A cross-diffusion system derived from a Fokker-Planck equation with partial averaging

A cross-diffusion system for two compoments with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for the probability density associated to a multi-dimensional It\={o} process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.


Introduction
The aim of this paper is the analysis of the following cross-diffusion system (1) ∂ t u i = ∆ a(u 1 /u 2 )u i + µ i u i , t > 0, u i (0) = u 0 i ≥ 0 in T d , i = 1, 2, where T d is the d-dimensional torus with d ≥ 1, a : (0, ∞) → (0, ∞) is a continuously differentiable function, and µ i ∈ R. This system can be formally derived [7] from a (d + 1)dimensional Fokker-Planck equation for the probability density f (x, y, t), where x ∈ R d , y ∈ R. The function u i is obtained from f by partial averaging, u i (x, t) = R f (x, y, t)e λ i y dy, i = 1, 2, µ i is a function of λ i , and a(u 1 /u 2 ) is related to the diffusion coefficients in the Fokker-Planck equation. Strictly speaking, equation (1) holds in R d (or on some subset of R d ) but we consider this equation on the torus for the sake of simplicity (and to avoid possible issues with boundary conditions). For details on the derivation, we refer to Section 2.
System (1) has been suggested by P.-L. Lions in [7], and the global-in-time existence of (weak) solutions has been identified as an open problem. In this paper, we solve this problem by applying the entropy method for diffusive equations.
The underlying Fokker-Planck equation for f (x, y, t) models the time evolution of the value of a financial product in an idealized financial market, depending on various underlying assets or economic values. The function u i is an average with respect to the variable y, which may be interpreted as the value of an economic parameter, and the exponential weight emphasizes large positive or large negative values of y, depending on the sign of λ i . We note that partial averaging is also employed to simplify chemical master equations [9]. Here, we are not interested in potential applications, but more in the refinement of mathematical tools to analyze (1).
We assume that there exist a 0 > 0 and p ≥ 0 such that for all r > 0, (2) a(r) ≥ r|a ′ (r)|, a(r) ≥ a 0 r p + r −p .
The first condition means that a grows at most linearly (see Lemma 6). The second condition is a technical assumption needed for the entropy method (see the proof of Lemma 5). Examples are a(r) = 1, which leads to uncoupled heat equations for u 1 and u 2 , a(r) = r α with 0 < α ≤ 1, a(r) = r β /(1 + r β−1 ) with β > 0, and a(r) = 1/r. The last example gives the equations Surprisingly, this system corresponds (up to a factor) to an energy-transport model for semiconductors. Indeed, introducing the electron density n := u 1 and the electron temperature θ := u 2 /u 1 , equations (3) can be written as A class of energy-transport models that includes the above example was analyzed in [13]. Another class of models which resembles (1) are the equations modeling the time evolution of population densities u i . These systems are analyzed in, e.g., [5,8], essentially for m = 2. In this application, p i is often given by the sum p i1 (u 1 )+p i2 (u 2 ), and consequently, the results of [5,8] do not apply and we need to develop new ideas. Our first main result is the global-in-time existence of weak solutions to (1).
Theorem 1 (Existence of weak solutions). Let (2) hold and let T > 0, α ≥ p + 4, µ 1 , If additionally µ i ≤ 0 for i = 1, 2, we have the uniform bounds As mentioned above, the proof of this theorem is based on entropy methods. These methods have been originally developed to understand the large-time behavior of solutions; see, e.g., [2,12]. The "entropy" of system (1) is often understood as a convex Lyapunov functional which provides suitable nonlinear gradient estimates. In many situations, and also in the financial context presented here, the "entropy" has no physical counterpart. However, we claim that this notion is appropriate since it naturally generalizes physical situations. For details, we refer to [8].
Our key idea is to employ the functional where α ≥ p + 4 and u = (u 1 , u 2 ) ∈ (0, ∞) 2 . We will show that for some constant C > 0 which vanishes if µ 1 = µ 2 = 0. In this situation, the mapping is bounded on finite time intervals. We infer from the inequality x + x −1 ≥ 2 for all x > 0 uniform bounds for u i (t) in H 1 (T d ), which are needed for the compactness argument. The entropy method gives more than just the a priori estimate (7). Indeed, let us write (1) in divergence form: where the ith component of div(A(u)∇u) equals d j=1 2 k=1 ∂ j (A ik (u)∂ j u k ), ∂ j = ∂/∂x j , and f (u) = (µ 1 u 1 , µ 2 u 2 ) ⊤ . The diffusion matrix is generally neither symmetric nor positive definite. Since the only eigenvalue of A(u) is given by λ = a(u 1 /u 2 ) > 0, the system is normally elliptic [1] and local-in-time existence of classical solutions can be expected. The difficulty is to prove the global-in-time existence. The entropy density h(u) allows us to formulate (1) in new variables with a positive semidefinite diffusion matrix. Then, together with the a priori estimates from (7), global existence will be deduced. Indeed, defining the so-called entropy variable w = (w 1 , w 2 ) by w i = ∂h/∂u i (i = 1, 2), equation (1) is equivalent to where B(w) = A(u)h ′′ (u) −1 is positive semidefinite (see Lemma 5) and h ′′ (u) is the Hessian matrix of h(u). With this formulation, we obtain The right-hand side can be bounded in terms of H[u] (see (14)), and the integral on the left-hand side is related to the corresponding integral in (7).
The proof of Theorem 1 is based on a regularization of (9), the fixed-point theorem of Leray-Schauder, and the de-regularization limit. The compactness is obtained from the entropy estimate (7). This technique is similar to those employed in our works [8,13]. The novelty here is the (nontrivial) observation that the cross-diffusion system (1) possesses a convex Lyapunov functional, defined by (6). Moreover, compared to [8,13], we are facing additional technical difficulties due to the quotient u 1 /u 2 .
The second result concerns the large-time asymptotics in the case µ i = 0 for i = 1, 2.
Theorem 2 (Large-time asymptotics). Let the assumptions of Theorem 1 hold and let If µ i < 0 for i = 1, 2, we prove the exponential convergence of u(t) to zero in H 1 (T d ) ′ , see Remark 9. For a discussion of the case µ i > 0, we refer to Remark 10.
The paper is organized as follows. In Section 2, we make precise the derivation of (1) from a Fokker-Planck equation. Some technical results are proved in Section 3. Section 4 is devoted to the proof of Theorem 1, and Theorem 2 is shown in Section 5. (1) We summarize the formal derivation of (1) from a Fokker-Planck equation as presented by P.-L. Lions in [7]. Consider the n-dimensional Itō process X t = (X 1 t , . . . , X n t ) on some probability space, driven by the n-dimensional Wiener process W t = (W 1 t , . . . , W n t ) with respect to some given filtration. We assume that X t solves the stochastic differential equation

