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Degenerate Parabolic Stochastic Partial Differential Equations

Martina Hofmanov´a1,∗

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic

ENS Cachan Bretagne, IRMAR, CNRS, UEB, av. Robert Schuman, 35 170 Bruz, France Institute of Information Theory and Automation of the ASCR, Pod Vod´arenskou vˇz´ı 4,

182 08 Praha 8, Czech Republic

Abstract

We study the Cauchy problem for a scalar semilinear degenerate parabolic par- tial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of exis- tence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approxima- tions.

Keywords: degenerate parabolic stochastic partial differential equation, kinetic solution

1. Introduction

In this paper, we study the Cauchy problem for a scalar semilinear degene- rate parabolic partial differential equation with stochastic forcing

du+ div B(u)

dt= div A(x)∇u

dt+Φ(u) dW, x∈TN, t∈(0, T),

u(0) =u0, (1)

whereW is a cylindrical Wiener process. Equations of this type are widely used in fluid mechanics since they model the phenomenon of convection-diffusion of

Correspondence address: ENS Cachan Bretagne, av. Robert Schuman, 35 170 Bruz, France

Email address: martina.hofmanova@bretagne.ens-cachan.fr(Martina Hofmanov´a)

1This research was supported in part by the GA ˇCR Grant no. P201/10/0752 and the GA UK Grant no. 556712.

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ideal fluid in porous media. Namely, the important applications including for instance two or three-phase flows can be found in petroleum engineering or in hydrogeology. For a thorough exposition of this area given from a practical point of view we refer the reader to [15] and to the references cited therein.

The aim of the present paper is to establish the well-posedness theory for solutions of the Cauchy problem (1) in any space dimension. Towards this end, we adapt the notion of kinetic formulation and kinetic solution which has already been studied in the case of hyperbolic scalar conservation laws in both deterministic (see e.g. [20], [23], [24], [26], or [27] for a general presentation) and stochastic setting (see [9]); and also in the case of deterministic degenerate parabolic equations of second-order (see [5]). To the best of our knowledge, in the degenerate case, stochastic equations of type (1) have not been studied yet, neither by means of kinetic formulation nor by any other approach.

The concept of kinetic solution was first introduced by Lions, Perthame, Tadmor in [24] for deterministic scalar conservation laws and applies to more general situations than the one of entropy solution as considered for example in [4], [12], [21]. Moreover, it appears to be better suited particularly for degen- erate parabolic problems since it allows us to keep the precise structure of the parabolic dissipative measure, whereas in the case of entropy solution part of this information is lost and has to be recovered at some stage. This technique also supplies a good technical framework to prove theL1-comparison principle which allows to prove uniqueness. Nevertheless, kinetic formulation can be de- rived only for smooth solutions hence the classical result [17] givingLp-valued solutions for the nondegenerate case has to be improved (see [19], [12]).

In the case of hyperbolic scalar conservation laws, Debussche and Vovelle [9] defined a notion of generalized kinetic solution and obtained a comparison result showing that any generalized kinetic solution is actually a kinetic solution.

Accordingly, the proof of existence simplified since only weak convergence of approximate viscous solutions was necessary.

The situation is quite different in the case of parabolic scalar conservation laws. Indeed, due to the parabolic term, the approach of [9] is not applicable: the comparison principle can be proved only for kinetic solutions (not generalized ones) and therefore strong convergence of approximate solutions is needed in order to prove the existence. Moreover, the proof of the comparison principle itself is much more delicate than in the hyperbolic case.

We note that an important step in the proof of existence, identification of the limit of an approximating sequence of solutions, is based on a new general method of constructing martingale solutions of SPDEs (see Propositions4.14, 4.15and the sequel), that does not rely on any kind of martingale representation theorem and therefore holds independent interest especially in situations where these representation theorems are no longer available. First applications were already done in [3], [25] and, in the finite-dimensional case, also in [18]. In the present work, this method is further generalized as the martingales to be dealt with are only defined for almost all times.

The exposition is organised as follows. In Section2 we review the basic set- ting and define the notion of kinetic solution. Section3 is devoted to the proof

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of uniqueness. We first establish a technical Proposition3.2 which then turns out to be the keystone in the proof of comparison principle in Theorem3.3. We next turn to the proof of existence in Sections4 and5. First of all, in Section 4, we make an additional hypothesis upon the initial condition and employ the vanishing viscosity method. In particular, we study certain nondegenerate prob- lems and establish suitable uniform estimates for the corresponding sequence of approximate solutions. The compactness argument then yields the existence of a martingale kinetic solution which together with the pathwise uniqueness gives the desired kinetic solution (defined on the original stochastic basis). In Section 5, the existence of a kinetic solution is shown for general initial data.

In the final section Appendix A, we formulate and prove an auxiliary result concerning densely defined martingales.

2. Notation and main result

We now give the precise assumptions on each of the terms appearing in the above equation (1). We work on a finite-time interval [0, T], T >0,and consider periodic boundary conditions: x∈ TN where TN is the N-dimensional torus.

The flux function

B= (B1, . . . , BN) :R−→RN

is supposed to be of classC1 with a polynomial growth of its derivative, which is denoted byb= (b1, . . . , bN). The diffusion matrix

A= (Aij)Ni,j=1:TN −→RN×N

is of class C, symmetric and positive semidefinite. Its square-root matrix, which is also symmetric and positive semidefinite, is denoted byσ.

