On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with L\'{e}vy noise

In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative L\'{e}vy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.


Introduction
Let Ω, P, F , {F t } t≥0 be a filtered probability space satisfying the usual hypothesis i.e. {F t } t≥0 is a right-continuous filtration such that F 0 contains all the P -null subsets of (Ω, F ). In addition, let E, E, m be a σ-finite measure space and N ( dt, dz) be a Poisson random measure on E, E with intensity measure m( dz) with respect to the same stochastic basis. The existence and construction of such general notion of Poisson random measure with a given intensity measure are detailed in [22]. We are interested in the Cauchy problem for a nonlinear degenerate parabolic stochastic PDE of the following type du(t, x) − ∆φ(u(t, x)) dt − div x f (u(t, x)) dt = E η(x, u(t, x); z)Ñ (dz, dt), (t, x) ∈ Π T , (1.1) with the initial condition where Π T = [0, T ) × R d with T > 0 fixed, u(t, x) is the unknown random scalar valued function, F : R → R d is given flux function, andÑ (dz, dt) = N (dz, dt) − m(dz) dt, the compensated Poisson random measure. Furthermore, (x, u, z) → η(x, u; z) is a real valued function defined on the domain R d × R × E and φ : R → R is a given non-decreasing Lipschitz continuous function. The stochastic integral in the RHS of (1.1) is defined in the Lévy-Itô sense.
Remark 1.1. Since φ is a real valued non-decreasing, Lipschitz continuous function, the set A = r ∈ R : φ ′ (r) = 0 is not empty in general and hence the problem is called degenerate. Even more, A is not negligible either and the problem is strongly degenerate in the sense of [10].
Remark 1.2. The analysis of this paper remains valid if the noise on the RHS of (1.1) is of jumpdiffusion type. In other words, the same analysis holds if we add a σ(x, u)dW t term in the RHS of (1.1) where W t is a cylindrical Brownian motion. Moreover, we will carry out our analysis under the structural assumption E = O × R * where O is a subset of the Euclidean space. The measure m on E is defined as λ × µ where λ is a Radon measure on O and µ is a so called Lévy measure on R * . In such a case, the noise of the RHS would be called an impulsive white noise with jump position intensity λ and jump size intensity µ. We refer to [22] for more on Lévy sheet and related impulsive white noise.
The equation (1.1) becomes a multidimensional deterministic degenerate parabolic-hyperbolic equation if η = 0. It is well-documented in the literature that the solution has to be interpreted in the weak sense and one needs an entropy formulation to prove wellposedness. We refer to [1,10,12,13,5,25] and references therein for more on entropy solution theory for deterministic degenerate parabolic-hyperbolic equations.
1.1. Studies on degenerate parabolic-hyperbolic equations with Brownian noise. The study of stochastic degenerate parabolic-hyperbolic equations has so far been limited to mainly equations with Brownian noise. In particular, hyperbolic conservation laws with Brownian noise are the examples of such problems that have attracted the attention of many. The first documented development in this direction is [19], where the authors established existence of path-wise weak solution (possibly non-unique) of one dimensional balance laws via splitting method. In a separate development, Khanin et al. [20] published their celebrated work that described some statistical properties of Burgers equations with noise. J. U. Kim [21] extended Kruzkov's entropy formulation and established the wellposedness for one dimensional balance laws that are driven by additive Brownian noise. Multidimensional case was studied by Vallet and Wittbold [23], and they established wellposedness of entropy solution with the theory of Young-measures but in a bounded domain.
This approach is not applicable for multiplicative noise case. This was studied by many authors ( [4,11,15,18]). In [18], Feng and Nualart came up with a way to recover the necessary information in the form of strong entropy condition from the parabolic regularisation and established the uniqueness of strong entropy solution in L p -framework for several space dimensions but the existence was for one space dimension. We also add here that Feng and Nualart [18] uses an entropy formulation which is strong in time but weak in space, which in our view may give rise to problems where the solutions are not shown to have continuous sample paths. We refer to [6], where a few technical questions are raised on the strong in time formulation and remedial measures have been proposed. In [15], the authors obtain the existence via kinetic formulation and [11] uses BV solution framework. In a recent paper, Vallet et al. [4], established the wellposedness via the Young's measure approach. The wellposedness result of the multidimensional degenerate parabolic-hyperbolic stochastic problem has been studied by Vovelle, Hofmanova and Debussche [14], and Vallet et al. [3]. In [14], they adapt the notion of kinetic formulation and develop a wellposedness theory. In [3], the authors revisited [1,10,12] and established the wellposedness of the entropy solution via Young's measure theory.

