Feller property has been introduced in Hairer-Mattingley[63] as a weaker and more natural replacement of the sufficient “strong Feller” property(Definition 2.2) in dissipative infinite-dimensional systems, the possession of which guaran-tees the uniqueness of an invariant measure. In the finite-dimensional case of SDEs, thereis a related notion of eventual strong Feller property, for which sufficient conditions aregiven in [5].The definition of asymptotic strong Feller property depends on a preliminary definition:
Definition 6.1 (Totally separating system) . A pseudo-metric is a function d : X → R +0 forwhich d ( x, x ) = 0 and the triangle inequality is satisfied, and d ≥ d if the inequality holdsfor all arguments ( x, y ) ∈ X . Let { d k } ∞ k =0 be an increasing sequence of pseudo-metrics ona Polish space X . Then { d k } ∞ k =0 is a totally separating system of pseudo-metrics iflim k → d k = 1pointwise everywhere off the diagonal on X .Then the asymptotic strong Feller property is defined as follows: Definition 6.2 (Asymptotic Strong Feller property) . A Markov transition semigroup P t on a Polish space X is asymptotically strong Feller at x if there exist both a totally sepa-rating system of pseudo-metrics { d k } ∞ k =0 and an increasing sequence of times t k such thatinf U ∈ nb( x ) lim sup k →∞ sup y ∈ U k P t k ( x, · ) − P t k ( y, · ) k d k = 0 , where nb( x ) is the collection of open sets containing x , P is the transition probabilitiesassociated to P , and k P − P k d k is the norm given by k P − P k d k = inf ˆ X d k ( w, z )Π(d w, d z ) , the infimum being taken over all positive measures on X with marginals P and P . The idea behind the asymptotic strong Feller condition is that ergodicity is preservedeven if the stochastic forcing is restricted to a few unstable modes, and dissipated in theothers. Using this idea, the ergodicity of the 2-D stochastic INSE with degenerate noise hasbeen established (see [63]). Some results of ergodicity for the 3-D INSEs driven by mildlydegenerate noise relying on the strong asymptotic Feller property have also established(see [93, 94] and the references cited therein).6.4.
The KPP equation and multiplicative noises.
The Kolmogorov-Petrovsky-Piskunovequation (KPP) is given by ∂ t u = ∇ · ( A ( x, t ) ∇ u ) + h ( u ) + g ( u ) ∂ t W,u | t =0 = ϕ. Attention is often restricted to the case in which g and h both vanish at the two points a, b ∈ R , and g, h > a, b ). In this way, the asymptotic size is controlled in L .It has been shown in Chueshov-Villermot [15–20] that, for the semilinear equation with h ( u ) = sg ( u ), evolution on a bounded, open domain with zero Neumann boundary con-dition is bounded in space. Moreover, the notion of stability in probability has also beenintroduced in [18]: Definition 6.3 (Stability in probability) . A function u f is stable in probability if, for every ǫ >
0, the following relation holds:lim k ϕ − u f k L ∞ → P (cid:8) ω ∈ Ω : sup t ∈ R + \{ } k u ϕ ( s, · , t, ω ) − u f k L ∞ > ǫ (cid:9) = 0 . Otherwise, u f is unstable in probability .A function u f is globally asymptotically stable in probability if it is stable in probabilityand P (cid:8) ω ∈ Ω : lim t →∞ k u ϕ ( s, · , t, ω ) − u f k L ∞ = 0 (cid:9) = 1 . By considering the moments of the spatial average, it has been shown that the constantfunctions u = a and u = b are fixed points whose stability in probability depends on thevalues of s . The results of [20] have been refined, say in [4], and the properties of the globalattractor, including the computation of exact Lyapunov exponents in a decay scenario havebeen derived.As we have remarked, the main reason that multiplicative noises complicate the analysisof stochastic PDEs is that one fails to have much control over the spatial average, exceptwhen additional restrictions on the noise and initial conditions are specified. When thenoise has a root, that constant is immediately a fixed point. This cannot be avoided evenwhen working over the non-compact domain R because the L p boundedness often relies onthe space that is compact, and is a difficulty we have to overcome in order to gain a deeperunderstanding of the asymptotic behaviors of solutions.6.5. Large deviation principles.
Beyond the existence and uniqueness of invariant mea-sures, large deviation principles touch on their specific properties. Whilst it goes some wayoutside the scope of this survey even to introduce the theory of large deviations, which at-tempts to characterize the limiting behavior of a family of probability measures (in our case,invariant measures) depending on some parameter by using a rate function , we should beremiss to neglect mentioning it altogether; two vintage references to the subject are [38,58].
NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 35
More modern treatments can be found in [36, 43, 68, 99] and the references cited therein.Of particular interest has been the “zero-noise” limit of stochastic equations in which onelooks at the stochastic equations with a small parameter ε multiplied to the noise. Ques-tions of large deviation type also arise in stochastic homogenization theory. Each of thesesubjects can justify an independent survey. Pertaining specifically to stochastic conserva-tion laws, the literature is, however, more sparse. Going some way outside the classicalFreidlin-Wentzell theory, some results have been announced pertaining to large deviationestimates for stochastic conservation laws. Specifically, in [85], large deviation principleshave been investigated and derived in the limit of jointly vanishing noise and viscosity byusing delicate scaling arguments. Notably, in [2], the bounds for the rate function have alsobeen derived in the vanishing viscosity limit only, so that the noise is allowed in the limit,and in the multidimensional setting. Finally, we mention the more recent work [41] andthe references cited therein, large deviation principles have been derived for the first-orderscalar conservation laws with small multiplicative noise on T d in the zero-noise limit byusing the Freidlin-Wentzell theory. Much still remains to be explored in this direction. Acknowledgements . The research of Gui-Qiang G. Chen was supported in part bythe UK Engineering and Physical Sciences Research Council Award EP/E035027/1 andEP/L015811/1, and the Royal Society–Wolfson Research Merit Award (UK). The researchof Peter Pang was supported in part by the UK Engineering and Physical Sciences ResearchCouncil Award EP/E035027/1 and EP/L015811/1, and an Oxford Croucher Scholarship.
References [1] L. Arnold.
Random Dynamical Systems . Springer-Verlag: Berlin Heidelberg, 1998.[2] J. Barr´e, C. Bernardin, and R. Chertrite. Density large deviations for multidimensional stochastichyperbolic conservation laws.
J. Stat. Phys. (2017), 466 – 491.[3] C. Bauzet, G. Vallet, and P. Wittbold. The Cauchy problem for conservation laws with a multiplicativestochastic perturbation.
J. Hyper. Diff. Eq. (2012), 661–709.[4] B. Berg´e and B. Saussereau. On the long-time behavior of a class of parabolic SPDEs: Monotonicitymethods and exchange of stability.
ESAIM: Probab. Stat. (2005), 254–276.[5] J. Birkens. Sufficient conditions for the eventual strong Feller property for degenerate stochastic evo-lutions. J. Math. Anal. Appl. (2011), 469–481.[6] F. Bouchut and L. Desvillettes. Averaging lemmas without time Fourier transform and application todiscretized kinetic equations.
Proc. Royal Soc. Edin. Sec. A , (1999), 19–36.[7] J. Bricmont, A. Kupiainen, and R. Lefevere. Ergodicity of the 2D Navier-Stokes equations with randomforcing. Comm. Math. Phys. (2001), 65–81.[8] J. Bricmont, A.Kupiainen, and R. Lefevere. Exponential mixing of the 2D Navier-Stokes dynamics.
Comm. Math. Phys. (2002), 87–132.[9] G.-Q. Chen, Q. Ding, and K. Karlsen. On nonlinear stochastic balance laws.
Arch. Ration. Mech. Anal. (2012), 707–743.[10] G.-Q. Chen and H. Frid. Large time behavior of entropy solutions in L ∞ for multidimensional conser-vation laws. In: Advances in Nonlinear PDEs and Related Areas , 28–44, World Scientific, 1998.[11] G.-Q. Chen and Y.-G. Lu. A study of approaches to applying the theory of compensated compactness.