Regarding the stochastic term, let (Ω,F,(Ft)t≥0,P) be a stochastic basis with a complete, right-continuous filtration. Let P denote the predictable σ- algebra on Ω×[0, T] associated to (Ft)t≥0. The initial datum may be random in general, i.e. F0-measurable, and we assumeu0∈Lp(Ω;Lp(TN)) for allp∈ [1,∞). The process W is a cylindrical Wiener process: W(t) =P

k≥1βk(t)ek with (βk)k≥1being mutually independent real-valued standard Wiener processes relative to (Ft)t≥0 and (ek)k≥1 a complete orthonormal system in a separable Hilbert spaceU. In this setting, we can assume, without loss of generality, that theσ-algebra F is countably generated and (Ft)t≥0 is the filtration generated by the Wiener process and the initial condition. For each z ∈ L2(TN) we consider a mapping Φ(z) : U → L2(TN) defined by Φ(z)ek = gk(·, z(·)). In particular, we suppose thatgk∈C(TN×R) and the following conditions

G2(x, ξ) =X

k≥1

gk(x, ξ)

2≤L 1 +|ξ|2

, (2)

X

k≥1

gk(x, ξ)−gk(y, ζ)

2≤L |x−y|2+|ξ−ζ|h(|ξ−ζ|)

, (3)

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are fulfilled for everyx, y∈TN, ξ, ζ∈R, wherehis a continuous nondecreasing function onR+ satisfying, for someα >0,

h(δ)≤Cδα, δ <1. (4)

The conditions imposed onΦ, particularly assumption (2), imply that Φ:L2(TN)−→L2(U;L2(TN)),

whereL2(U;L2(TN)) denotes the collection of Hilbert-Schmidt operators from Uto L2(TN). Thus, given a predictable process u ∈L2(Ω;L2(0, T;L2(TN))), the stochastic integralt7→Rt

0Φ(u)dW is a well defined process taking values in L2(TN) (see [8] for detailed construction).

Finally, define the auxiliary spaceU0⊃Uvia U0=

v=X

k≥1

αkek; X

k≥1

α2k k2 <∞

,

endowed with the norm

kvk2U0 =X

k≥1

α2k

k2, v=X

k≥1

αkek.

Note that the embeddingU,→U0is Hilbert-Schmidt. Moreover, trajectories of W areP-a.s. inC([0, T];U0) (see [8]).

In the present paper, we use the bracketsh·,·ito denote the duality between the space of distributions overTN ×RandCc(TN ×R). We denote similarly the integral

hF, Gi= Z

TN

Z

R

F(x, ξ)G(x, ξ) dxdξ, F ∈Lp(TN ×R), G∈Lq(TN×R), wherep, q ∈[1,∞] are conjugate exponents. The differential operators of gra- dient∇, divergence div and Laplacian ∆ are always understood with respect to the space variablex.

As the next step, we introduce the kinetic formulation of (1) as well as the basic definitions concerning the notion of kinetic solution. The motivation for this approach is given by the nonexistence of a strong solution and, on the other hand, the nonuniqueness of weak solutions, even in simple cases. The idea is to establish an additional criterion – the kinetic formulation – which is automatically satisfied by any strong solution to (1) and which permits to ensure the well-posedness.

Definition 2.1 (Kinetic measure). A mappingmfrom Ω to the set of non- negative finite measures over TN ×[0, T]×R is said to be a kinetic measure provided

(i) mis measurable in the following sense: for each ψ∈C0(TN ×[0, T]×R) the mappingm(ψ) : Ω→Ris measurable,

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(ii) mvanishes for largeξ: ifBRc ={ξ∈R;|ξ| ≥R}then

R→∞lim Em TN ×[0, T]×BRc

= 0, (5)

(iii) for anyψ∈C0(TN×R) Z

TN×[0,t]×R

ψ(x, ξ) dm(x, s, ξ)∈L2(Ω×[0, T]) admits a predictable representative2.

Definition 2.2 (Kinetic solution). Assume that, for allp∈[1,∞), u∈Lp(Ω×[0, T],P,dP⊗dt;Lp(TN))

and

(i) there existsCp>0 such that Eess sup

0≤t≤T

ku(t)kpLp(TN)≤Cp, (6) (ii) σ∇u∈L2(Ω×[0, T];L2(TN)).

Let n1 be a mapping from Ω to the set of nonnegative finite measures over TN×[0, T]×Rdefined for any Borel setD∈ B(TN ×[0, T]×R) as3

n1(D) = Z

TN×[0,T]

Z

R

1D(x, t, ξ) dδu(x,t)(ξ)

σ(x)∇u

2dxdt, P-a.s., (7) and let

f =1u>ξ: Ω×TN×[0, T]×R−→R.

Then u is said to be a kinetic solution to (1) with initial datum u0 provided there exists a kinetic measurem ≥ n1 a.s., such that the pair (f =1u>ξ, m) satisfies, for allϕ∈Cc(TN×[0, T)×R),P-a.s.,

Z T 0

f(t), ∂tϕ(t) dt+

f0, ϕ(0) +

Z T 0

f(t), b(ξ)· ∇ϕ(t) dt +

Z T 0

f(t),div A(x)∇ϕ(t) dt

=−X

k≥1

Z T 0

Z

TN

gk x, u(x, t)

ϕ x, t, u(x, t)

dxdβk(t)

−1 2

Z T 0

Z

TN

G2 x, u(x, t)

ξϕ x, t, u(x, t)

dxdt+m(∂ξϕ).

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2Throughout the paper, the termrepresentativestands for an element of a class of equiva- lence.