1.2.
Relevant studies on problems with Lévy noise. Over the last decade there has been many contributions on the larger area of stochastic partial differential equations that are driven by Lévy noise. An worthy reference on this subject is [22]. However, very little is available on the specific problem of degenerate parabolic problems with Lévy noise such as (1.1). This article marks an important step in our quest to develop a comprehensive theory of stochastic degenerate parabolic equations that are driven by jump-diffusions. The relevant results in this context are made available recently and they are on conservation laws that are perturbed by Lévy noise. In recent articles [7,8], Biswas et al. established existence, uniqueness of entropy solution for multidimensional conservation laws with Poisson noise via Young measure approach. In [8], the authors developed a continuous dependence theory on nonlinearities within BV solution setting.
Stochastic degenerate parabolic-hyperbolic equations are one of the most important classes of nonlinear stochastic PDEs. Nonlinearity and degeneracy are two main features of these equations and yield several striking phenomena. Therefore, it requires new mathematical ideas, approaches, and theories. It is well-known that due to presence of nonlinear flux term, solutions to (1.1) are not smooth even for smooth initial data u 0 (x). Therefore the solutions must be interpreted in the weak sense. Before introducing the concept of weak solutions, we first recall the notion of predictable σ-field. By a predictable σ-field on [0, T ] × Ω, denoted by P T , we mean that the σ-field generated by the sets of the form: {0} × A and (s, t] × B for any A ∈ F 0 ; B ∈ F s , 0 < s, t ≤ T . The notion of stochastic weak solution is defined as follows. x) is said to be a weak solution to our problem (1.1) provided For almost every t ∈ [0, T ] and P − a.s, the following variational formulation holds: However, it is well-known that weak solutions may be discontinuous and they are not uniquely determined by their initial data. Consequently, an admissibility criterion for so called entropy solution (see Section 2 for the definition of entropy solution) must be imposed to single out the physically correct solution.
1.3. Goal of the study and outline of the paper. The case of a strongly degenerate stochastic problem driven by Brownian noise is studied by Bauzet et al. [3]. In this article, drawing primary motivation from [3,7,10], we propose to establish the wellposedness of the entropy solution to degenerate Cauchy problem (1.1) by using vanishing viscosity method along with few a priori bounds.
The rest of the paper is organized as follows. We state the assumptions, details of the technical framework and state the main results in Section 2. Section 3 is devoted to prove the existence of weak solution for viscous problem via implicit time discretization scheme and to derive some a priori estimates for viscous solution. In section 4, we first establish uniqueness of the limit of viscous solutions as viscous parameter goes to zero via Young measure theory and then we establish existence of entropy solution. The uniqueness of the entropy solution is presented in the final section.