Chinese Sci. Bull. (1989), 15–19.[12] G.-Q. Chen and P. Pang. On nonlinear anisotropic degenerate parabolic-hyperbolic equations withstochastic forcing. Preprint arXiv:1903.02693, March 2019.[13] G.-Q. Chen and B. Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equa-tions. Ann. l’I.H.P. Anal. Non-Lin´eaires , (2003), 645–668.[14] G.-Q. Chen and B. Perthame. Large-time behavior of periodic entropy solutions to anisotropic degen-erate parabolic-hyperbolic equations. Proc. A.M.S. , (2009), 3003–3011. [15] I. Chueshov and P. Vuillermot. On the large-time dynamics of a class of parabolic equations subjectedhomogeneous white noise: Stratonovitch’s case. C. R. Acad. Sci. Paris , (1996), 29–33.[16] I. Chueshov and P. Vuillermot. On the large-time dynamics of a class of random parabolic equations. C. R. Acad. Sci. Paris , (1996), 1181–1186.[17] I. Chueshov and P. Vuillermot. Long-time behavior of solutions to a class of quasilinear parabolicequations with random coefficients. Ann. I.H.P. Anal. Non-Lin´eaires , (1998), 191–232.[18] I. Chueshov and P. Vuillermot. Long-time behavior of solutions to a class of stochastic parabolicequations with homogeneous white noise: Straonovitch’s case. Probab. Theory Related Fields , (1998), 149–202.[19] I. Chueshov and P. Vuillermot. On the large-time dynamics of a class of parabolic equations subjectedto homogeneous white noise: Itˆo’s case. C. R. Acad. Sci. Paris , (1998), 1299–1304.[20] I. Chueshov and P. Vuillermot. Long-time behavior of solutions to a class of stochastic parabolicequations with homogeneous white noise: Itˆo’s case. Stoch. Anal. Appl. (2000), 581–615.[21] M. Coti-Zelati, N. Glatt-Holtz, and K. Trivisa. Invariant measures for the stochastic one-dimensionalcompressible Navier-Stokes equations. arXiv:1802.04000v1 , 2018.[22] H. Crauel. Markov measures for random dynamical systems.
Stochastics and Stochastic Reports , (1991), 153–173.[23] H. Crauel. Random Probability Measures on Polish Spaces . Taylor and Francis: London, 2002.[24] H. Crauel, A. Debussche, and F. Flandoli. Random attractors.
J. Dynam. Diff. Eq. (1997),307–341.[25] H. Crauel and F. Flandoli. Attractors for random dynamical systems.
Probab. Theory Relat. Fields , (1994), 365–393.[26] G. Da Prato and A. Debussche. Two-dimensional Navier-Stokes equations driven by a space-time whitenoise. J. Funct. Anal. (2002), 180–210.[27] G. Da Prato and A. Debussche. Ergodicity for the 3D stochastic Navier-Stokes equations.
J. Math.Pures Appl. (2003), 877–947.[28] G. Da Prato, A. Debussche, and R. Temam. Stochastic Burger’s equation.
No.D.E.A. (1994), 389–402.[29] G. Da Prato, F. Flandoli, E. Priola, and M. R¨ockner. Strong uniqueness for stochastic evolutionequations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Prob. (2013), 3306–3344.[30] G. Da Prato and J. Zabczyk.
Stochastic Equations in Infinite Dimensions . Cambridge University Press:Cambridge, 2008.[31] A. Debussche. On the finite dimensionality of random attractors.
Stoch. Anal. Appl. (1997),473–492.[32] A. Debussche. Hausdorff dimension of a random invariant set.
J. Math. Pures Appl. (1998),967–988.[33] A. Debussche, M. Hofmanov´a, and J. Vovelle. Degenerate parabolic stochastic partial differential equa-tions: Quasilinear case.
Ann. Probab. (2016), 1916–1955.[34] A. Debussche and J. Vovelle. Scalar conservation laws with stochastic forcing.
J. Funct. Anal. (2010), 1014–1042.[35] A. Debussche and J. Vovelle. Invariant measure of scalar first-order conservation laws with stochasticforcing.
Probab. Theory Relat. Fields , (2015), 575–611.[36] A. Dembo, and O. Zeitouni. Large Deviations Techniques and Applications.
New York, Springer, 2ndEd., 1997.[37] F. Delarue, F. Flandoli, and D. Vincenzi. Noise prevents collapse of Vlasov-Poisson point charges.
Comm. Pure Appl. Math. (2014), 1700–1736.[38] J.-D. Deuschel, and D. Stroock. Large Deviations. Academic Press: London 1989.[39] W. Doeblin. ´El´ements d’une th´eorie g´en´erale des chaˆınes simples constantes de Markoff.
Ann. Sci.´Ecole Normale Superieur , (1940), 61–111.[40] W. Doeblin and R. Fortet. Sur le chaˆınes `a liaisons compl`ete. Bull. Soc. Math. France , (1937),132–148.[41] Z. Dong, J.-L. Wu, R.-R. Zhang, and T.-S. Zhang. Large deviation principles for first-order scalarconservation laws with stochastic forcing. arXiv:1806.02955v1 , 2018. NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 37 [42] J. Doob. Asymptotic property of Markoff transition probability.
Trans. Amer. Math. Soc. , (1948), 393–421.[43] P. Dupuis, R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations.
NewYork,Wiley, 1997.[44] W. E, K. Khanin, A Mazel, and Ya. Sinai. Probability distribution functions for the random forcedBurgers equation.