3We will write shortly dn1(x, t, ξ) = σ(x)∇u

2u(x,t)(ξ) dxdt.

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Remark 2.3. We emphasize that a kinetic solution is, in fact, a class of equi- valence in Lp(Ω×[0, T],P,dP⊗dt;Lp(TN)) so not necessarily a stochastic process in the usual sense. Nevertheless, it will be seen later (see Corollary 3.4) that, in this class of equivalence, there exists a representative with good continuity properties, namely, u ∈ C([0, T];Lp(TN)), P-a.s., and therefore, it can be regarded as a stochastic process.

Remark 2.4. Let us also make an observation which clarifies the point (ii) in the above definition: if u∈L2(Ω×[0, T];L2(TN)) then it can be shown that σ∇uis well defined inL2(Ω×[0, T];H−1(TN)) since the square-root matrixσ belongs toW1,∞(TN) according to [14], [28].

Byf =1u>ξwe understand a real function of four variables, where the addi- tional variableξis called velocity. In the deterministic case, i.e. corresponding to the situationΦ= 0, the equation (8) in the above definition is the so-called kinetic formulation of (1)

t1u>ξ+b(ξ)· ∇1u>ξ−div A(x)∇1u>ξ

=∂ξm

where the unknown is the pair (1u>ξ, m) and it is solved in the sense of distri- butions overTN×[0, T)×R. In the stochastic case, we write formally4

t1u>ξ+b(ξ)· ∇1u>ξ−div A(x)∇1u>ξ

u=ξΦ(u) ˙W+∂ξ

m−1 2G2δu=ξ

.

(9) It will be seen later that this choice is reasonable since for anyubeing a strong solution to (1) the pair (1u>ξ, n1) satisfies (8) and consequently uis a kinetic solution to (1). The measure n1 relates to the diffusion term in (1) and so is called parabolic dissipative measure. It gives us better regularity of solutions in the nondegeneracy zones of the diffusion matrix A which is exactly what one would expect according to the theory of (nondegenerate) parabolic SPDEs.

Indeed, for the case of a nondegenerate diffusion matrixA, i.e. when the second order term defines a strongly elliptic differential operator, the kinetic solutionu belongs toL2(Ω;L2(0, T;H1(TN))) (cf. Definition2.2(ii)). Thus, the measure n2=m−n1 which takes account of possible singularities of solution vanishes in the nondegenerate case.

We now derive the kinetic formulation in case of a sufficiently smooth u satisfying (1), namely, u ∈ C([0, T];C2(TN)), P-a.s.. Note, that also in this case, the measure n2 vanishes. For almost every x ∈ TN, we aim at finding the stochastic differential of θ(u(x, t)), where θ ∈ C(R) is an arbitrary test

4Hereafter, we employ the notation which is commonly used in papers concerning the kinetic solutions to conservation laws and writeδu=ξfor the Dirac measure centered atu(x, t).

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function. Such a method can be performed by the Itˆo formula since u(x, t) =u0(x)−

Z t 0

div B(u(x, s)) ds+

Z t 0

div A(x)∇u(x, s) ds

+X

k≥1

Z t 0

gk x, u(x, s)

k(s), a.e. (ω, x)∈Ω×TN,∀t∈[0, T].

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In the following we denote byh·,·iξthe duality between the space of distributions overRandCc(R). Fixx∈TN such that (10) holds true and consider1u(x,t)>ξ

as a (random) distribution onR. Then h1u(x,t)>ξ, θ0iξ =

Z

R

1u(x,t)>ξθ0(ξ) dξ=θ(u(x, t)) and the application of the Itˆo formula yields:

dh1u(x,t)>ξ, θ0iξ0(u(x, t))h

−div B(u(x, t))

dt+ div A(x)∇u(x, t) dt

+X

k≥1

gk(x, u(x, t)) dβk(t)i +1

00(u(x, t))G2(u(x, t))dt.

Afterwards, we proceed term by term and employ the fact that all the necessary derivatives ofuexists as functions

θ0(u(x, t)) div B(u(x, t))

0(u(x, t))b(u(x, t))· ∇u(x, t)

= div

Z u(x,t)

−∞

b(ξ)θ0(ξ)dξ

= div hb1u(x,t)>ξ, θ0iξ θ0(u(x, t)) div A(x)∇u(x, t)

=

N

X

i,j=1

xi

Aij(x)θ0(u(x, t))∂xju(x, t)

N

X

i,j=1

θ00(u(x, t))∂xiu(x, t)Aij(x)∂xju(x, t)

=

N

X

i,j=1

xi

Aij(x)∂xj

Z u(x,t)

−∞

θ0(ξ)dξ

+

ξn1(x, t), θ0

ξ

= div

A(x)∇h1u(x,t)>ξ, θ0iξ

+

ξn1(x, t), θ0

ξ

θ0(u(x, t))gk(x, u(x, t)) =hgk(x, ξ)δu(x,t)=ξ, θ0iξ

θ00(u(x, t))G2(x, u(x, t)) =hG2(x, ξ)δu(x,t)=ξ, θ00iξ

=−

ξ(G2(x, ξ)δu(x,t)=ξ), θ0

ξ

Note, that according to the definition of the parabolic dissipative measure (7) it makes sense to write∂ξn1(x, t), i.e for fixed x, twe regardn1(x, t) as a random measure onR: for any Borel setD1∈ B(R)

n1(x, t, D1) =

σ(x)∇u(x, t)

2δu(x,t)(D1), P-a.s..