Technical framework and statements of the main results
Here and in the sequel, we denote by N 2 ω (0, T, L 2 (R d )) the space of predictable L 2 (R d )-valued processes u such that E ΠT |u| 2 dt dx < +∞. Moreover, we use the letter C to denote various generic constants. There are situations where constants may change from line to line, but the notation is kept unchanged so long as it does not impact the primary implication. We denote c φ and c f the Lipschitz constants of φ, and f respectively. Also, we use , to denote the pairing between H 1 (R d ) and H −1 (R d ).
2.1. Entropy inequalities. We begin this subsection with a formal derivation of entropy inequalitiesà la Kruzkov. Remember that we need to replace the traditional chain rule for deterministic calculus by Itô-Lévy chain rule. Definition 2.1 (Entropy flux triple). A triplet (β, ζ, ν) is called an entropy flux triple if β ∈ C 2 (R), Lipschitz and β ≥ 0, ζ = (ζ 1 , ζ 2 , ....ζ d ) : R → R d is a vector valued function, and ν : R → R is a scalar valued function such that For a small positive number ε > 0, assume that the parabolic perturbation of (1.1) has a unique weak solution u ε (t, x). Note that this weak solution u ε ∈ L 2 ((0, T ) × Ω; H 1 (R d )). Moreover, for the time being, we assume that it satisfies the initial condition in the sense of (A.2). This enables one to derive a weak version of Itô -Lévy formula for the solutions to (1.1), as detailed in the Theorem A.1 in the Appendix.
Let (β, ζ, ν) be an entropy flux triple. Given a nonnegative test function ψ ∈ C 1,2 c ([0, ∞) × R d ), we apply generalised version of the Itô-Lévy formula to have, for almost every T > 0, Let G be the associated Kirchoff's function of φ, given by G(x) = x 0 φ ′ (r) dr. A simple calculation shows that |∇G(u ε (t, x))| 2 = φ ′ (u ε (t, x))|∇u ε (t, x)| 2 . Since β and ψ are nonnegative functions, we obtain Clearly, the above inequality is stable under the limit ε → 0, if the family {u ε } ε>0 has L p loc -type stability. Just as the deterministic equations, the above inequality provides us with the entropy condition. We now formally define the entropy solution.
The primary aim of this paper is to establish the existence and uniqueness of entropy solutions for the Cauchy problem (1.1) in accordance with Definition 2.2, and we do so under the following assumptions: is not a constant function with respect to the space variable x, t −→ φ ′ (t) has a modulus of continuity ω φ such that The space E is of the form O × R * and the Borel measure m on E has the form λ × µ where λ is a Radon measure on O and µ is a so-called one dimensional Lévy measure. (A.4) There exist positive constants K > 0, λ * ∈ (0, 1) and h 1 (z) ∈ L 2 (E, m) with 0 ≤ h 1 (z) ≤ 1 such that (A.5) There exists a nonnegative function g ∈ L ∞ (R d ) ∩ L 2 (R d ) and h 2 (z) ∈ L 2 (E, m) such that for all (x, u, z) ∈ R d × R × E, The above definition does not say anything explicitly about the entropy solution satisfying the initial condition. However, the initial condition is satisfied in a certain weak sense. Here we state the lemma whose proof follows a simple line argument as in the Lemma 2.3 of [7]. Lemma 2.2. Any entropy solution u(t, ·) of (1.1) satisfies the initial condition in the following sense: for every non negative test function ψ ∈ C 2 Next, we describe a special class of entropy functions that plays an important role in later analysis. Let β : R → R be a C ∞ and Lipschitz function satisfying For any ϑ > 0, define β ϑ : R → R by β ϑ (r) = ϑβ( r ϑ ). Then By simply dropping ϑ, for β = β ϑ we define We conclude this section by stating the main results of this paper.
Then, the entropy solution of (1.1) is unique.
We sketch a justification of this claim in Section 4.

Existence of weak solution for viscous problem
Just as the deterministic problem, here also we study the corresponding regularized problem by adding a small diffusion operator and derive some a priori bounds. Due to the nonlinear function φ and related degeneracy, one cannot expect classical solution and instead seeks an weak solution.
3.1. Existence of weak solution to viscous problem. For a small parameter ε > 0, we consider the viscous approximation of (1.1) as In this subsection, we establish the existence of a weak solution for the problem (3.1). To do this, we use an implicit time discretization scheme. Let ∆t = T N for some positive integer N ≥ 1. Set t n = n ∆t for n = 0, 1, 2 · · · , N . Define Proposition 3.1. Assume that ∆t is small. For any given u n ∈ H n , there exists a unique u n+1 ∈ N n+1 with φ(u n+1 ) ∈ N n+1 such that P −a.s. for any v ∈ H 1 (R d ), the following variational formula holds: η(x, u n ; z) vÑ (dz, ds) dx.
Before proving the proposition, first we state a key deterministic lemma, related to the weak solution of degenerate parabolic equations. We have the following lemma, a proof of which could be found in [ page 19, [9] ]. Lemma 3.2. Assume that ∆t is small and X ∈ L 2 (R d ). Then, for fixed positive parameter ε > 0, (2) There exists a constant C = C(∆t) > 0 such that the following a priori estimate holds Proof of the Proposition 3.1 Let u n ∈ N n . Take X = u n + tn+1 tn E η(x, u n ; z)Ñ (dz, ds). Then, by the assumption (A.5), we obtain This shows that for a.s. ω ∈ Ω, X ∈ L 2 (R d ). Therefore, one can use the Lemma 3.2, and conclude that for almost surely ω ∈ Ω, there exist unique u(ω) satisfying the variational equality (3.2). Moreover, by construction X ∈ H n+1 . Thus, due to the continuity of Θ for the F (n+1)∆t measurability and to a priori estimate (3.4), we conclude that u ∈ N n+1 with φ(u) ∈ N n+1 . We denote this solution u by u n+1 . Hence the proof of the proposition follows.
In view of the assumption (A.5), the inequality (3.5), Itô-Lévy isometry, and the fact that for any Since α > 0 is arbitrary, one can choose α > 0 so that Thanks to discrete Gronwall's lemma, one has from (3.6), For fixed ∆t = T N , we define A straightforward calculation shows that Since φ is a Lipschitz continuous function with φ(0) = 0, in view of the above definitions and a priori estimate (3.7), we have the following proposition.
Next, we want to find some upper bound forB ∆t (t). Regarding this, we have the following proposition.
Proof. First we prove the boundedness ofB ∆t (t). By using definition ofB ∆t (t), the assumption (A.5), and boundedness of u ∆t in L ∞ (0, T ; H), we obtain To prove second part of the proposition, we see that for any t ∈ n∆t, (n + 1)∆t , Therefore, in view of (3.7) and assumption (A.5), we have This completes the proof.