Phys. Rev. Lett. (1997), 1904–1907.[45] W. E, K. Khanin, A Mazel, and Ya. Sinai. Invariant measures for Burgers equation with stochasticforcing.
Ann. Math. (2000), 877–960.[46] W. E, J. Mattingly, and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forcedNavier-Stokes equation.
Comm. Math. Phys. (2001), 83 – 106.[47] E. Fedrizzi, F. Flandoli, E. Priola, and J. Vovelle. Regularity of stochastic kinetic equations.
Electron.J. Probab. , (2017), 1–42.[48] B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear, conservativenoise. Preprint arXiv:1712.05775, 2019[49] J. Feng and D. Nualart. Stochastic scalar conservation laws. J. Funct. Anal. (2008), 313–373.[50] F. Flandoli. Stochastic flows and Lyapunov exponents for abstract stochastic PDEs of parabolic type.In:
Lyapunov Exponents Proceedings ( L. Arnold, H. Crauel, J.-P. Eckamann eds. ), LNM (1991),196–205.[51] F. Flandoli. Dissipativity and invariant measures for stochastic Navier-Stokes equations.
No.D.E.A. (1994), 403–423.[52] F. Flandoli and D. Gatarek. Martingale and stationary solutions for stochastic Navier-Stokes equations.
Probab. Theory Relat. Fields , (1995), 367–391.[53] F. Flandoli, M. Gubinelli, and E. Priola. Well-posedness of the transport equation by stochastic per-turbation. Invent. Math. (2010), 1–53.[54] F. Flandoli and B. Maslowski. Ergodicity of the 2-D Navier-Stokes equations under random perturba-tions.
Comm. Math. Phys. (1995), 119–141.[55] Franco Flandoli.
Regularity Theory and Stochastic Flows for Parabolic SPDEs . Gordon and BreachScience Publishers: Singapore, 1995.[56] J. F¨oldes, S. Friedlander, N. Glatt-Holtz, and G. Richards. Asymptotic analysis for randomly forcedMHD. arXiv:1604.06352 , 2016.[57] J. F¨oldes, N. Glatt-Holtz, G. Richards, and J.P. Whitehead. Ergodicity in randomly forced Rayleigh-B´enard convection. arXiv:1511.01247 , 2015.[58] M.I. Freidlin, and A.D. Wentzell.
Random Perturbation of Dynamical Systems . Springer: New York,2nd Ed., 1998.[59] B. Gess and P. Souganidis. Long-time behavior, invariant measures and regularizing effects for stochas-tic scalar conservation laws. arXiv:1411.3939 , 2014.[60] B. Gess and P. Souganidis. Stochastic non-isotropic degenerate parabolic-hyperbolic equations. arXiv:1611.01303 , 2016.[61] Y. Giga and T. Miyakawa. A kinetic construction of global solutions of first order quasilinear equations.
Duke Math. J. (1989), 505–515.[62] I. Gy¨ongy and D. Nualart. On the stochastic Burgers equation in the real line.
Ann. Probab. (1999), 782–802.[63] M. Hairer and J. Mattingly. Ergodicity of the 2D Navier-Stokes equations with degenerate stochasticforcing.
Ann. Math. (2006), 993–1032.[64] M. Hairer and J. Mattingly. Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokesequations.
Ann. Probab. (2008), 2050–2091.[65] M. Hairer and J. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochasticPDEs.
Electr. J. Probab. (2011), 658–738.[66] M. Hairer and J. Voss. Approximations to the stochastic Burgers equation.
J. Nonlin. Sci.
21 (6) (2011), 897 – 920.[67] M. Hofmanov´a. Degenerate parabolic stochastic partial differential equations.
Stochastic Process. Appl. (2013), 4294–4336.[68] F. den Hollander.
Large Deviations.
Fields Institute Monographs, A.M.S.: Providence, RJ, 2000. [69] Y. Le Jan. ´Equilibre statistique pour les produits de diff´eomorphismes al´eatoires ind´ependants.
Ann.l’I.H.P. Probab. Stat. (1987), 111–120.[70] K. H. Karlsen and E. Storrøsten. On stochastic conservation laws and Malliavin calculus. arXiv:1507.05518v2 , 2015.[71] R. Khas’minskii. Ergodic properties of recurrent diffusion processes and stabilization of the solutionsto the Cauchy problem for parabolic equations.
Theory Probab. Appl. (1960), 179–196.[72] N. Krylov and M. R¨ockner. Strong solutions of stochastic equations with singular time dependent drift
Probab. Theory Relat. Fields , (2005), 154 – 196.[73] S. Kuksin and A. Shirikyan. Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2Space Dimensions . European Mathematical Society: Z¨urich, 2006.[74] S. Kuksin and A. Shirikyan.