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In the following, we distinguish between two situations. In the first case, we intend to use test functions independent ont. We setθ(ξ) =Rξ

−∞ϕ1(ζ) dζ for some test functionϕ1 ∈Cc(R) and test the above against ϕ2 ∈C(TN).

Since linear combinations of the test functions ψ(x, ξ) = ϕ1(ξ)ϕ2(x) form a dense subset of Cc(TN ×R) we obtain for any ψ ∈Cc(TN ×R), t∈ [0, T], P-a.s.,

f(t), ψ

− f0, ψ

− Z t

0

f(s), b(ξ)· ∇ψ ds−

Z t 0

f(s),div A(x)∇ψ ds

= Z t

0

δu=ξΦ(u) dW, ψ +1

2 Z t

0

δu=ξG2, ∂ξψ ds−

n1, ∂ξψ ([0, t)), where

n1, ∂ξψ

([0, t))1ξψ1[0,t)

. In order to allow test functions fromCc(TN× [0, T)×R), take ϕ3 ∈ Cc([0, T)) and apply the Itˆo formula to calculate the stochastic differential of the producthf(t), ψiϕ3(t). We have,P-a.s.,

f(t), ψ

ϕ3(t)− f0, ψ

ϕ3(0)− Z t

0

f(s), b(ξ)· ∇ψ

ϕ3(s) ds

− Z t

0

f(s),div A(x)∇ψ

ϕ3(s) ds

= Z t

0

δu=ξΦ(u)ϕ3(s) dW, ψ +1

2 Z t

0

δu=ξG2, ∂ξψ

ϕ3(s) ds

−n1ξψ1[0,t)ϕ3 +

Z t 0

f(s), ψ

sϕ3(s) ds.

Evaluating this process at t = T and setting ϕ(x, t, ξ) = ψ(x, ξ)ϕ3(t) yields the equation (8) hence f = 1u>ξ is a distributional solution to the kinetic formulation (9) withn2 = 0. Therefore any strong solution of (1) is a kinetic solution in the sense of Definition2.2.

Concerning the point (ii) in Definition 2.2, it was already mentioned in Remark2.4thatσ∇uis well defined inL2(Ω×[0, T];H−1(TN)). As we assume more in Definition2.2(ii) we obtain the following chain rule formula, which will be used in the proof of Theorem3.3,

σ∇f =σ∇u δu=ξ in D0(TN ×R),a.e. (ω, t)∈Ω×[0, T]. (11) It is a consequence of the next result.

Lemma 2.5. Assume thatv∈L2(TN)andσ(∇v)∈L2(TN). Ifg=1v>ξthen it holds true

σ∇g=σ∇v δv=ξ in D0(TN×R).

Proof. In order to prove this claim, we denote by σi the ith row ofσ. Let us fix test functionsψ1∈C(TN), ψ2 ∈Cc(R) and defineθ(ξ) =Rξ

−∞ψ2(ζ) dζ.

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We denote byh·,·ixthe duality between the space of distributions overTN and C(TN). It holds

σi∇g, ψ1ψ2

=−D

div(σiψ1), Z v

−∞

ψ2(ξ) dξE

x=−

div σiψ1

, θ(v)

x

=

σi∇θ(v), ψ1

x. If the following was true

σi∇θ(v) =θ0(v)σi∇v in D0(TN), (12) we would obtain

σi∇g, ψ1ψ2

=

θ0(v)σi∇v, ψ1

x=

σi∇v δv=ξ, ψ1ψ2

and the proof would be complete.

Hence it remains to verify (12). Towards this end, let us consider an ap- proximation to the identity on TN, denoted by (%τ). To be more precise, let

˜

% ∈ Cc(RN) be nonnegative symmetric function satisfying R

RN%˜ = 1 and supp ˜% ⊂ B(0,1/2). This function can be easily extended to become ZN- periodic, let this modification denote by ¯%. Now it is correct to define%= ¯%◦q−1, whereqdenotes the quotient mappingq:RN →TN =RN/ZN, and finally

%τ(x) = 1 τN %x

τ

.

Since the identity (12) is fulfilled by any sufficiently regular v, let us consider vτ, the mollifications ofv given by (%τ). We have

σi∇θ(vτ)−→σi∇θ(v) in D0(TN).

In order to obtain convergence of the corresponding right hand sides, i.e.

θ0(vτi∇vτ−→θ0(v)σi∇v in D0(TN),

we employ similar arguments as in the commutation lemma of DiPerna and Li- ons (see [10, Lemma II.1]). Namely, sinceσi(∇v)∈L2(TN) it is approximated inL2(TN) by its mollifications [σi∇v]τ. Consequently,

θ0(vτ) σi∇vτ

−→θ0(v)σi∇v in D0(TN).

Thus, it is enough to show that θ0(vτ)

σi∇vτ

σi∇vτ

−→0 in D0(TN). (13) It holds

σi(x)∇vτ(x)− σi∇vτ

(x)

= Z

TN

v(y)σi(x)(∇%τ)(x−y) dy+ Z

TN

v(y) divy σi(y)%τ(x−y) dy

=− Z

TN

v(y) σi(y)−σi(x)

(∇%τ)(x−y)dy+ Z

TN

v(y) div σi(y)

%τ(x−y)dy.