3.1.2.
Convergence of u ∆t (t, x). Thanks to Proposition 3.3 and Lipschitz property of f and φ, there exist u, φ u and f u such that (up to a subsequence) Next, we want to identify the weak limits φ u and f u . Note that, for any v ∈ H 1 (R d ), we can rewrite (3.2), in terms of u ∆t ,ũ ∆t andB ∆t as Proof. Consider two positive integers N and M and denote ∆t = T N , ∆s = T M . Then, for any v ∈ H 1 (R d ), one gets from (3.9), Similarly, we also have Combining (3.11), (3.12) and (3.13) in for some α and β > 0. Since α, β > 0 are arbitrary, there exist positive constants C 1 , C 2 and C 3 such that In view of the Proposition 3.3, we notice that So we need to estimate the term where we have used Proposition 3.4 and the assumption (A.5). Thus, we get We combine (3.15) and (3.16) in (3.14) and have Hence, an application of the Gronwall's lemma gives We are now in a position to identify the weak limits φ u and f u . We have shown that u ∆t ⇀ u and u ∆t is a Cauchy sequence in L 2 (Ω × Π T ). Thanks to the Lipschitz continuity of φ and f , one can easily conclude that φ u = φ(u) and f u = f (u).
In view of the variational formula (3.9), one needs to show the boundedness of ∂ ∂t (ũ ∆t −B ∆t ) in L 2 (Ω × (0, T ); H −1 (R d )) and then identify the weak limit. Regarding this, we have the following lemma.
where u is given by (3.8).
Proof. To prove the lemma, let Γ = Ω × [0, T ] × E, G = P T × L(E) and ς = P ⊗ ℓ t ⊗ m, where P T represents predictable σ-algebra on Ω × [0, T ] and L(E) represents a Lebesgue σ-algebra on E. Then L 2 (Γ, G, ς); R consists of all square integrable predictable processes which are Borel measurable functions of z-variable. The space L 2 (Γ, G, ς); R represents the space of square integrable predictable integrands for Itô-Lévy integrals with respect to the compensated compound Poisson random measureÑ ( dz, dt). Moreover, Itô-Lévy integral defines a linear operator from L 2 (Γ, G, ς); R to L 2 (Ω, F T ); R and it preserves the norm (cf. for example [22]). Thank to Propositions 3.3 and 3.5, u ∆t (t − ∆t) converges to u in L 2 (Ω × Π T ). Therefore, in view of the Proposition 3.4, Lipschitz property of η, and the above discussion we conclude that Again, note that and hence . To prove the second part of the lemma, we recall thatB ∆t ⇀ · 0 E η(x, u; z)Ñ (dz, ds) and u ∆t ⇀ u in L 2 (Ω × Π T ). In view of the first part of this lemma, one can conclude that, up to a subsequence This completes the proof.
3.1.3. Existence of weak solution. As we have emphasized, our aim is to prove the existence of weak solution for viscous problem. For this, it is required to pass the limit as ∆t → 0. To this end, let α ∈ L 2 ((0, T )) and β ∈ L 2 (Ω). Then, in view of variational formula (3.9), we obtain We make use of (3.8), Lemmas 3.5 and 3.6 to pass to the limit as ∆t → 0 in the above variational formulation and arrive at η(x, u; z)Ñ (dz, ds) , v α β dt dP + ε Ω×ΠT ∇u · ∇v αβ dx dt dP + Ω×ΠT ∇φ(u) · ∇v + f (u) · ∇v αβ dx dt dP = 0. (3.17) Since H 1 (R d ) is a separable Hilbert space, the above formulation (3.17) yields for almost surely ω ∈ Ω, This proves that u is a weak solution of (3.1). Note that, for every φ ∈ H 1 (R d ), it is easily seen that Therefore, Now, by letting ∆t ↓ 0, we have