Mathematics of Two-Dimensional Turbulence . Cambridge University Press:Cambridge, 2002.[75] F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. In:
Lyapunov ExponentsProceedings ( L. Arnold and V. Wihstutz, eds. ) LNM (1986), 56–73.[76] J. Le`on, D. Nualart, and R. Pettersson The stochastic Burgers equation: Finite moments and smooth-ness of the density.
Infinite Dimensional Analysis , (2000), 363 – 385.[77] T. Lindvall. Lectures on the Coupling Method . John Wiley & Sons: New York, 1992.[78] P.-L. Lions, B. Perthame, and P. Souganidis. Scalar conservation laws with rough (stochastic) fluxes.
Stochastic PDEs : Anal. Comput. (2013), 664–686.[79] P.-L. Lions, B. Perthame, and P. Souganidis. Scalar conservation laws with rough (stochastic) fluxes:the spatially dependent case. Stochastic PDEs : Anal. Comput. (2014), 517–538.[80] P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservationlaws and related equations.
J. Amer. Math. Soc. (1994), 169–191.[81] P.-L. Lions, B. Perthame, and E. Tadmor. Kinetic formulation of the isentropic gas dynamics and p -systems. Comm. Math. Phys. (1994), 415–431.[82] A. Majda.
Introduction to Turbulent Dynamical Systems in Complex Systems . Springer: Cham, 2016.[83] A. Majda, and X.-T. Tong Ergodicity of truncated stochastic Navier Stokes with deterministic forcingand dispersion.
J. Nonlin. Sci.
26 (5) (2016), 1483–1506.[84] A. Majda and X.-M. Wang.
Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows .Cambridge University Press: Cambridge, 2006.[85] M. Mariani. Large deviations principles for stochastic scalar conservation laws.
Probab. Theory Relat.Fields (2010), 607 – 648.[86] B. Maslowski and J. Seidler. Invariant measures for nonlinear SPDE’s: uniqueness and stability.
Archivum Mathematicum , (1998), 153–172.[87] N. Masmoudi and L.-S. Young. Ergodic theory of infinite dimensional systems with applications todissipative parabolic PDEs. Comm. Math. Phys. (2002), 461–481.[88] J. C. Mattingly. Exponential convergence for the stochastically forced Navier-Stokes equations andother partially dissipative dynamics.
Comm. Math. Phys. (2002), 421–462.[89] J. C. Mattingly. Recent progress for the stochastic Navier-Stokes equations.
Journe´e ´EDP.
Kinetic Formulation of Conservation Laws . Oxford University Press: Oxford, 2002.[91] S. Peszat and J. Zabczyk. Strong Feller property and irreducibility for diffusions on Hilbert spaces.
Ann. Probab. (1995), 157–172.[92] S. Peszat and J. Zabczyk.
Stochastic Partial Differential Equations with L´evy Noise: An EvolutionEquation Approach . Cambridge University Press: Cambridge, 2007.[93] M. R¨ockner and X.-C. Zhang. Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness andergodicity.
Prob. Theory Relat. Fields , (2009), 211 – 267.[94] M. Romito and L.-H. Xu. Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildlydegenerate noise. Stoc. Proc. Appl. (2011), 673 – 700.[95] B. Schmalfuss. The random attractor of the stochastic Lorenz system.
ZAMP (1997), 951 –975.[96] Ya. Sinai. Two results concerning asymptotic behavior of solutions of the Burgers equation with force.
J. Stat. Phys. (1991), 1–12.
NVARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LAWS 39 [97] E. Tadmor and T. Tao. Velocity averaging, kinetic formulations and regularizing effects in quasi-linearpdes.
Comm. Pure Appl. Math. (2007), 1488–1521.[98] R. Temam.
Infinite Dimensional Dynamical Systems in Mechanics and Physics . Springer: New York,1997.[99] S.R.S. Varadhan.
Large Deviations.
Courant Lecture Notes, Amer. Math. Soc., 2016.[100] A. Veretennikov. On strong solutions and explicit formulas for solutions of stochastic integral equa-tions.
Math. USSR Sbornik , (1981).[101] C. Villani. Optimal Transport , Grundlehren der mathematischen Wissenschaften 338, Springer, 2009.[102] R. Zhu and X. Zhu. Three-dimensional Navier-Stokes equations driven by space-time white noise.
J.Diff. Eq. (2015), 4443–4508.
Gui-Qiang G. Chen, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
E-mail address : [email protected] Peter Pang, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
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