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The second term on the right hand side is the mollification ofvdivσi∈L2(TN) hence converges inL2(TN) tovdivσi. We will show that the first term converges inL1(TN) to−vdivσi. Sinceτ|∇%τ|(·)≤C%(·) with a constant independent onτ, we obtain the following estimate

Z

TN

v(y) σi(y)−σi(x)

(∇%τ)(x−y) dy L2(

TN)

≤CkσikW1,∞(TN)kvkL2(TN).

Due to this estimate, it is sufficient to considervandσismooth and the general case can be concluded by a density argument. We infer5

− Z

TN

v(y) σi(y)−σi(x)

(∇%τ)(x−y) dy

=− 1 τN+1

Z

TN

Z 1

0

v(y) Dσi x+r(y−x)

(y−x)·(∇%)x−y τ

drdy

= Z

TN

Z 1

0

v(x−τ z) Dσi(x−rτ z)z·(∇%)(z) drdz

−→v(x) Dσi(x) : Z

TN

z⊗(∇%)(z) dz, ∀x∈TN. Integration by parts now yields

Z

TN

z⊗(∇%)(z) dz=−Id (14)

hence

v(x) Dσi(x) : Z

TN

z⊗(∇%)(z) dz=−v(x) div σi(x)

, ∀x∈TN,

and the convergence inL1(TN) follows by the Vitali convergence theorem from the above estimate. Employing the Vitali convergence theorem again, we obtain (13) and consequently also (12) which completes the proof.

We proceed by two related definitions, which will be useful especially in the proof of uniqueness.

Definition 2.6 (Young measure). Let (X, λ) be a finite measure space. A mappingν fromX to the set of probability measures onRis said to be a Young measure if, for allψ∈Cb(R), the mapz7→νz(ψ) fromX intoRis measurable.

We say that a Young measureν vanishes at infinity if, for allp≥1, Z

X

Z

R

|ξ|pz(ξ) dλ(z)<∞.

5By : we denote the component-wise inner product of matrices and bythe tensor product.

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Definition 2.7 (Kinetic function). Let (X, λ) be a finite measure space. A measurable functionf :X×R→[0,1] is said to be a kinetic function if there exists a Young measureνonX vanishing at infinity such that, forλ-a.e. z∈X, for allξ∈R,

f(z, ξ) =νz(ξ,∞).

Remark 2.8. Note, that if f is a kinetic function then ∂ξf = −ν for λ-a.e.

z∈X. Similarly, letube a kinetic solution of (1) and consider f =1u>ξ. We have ∂ξf = −δu=ξ, whereν =δu=ξ is a Young measure on Ω×TN ×[0, T].

Therefore, the expression (8) can be rewritten in the following form: for all ϕ∈Cc(TN ×[0, T)×R),P-a.s.,

Z T 0

f(t), ∂tϕ(t) dt+

f0, ϕ(0) +

Z T 0

f(t), b(ξ)· ∇ϕ(t) dt +

Z T 0

f(t),div A(x)∇ϕ(t) dt

=−X

k≥1

Z T 0

Z

TN

Z

R

gk(x, ξ)ϕ(x, t, ξ)dνx,t(ξ) dxdβk(t)

−1 2

Z T 0

Z

TN

Z

R

G2(x, ξ)∂ξϕ(x, t, ξ)dνx,t(ξ) dxdt+m(∂ξϕ).

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For a general kinetic functionf with corresponding Young measureν, the above formulation leads to the notion of generalized kinetic solution as used in [9]. Al- though this concept is not established here, the notation will be used throughout the paper, i.e. we will often writeνx,t(ξ) instead of δu(x,t)=ξ.

Lemma 2.9. Let(X, λ)be a finite measure space such thatL1(X)is separable.6 Let {fn;n ∈ N} be a sequence of kinetic functions on X ×R, i.e. fn(z, ξ) = νzn(ξ,∞)whereνn are Young measures onX. Suppose that, for some p≥1,

sup

n∈N

Z

X

Z

R

|ξ|pzn(ξ) dλ(z)<∞.

Then there exists a kinetic functionf onX×Rand a subsequence still denoted by{fn;n∈N} such that

fn w

−→f, in L(X×R)-weak. Proof. The proof can be found in [9, Corollary 6].

To conclude this section we state the main result of the paper.

6According to [7, Proposition 3.4.5], it is sufficient to assume that the correspondingσ- algebra is countably generated.

(12)

Theorem 2.10. Let u0 ∈Lp(Ω;Lp(TN)), for all p∈[1,∞). Under the above assumptions, there exists a unique kinetic solution to the problem(1)and it has almost surely continuous trajectories in Lp(TN), for all p∈[1,∞). Moreover, ifu1, u2are kinetic solutions to (1)with initial datau1,0 andu2,0, respectively, then for allt∈[0, T]

Eku1(t)−u2(t)kL1(TN)≤Eku1,0−u2,0kL1(TN). 3. Uniqueness

We begin with the question of uniqueness. Due to the following proposition, we obtain an auxiliary property of kinetic solutions, which will be useful later on in the proof of the comparison principle in Theorem3.3.

Proposition 3.1(Left- and right-continuous representatives). Letube a kinetic solution to (1). Then f =1u>ξ admits representatives f and f+ which are almost surely left- and right-continuous, respectively, at all pointst∈[0, T]in the sense of distributions overTN×R. More precisely, for all t∈[0, T]there exist kinetic functions f∗,± on Ω×TN ×R such that setting f±(t) = f∗,±

yieldsf±=f almost everywhere and f±(t± ε), ψ

−→

f±(t), ψ

ε↓0 ∀ψ∈Cc2(TN×R) P-a.s..