3.2.
A priori bounds for viscous solutions. Note that for fixed ε > 0, there exists a weak solution, denoted as u ε ∈ H 1 (R d ), which satisfies the following variational formulation: P -almost surely in Ω, and almost every t ∈ (0, T ), Applying Itô-Lévy formula (cf. Theorem A.1 in the Appendix) to β(u), one gets that for any t > 0 ds.
An application of Gronwall's inequality yields The achieved results can be summarized into the following theorem.
Theorem 3.7. For any ε > 0, there exists a weak solution u ε to the problem (3.1). Moreover, it satisfies the following estimate:

19)
where G is an associated Kirchoff 's function of φ, defined by G(x) = x 0 φ ′ (r) dr.
Remark 3.8. Let us remark that since any solution to (3.1) is an entropy solution, the solution u ǫ in unique.

Existence of entropy solution
In this section, we will prove the existence of entropy solution. In view of the a priori estimates as in (3.19), we can apply Lemmas 4.2 and 4.3 of [7](see also [4]) and show the existence of Young measure-valued limit process solution u(t, x, α), α ∈ (0, 1) associated to the sequence {u ε (t, x)} ε>0 .
The basic strategy in this case is to apply Young measure technique and adapt Kruzkov's doubling method in the presence of noise for viscous solutions with two different parameters and then send the viscous parameters goes to zero. One needs a version of classical L 1 contraction principle( for conservation laws) to get the uniqueness of Young measure valued limit and show that Young measure valued limit process is independent of the additional (dummy) variable and hence it will imply the point-wise convergence of viscous solutions. 4.1. Uniqueness of Young measure valued limit process. To do this, we follow the same line of argument as in [3] for the degenerate parabolic part and [7] for the Lévy noise. For the convenience of the reader, we have chosen to provide detailed proofs of a few crucial technical lemmas and the rest are referred to the appropriate resources. In [4,7], the authors used the fact that ∆u ε ∈ L 2 (Ω × Π T ). Note that, in this case, u ε ∈ H 1 (R d ). Therefore, we need to regularize u ε by convolution. Let {τ κ } be a sequence of mollifiers in R d . Since u ε is a viscous solution to the problem (3.1), as shown in the proof of Theorem A.1, u ε * τ κ is a solution to the problem Note that, for fixed ε > 0, ∆(u ε * τ κ ) ∈ L 2 (Ω × Π T ).
By using the properties of Lebesgue points, convolutions, and approximations by mollifications one can able to pass to the limit in I 0,2 and J 0,2 , and conclude the following lemma.
Proof. In view of Lemma 4.1, we see that Let u(t, y, α) andũ(t, x, γ) be Young measure-valued narrow limit associated to the sequences {u ε (t, y)} ε>0 and {u θ (t, x)} θ>0 respectively. With these at hand, one can use a similar argument as in [3,Lemma 3.4] and arrive at the conclusion that (4.5) holds.
Let us consider the terms (I 1 + J 1 ) coming from initial conditions. Note that I 1 = 0 as supp ρ δ0 ⊂ [−δ 0 , 0). Under a slight modification of the same line arguments as in [7], we arrive at the following lemma We now turn our attention to (I 2 + J 2 ). Note that ∂ t ρ δ0 (t − s) = −∂ s ρ δ0 (t − s) and β, J l are even functions. A simple calculation gives One can pass to the limit in (I 2 + J 2 ) and have the following conclusion. Let us consider I 6 and J 5 . Regarding this, we have the following lemma φ β u(s, y, α),ũ(s, x, γ) ∆ y ψ(s, y)̺ δ (x − y) dγ dα dx dy ds .
Proof. Let us consider the passage to the limits in I 6 . To do this, we define, Note that, for all a, b, c ∈ R, By using (4.6), we have where K δ ⊂ R d is a compact set depending on ψ and δ. Note that, lim x)| dx dt → 0 almost surely for all r ∈ [0, 1]. Therefore, by dominated convergence theorem, we have lim δ0→0 Moreover, one can use the property of convolution to conclude lim κ→0 lim δ0→0 Passage to the limit as l → 0: Note that for all a, b, c ∈ R, (4.7) Therefore, by (4.7) we have One can justify the passage to the limit as ε → 0 and θ → 0 in the sense of Young measures as in [4,7] and conclude This proves the first part of the lemma.
To prove the second part, let us recall that