Moreover,f+=f for allt∈[0, T]except for some at most countable set.

Proof. As the space Cc2(TN ×R) (endowed with the topology of the uniform convergence on any compact set of functions and their first and second deriva- tives) is separable, let us fix a countable dense subset D1. Let ψ ∈ D1 and α ∈ Cc1([0, T)) and set ϕ(x, t, ξ) = ψ(x, ξ)α(t). Integration by parts and the stochastic version of Fubini’s theorem applied to (15) yield

Z T 0

gψ(t)α0(t)dt+hf0, ψiα(0) =hm, ∂ξψi(α) P-a.s.

where gψ(t) =

f(t), ψ

− Z t

0

f(s), b(ξ)· ∇ψ ds−

Z t 0

f(t),div A(x)∇ψ ds

−X

k≥1

Z t 0

Z

TN

Z

R

gk(x, ξ)ψ(x, ξ) dνx,s(ξ) dxdβk(s)

−1 2

Z t 0

Z

TN

Z

R

ξψ(x, ξ)G2(x, ξ) dνx,s(ξ) dxds.

(16)

Hence ∂tgψ is a (pathwise) Radon measure on [0, T] and by the Riesz repre- sentation theoremgψ ∈BV([0, T]). Due to the properties ofBV-functions [2, Theorem 3.28], we obtain thatgψ admits left- and right-continuous representa- tives which coincide except for an at most countable set. Moreover, apart from

(13)

the first one all terms in (16) are almost surely continuous in t. Hence, on a set of full measure, denoted by Ωψ,hf, ψialso admits left- and right-continuous representatives which coincide except for an at most countable set. Let them be denoted byhf, ψi± and set Ω0=∩ψ∈D1ψ.Note, that asD1is countable, Ω0is also a set of full measure. Besides, forψ∈ D1, (t, ω)7→ hf(t, ω), ψi+ has right continuous trajectories in time and is thus measurable with respect to (t, ω).

For ψ ∈ Cc2(TN ×R), we define hf(t, ω), ψi+ on [0, T]×Ω0 as the limit of hf(t, ω), ψni+ for any sequence (ψn) in D1 converging to ψ. Then clearly hf(·,·), ψi+ is also measurable in (t, ω) and has right continuous trajectories.

It is now straightforward to definef+ byhf+, ψi=hf, ψi+. Thenf+: Ω× [0, T]→L(TN×R). Moreover, seen as a functionf+: Ω×[0, T]→Lploc(TN× R), for some p ∈ [1,∞), it is weakly measurable and therefore measurable.

According to Fubini theorem f+, as a function of four variables ω, x, t, ξ, is measurable.

Besides, f+ is a.s. right-continuous in the required sense. Next, we show that f+ is a representative (in time) of f, i.e. for a.e. t ∈ [0, T) it holds thatf+(t) =f(t), where the equality is understood in the sense of classes of equivalence inω, x, ξ. Indeed, due to the Lebesgue differentiation theorem,

ε→0lim 1 ε

Z t t

f(ω, x, t, ξ) dt=f(ω, x, t, ξ) a.e. (ω, x, t, ξ) hence by the dominated convergence theorem

ε→0lim 1 ε

Z t t

f(t, ω), ψ dt=

f(t, ω), ψ

a.e. (ω, t), for anyψ∈Cc2(TN×R). Since this limit is equal to

f+(t, ω), ψ

fort∈[0, T] andω∈Ω0, the conclusion follows.

Now, it only remains to show that f+(t) is a kinetic function on Ω×TN for allt∈[0, T). Towards this end, we observe that for allt∈[0, T)

fn(x, t, ξ) := 1 εn

Z tn

t

f(x, t, ξ) dt

is a kinetic function onX = Ω×TN and by (6) the assumptions of Lemma2.9 are fulfilled. Accordingly, there exists a kinetic functionf∗,+and a subsequence (nk) (which also depends ont) such that

fnk(t) w

−→f∗,+ in L(Ω×TN ×R)-w.

Note, that the domain of definition off∗,+does not depend onψ. Thus on the one hand we have

fnk(t), ψ w

−→

f∗,+, ψ

in L(Ω)-w, and on the other hand, due to the definition off+,

fnk(t), ψ w

−→

f+(t), ψ

in L(Ω)-w.

(14)

The proof of existence of the left-continuous representativefcan be carried out similarly and so will be left to the reader.

The fact thatf+(t) and f(t) coincide for all t ∈(0, T)\I, where I ⊂ (0, T) is countable, follows directly from their definition since the representatives hf(t, ω), ψi+ and hf(t, ω), ψi coincide except for an at most countable set forψ∈ D1.

From now on, we will work with these two fixed representatives of f and we can take any of them in an integral with respect to time or in a stochastic integral.

As the next step towards the proof of the comparison principle, we need a technical proposition relating two kinetic solutions of (1). We will also use the following notation: if f :X×R→[0,1] is a kinetic function, we denote by ¯f the conjugate function ¯f = 1−f.

Proposition 3.2. Let u1, u2 be two kinetic solutions to (1) and denote f1 = 1u1, f2 = 1u2. Then for t ∈ [0, T] and any nonnegative functions % ∈ C(TN), ψ∈Cc(R)we have

E Z

(TN)2

Z

R2

%(x−y)ψ(ξ−ζ)f1±(x, t, ξ) ¯f2±(y, t, ζ) dξdζdxdy

≤E Z

(TN)2

Z

R2

%(x−y)ψ(ξ−ζ)f1,0(x, ξ) ¯f2,0(y, ζ) dξdζdxdy+ I + J + K, (17) where

I = E Z t

0

Z

(TN)2

Z

R2

f12 b(ξ)−b(ζ)

· ∇xα(x, ξ, y, ζ) dξdζdxdyds,

J =E Z t

0

Z

(TN)2

Z

R2

f12 N

X

i,j=1

yj Aij(y)∂yiα

dξdζdxdyds

+E Z t

0

Z

(TN)2

Z

R2

f12

N

X

i,j=1

xj Aij(x)∂xiα

dξdζdxdyds

−E Z t

0

Z

(TN)2

Z

R2

α(x, ξ, y, ζ) dνx,s1 (ξ) dxdn2,1(y, s, ζ)

−E Z t

0

Z

(TN)2

Z

R2

α(x, ξ, y, ζ) dνy,s2 (ζ) dydn1,1(x, s, ξ), K = 1

2E Z t

0

Z

(TN)2

Z

R2

α(x, ξ, y, ζ)X

k≥1

gk(x, ξ)−gk(y, ζ)

2x,s1 (ξ)dνy,s2 (ζ)dxdyds, and the functionαis defined as α(x, ξ, y, ζ) =%(x−y)ψ(ξ−ζ).

(15)

Proof. Let us denote by hh·,·ii the scalar product inL2(TNx ×TNy ×Rξ×Rζ).

In order to prove the statement in the case off1+,f¯2+, we employ similar calcu- lations as in [9, Proposition 9] to obtain

E

f1+(t) ¯f2+(t), α

=E

f1,02,0, α +E

Z t 0

Z

(TN)2

Z

R2

f12 b(ξ)−b(ζ)

· ∇xαdξdζdxdyds

+E Z t

0

Z

(TN)2

Z

R2

f12 N

X

i,j=1

yj Aij(y)∂yiα

dξdζdxdyds

+E Z t

0

Z

(TN)2

Z

R2

f12

N

X

i,j=1

xj Aij(x)∂xiα

dξdζdxdyds

+1 2E

Z t 0

Z

(TN)2

Z

R2

2ξα G21x,s1 (ξ) dζdydxds

−1 2E

Z t 0

Z

(TN)2

Z

R2

f1ζα G22y,s2 (ζ) dξdydxds

−E Z t

0

Z

(TN)2

Z

R2

G1,2αdνx,s1 (ξ) dνy,s2 (ζ) dxdyds

−E Z t

0

Z

(TN)2

Z

R2

2ξαdm1(x, s, ξ) dζdy

+E Z t

0

Z

(TN)2

Z

R2

f1+ζαdm2(y, s, ζ) dξdx.

(18)

In particular, sinceα≥0, the last term in (18) satisfies E

Z t 0

Z

(TN)2

Z

R2

f1+ζαdm2(y, s, ζ) dξdx

=−E Z t

0

Z

(TN)2

Z

R2

αdνx,s1 (ξ) dxdn2,1(y, s, ζ)

−E Z t

0

Z

(TN)2

Z

R2

αdνx,s1 (ξ) dxdn2,2(y, s, ζ)

≤ −E Z t

0

Z

(TN)2

Z

R2

αdνx,s1 (ξ) dxdn2,1(y, s, ζ) and by symmetry

−E Z t

0

Z

(TN)2

Z

R2

2ξαdm1(x, s, ξ) dζdy

≤ −E Z t

0

Z

(TN)2

Z

R2

αdνy,s2 (ζ) dydn1,1(x, s, ξ).

(16)

Thus, the desired estimate (17) follows.

In the case off1, f¯2 we take tn ↑ t, write (17) forf1+(tn),f¯2+(tn) and let n→ ∞.

Theorem 3.3(Comparison principle). Letube a kinetic solution to (1). Then there exist u+ and u, representatives of u, such that, for all t ∈ [0, T], f±(x, t, ξ) = 1u±(x,t)>ξ for a.e. (ω, x, ξ). Moreover, if u1, u2 are kinetic so- lutions to (1)with initial datau1,0 andu2,0, respectively, then for allt∈[0, T]

Eku±1(t)−u±2(t)kL1(TN)≤Eku1,0−u2,0kL1(TN). (19) Proof. Denotef1=1u1, f2=1u2. Let (ψδ),(%τ) be approximations to the identity onRand TN, respectively. Namely, let ψ∈Cc(R) be a nonnegative symmetric function satisfyingR

Rψ= 1, suppψ⊂(−1,1) and set ψδ(ξ) =1

δψξ δ

.

For the space variable x∈ TN, we employ the approximation to the identity defined in Lemma2.5. Then we have

E Z

TN

Z

R

f1±(x, t, ξ) ¯f2±(x, t, ξ) dξdx

=E Z

(TN)2

Z

R2

%τ(x−y)ψδ(ξ−ζ)f1±(x, t, ξ) ¯f2±(y, t, ζ) dξdζdxdy+ηt(τ, δ), where limτ,δ→0ηt(τ, δ) = 0. With regard to Proposition 3.2, we need to find suitable bounds for terms I,J,K.

Sincebhas at most polynomial growth, there existC >0, p >1 such that b(ξ)−b(ζ)

≤Γ(ξ, ζ)|ξ−ζ|, Γ(ξ, ζ)≤C 1 +|ξ|p−1+|ζ|p−1 .

Hence

|I| ≤E Z t

0

Z

(TN)2

Z

R2

f12Γ(ξ, ζ)|ξ−ζ|ψδ(ξ−ζ) dξdζ

x%τ(x−y)

dxdyds.