dk dx dt dy ds
A classical properties of Lebesgue points and convolution yields lim κ→0 lim δ0→0 Making use of Green's type formula along with Young measure theory and keeping in mind that u(s, y, α) andũ(s, x, γ) are Young measure-valued limit processes associated to the sequences {u ε (s, y)} ε>0 and {u θ (s, x)} θ>0 respectively, we arrive at the following conclusion This completes the proof of the lemma.
In view of the above, we now want to pass the limit as (ϑ, δ) −→ (0, 0). We use similar line argument as in the proof of the second part of the Lemmas 5.7 and 5.8 in [7] and arrive at the following lemma. Let us consider the term I 7 + J 8 . Regarding this, we have the following Proof. Note that (By Cauchy-Schwartz inequality), Similarly, we have |I 7 | ≤ C(β, ψ, δ) θ 1 2 . This completes the proof.
Next we consider the stochastic terms I 3 + J 3 . To this end, we cite [7] and assert that for two constants T 1 , T 2 ≥ 0 with T 1 < T 2 , where J is a predictable integrand and X is an adapted process.
This completes the proof.
Lemma 4.13. The following holds: Moreover, if √ φ ′ has a modulus of continuity ω φ , then Let us back to the expression H. Regarding this, we have following: Proof. Let ω φ be a modulus of continuity of √ φ ′ . Then, thanks to Lemma 4.13, we obtain and Hence, we have Put δ = ϑ All of the above results can now be combined into the following proposition.

4.2.
Existence of entropy solution. In this subsection, using strong convergence of viscous solutions and a priori bounds (3.19) we establish the existence of entropy solution to the underlying problem (1.1).
We now close this section with a sketch of the justification of our claim in Remark 2.5. To see this, let h δ denote a smooth even convex approximation of |.| p define for positive x by: h δ vanishes at 0 and uniquely recovered from its second order derivative defined as Note that the weak Itô-Lévy formula in Theorem A.1 makes sense for β = h δ , as h ′′ δ is bounded. This enables us write, for almost every t > 0, We can now use the properties of h δ and the assumptions on η to arrive at and, by a weak Gronwall inequality, E R d h δ (u ǫ )dx ≤ e Kηt E R d |u 0 | p dx for all almost all t. This implies E R d |u ǫ | p dx ≤ e Cηt E R d |u 0 | p dx by monotone convergence theorem. The solution u will inherit the same property by Fatou's lemma.
If u 0 is bounded and η(x, u; z) = 0 for |u| ≥ M , M been given, then, consider non-negative regular convex function Since h(u 0 ) = 0 and h vanishes where η is active, the Itô formula Yields E R d |h(u ǫ )|dx = 0 and u ǫ is uniformly bounded by K. Again, the solution u will inherit the same property by passing to the limit.

Uniqueness of entropy solution
To prove the uniqueness of entropy solution, we compare any entropy solution to the viscous solution via Kruzkov's doubling variables method and then pass to the limit as viscous parameter goes to zero. We have already shown that limit of the viscous solutions serve for existence of entropy solution for the underlying problem. Now let v(t, x) be any entropy solution and u ε (t, x) be viscous solution for the problem (3.1). Then one can use exactly the same argument as in Section 4, and end up with the following equality This implies that, for almost every t ∈ [0, ∞), v(t, x) = u(t, x, α) for almost every x ∈ R d , (ω, α) ∈ Ω × (0, 1). In other words, this proves the uniqueness for entropy solutions.

It follows from straightforward computation that
T 0 E |I k (s, z) − I(s, z)| 2 m(dz) ds → 0 as k → 0. Therefore, we can invoke Itô-Lévy isometry and pass to the limit k → 0, in the martingale term in (A.6). This competes the validation of passage to the limit as k → 0 in every term of (A.6). The assertion is now concluded by simply letting k → 0 in (A.6) and rearranging the terms.