As the next step we apply integration by parts with respect toζ, ξ. Focusing only on the relevant integrals we get

Z

R

f1(ξ) Z

R

2(ζ)Γ(ξ, ζ)|ξ−ζ|ψδ(ξ−ζ)dζdξ

= Z

R

f1(ξ) Z

R

Γ(ξ, ζ0)|ξ−ζ0δ(ξ−ζ0)dζ0

− Z

R2

f1(ξ) Z ζ

−∞

Γ(ξ, ζ0)|ξ−ζ0δ(ξ−ζ0)dζ0dξdνy,s2 (ζ)

= Z

R2

f1(ξ) Z

ζ

Γ(ξ, ζ0)|ξ−ζ0δ(ξ−ζ0)dζ0dξdνy,s2 (ζ)

= Z

R2

Υ(ξ, ζ)dνx,s1 (ξ)dνy,s2 (ζ)

(20)

(17)

where

Υ(ξ, ζ) = Z ξ

−∞

Z

ζ

Γ(ξ0, ζ0)|ξ0−ζ0δ0−ζ0)dζ00. Therefore, we find

|I| ≤E Z t

0

Z

(TN)2

Z

R2

Υ(ξ, ζ) dνx,s1 (ξ)dνy,s2 (ζ)

x%τ(x−y)

dxdyds.

The functionΥ can be estimated using the substitutionξ000−ζ0 Υ(ξ, ζ) =

Z

ζ

Z

00|<δ, ξ00<ξ−ζ0

Γ(ξ000, ζ0)|ξ00δ00) dξ000

≤Cδ Z ξ+δ

ζ

00|<δ, ξmax00<ξ−ζ0Γ(ξ000, ζ0) dζ0

≤Cδ Z ξ+δ

ζ

1 +|ξ|p−1+|ζ0|p−10

≤Cδ 1 +|ξ|p+|ζ|p hence, sinceν1, ν2 vanish at infinity,

|I| ≤Ctδ Z

TN

x%τ(x)

dx≤Ctδτ−1. We recall thatf1=1u1(x,t)>ξ, f2=1u2(y,t)>ζ hence

ξf1=−ν1=−δu1(x,t)=ξ, ∂ζf2=−ν2=−δu2(y,t)=ζ

and as bothu1, u2possess some regularity in the nondegeneracy zones ofAdue to Definition2.2(ii), we obtain as in (11)

σ∇f1=σ∇u1δu1(x,s)=ξ, σ∇f¯2=−σ∇u2δu2(y,s)=ζ

in the sense of distributions overTN×R. The first term in J can be rewritten in the following manner using integration by parts (and considering only relevant integrals)

Z

TN

f1 Z

TN

2yj Aij(y)∂yi%τ(x−y) dydx

= Z

(TN)2

f1(x, s, ξ)Aij(y)∂yj2(y, s, ζ)∂xi%τ(x−y)dxdy.

and similarly for the second term. Let us define Θδ(ξ) =

Z ξ

−∞

ψδ(ζ) dζ.

(18)

Then we have J = J1+ J2+ J3with J1=−E

Z t 0

Z

(TN)2

(∇xu1)σ(x)σ(x)(∇%τ)(x−y)Θδ u1(x, s)−u2(y, s)

dxdyds, J2=−E

Z t 0

Z

(TN)2

(∇yu2)σ(y)σ(y)(∇%τ)(x−y)Θδ u1(x, s)−u2(y, s)

dxdyds, J3=−E

Z t 0

Z

(TN)2

|σ(x)∇xu1|2+|σ(y)∇yu2|2

%τ(x−y)

×ψδ u1(x, s)−u2(y, s)

dxdyds.

Let H =E

Z t 0

Z

(TN)2

(∇xu1)σ(x)σ(y)(∇yu2)%τ(x−y)ψδ u1(x, s)−u2(y, s)

dxdyds.

We intend to show that J1= H +o(1),J2= H +o(1), whereo(1)→0 asτ →0 uniformly inδ, and consequently

J =−E Z t

0

Z

(TN)2

σ(x)∇xu1−σ(y)∇yu2

2%τ(x−y)

×ψδ u1(x, s)−u2(y, s)

dxdyds+o(1)≤o(1).

(21)

We only prove the claim for J1since the case of J2 is analogous. Let us define g(x, y, s) = (∇xu1)σ(x)Θδ u1(x, s)−u2(y, s)

.

Here, we employ again the assumption (ii) in Definition2.2. Recall, that it gives us some regularity of the solution in the nondegeneracy zones of the diffusion matrixAand henceg∈L2(Ω×TNx ×TNy ×[0, T]). It holds

J1=−E Z t

0

Z

(TN)2

g(x, y, s)

σ(x)−σ(y)

(∇%τ)(x−y) dxdyds

−E Z t

0

Z

(TN)2

g(x, y, s)σ(y)(∇%τ)(x−y) dxdyds, H =E

Z t 0

Z

(TN)2

g(x, y, s) divy

σ(y)%τ(x−y)

dxdyds

=E Z t

0

Z

(TN)2

g(x, y, s) div σ(y)

%τ(x−y) dxdyds

−E Z t

0

Z

(TN)2

g(x, y, s)σ(y)(∇%τ)(x−y) dxdyds,

where divergence is applied row-wise to a matrix-valued function. Therefore, it is enough to show that the first terms in J1 and H have the same limit value if

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