Derivation of the cross-diffusion system
. . , µ n t ), and σ t = (σ ij t ) i,j,=1,...,n is an n × n matrix. It is well known [10, Theorems 7.3.3, 8.2.1] that the probability density f (x 1 , . . . , x n , t) for X t satisfies the Fokker-Planck (or forward Kolmogorov) equation In the following, we set µ t = 0 and σ t = diag(σ 1 , . . . , σ n ). This means that we neglect correlations between the processes. Taking them into account will lead to first-order terms in the final equations; see Remark 3. Under the above simplifications, the Fokker-Planck equation becomes We assume that σ j is a function of the partial averages where λ i are some given (pairwise different) parameters. Temporal averages appear, for instance, in the modeling of Asian options. Here, u i may be interpreted as an average with respect to the ecocnomic parameter x n . We may employ other weights than the exponential one but this one is mathematically extremely convenient because of the property ∂u i /∂x n = λ i u i (see Remark 3). Multiplying (10) by e λ i xn and integrating with respect to x n ∈ R, a straightforward calculation shows that u i solves We allow σ j to depend on the partial averages, σ j = σ j (u 1 , . . . , u m ). We consider only the special case m = 2, σ := σ j for j = 1, . . . , n − 1, and σ n is constant and positive. Setting u = (u 1 , u 2 ), µ i := λ 2 i σ n /2, we find that In divergence form, this system is equivalent to where ∂ i σ = ∂σ/∂u i , i = 1, 2. This system is of parabolic type in the sense of Petrovski if the real parts of the eigenvalues of A are nonnegative [1], i.e. if σ + ∂ 1 σu 1 + ∂ 2 σu 2 ≥ 0 for all u ∈ R 2 . This requirement is fulfilled if, for instance, σ depends on the quotient u 1 /u 2 only. Therefore, we set σ(u) 2 = 2a(u 1 /u 2 ). Then is of parabolic type in the sense of Petrovski, and these equations correspond to (1).
Remark 3 (Generalizations). The general model for nonvanishing µ i t and nondiagonal σ t is derived as above, and the result reads as Compared to (11), this equation also contains first-order terms. If µ i t = 0 and σ t is diagonal, we obtain m equations of the type (12). The analysis of cross-diffusion systems with more than two components is expected to be much more involved than for those with two components. For instance, the analysis of the cross-diffusion model (4) is rather well understood only in the case of m = 2 components, while the case of m ≥ 3 equations requires additional properties [3].
Another generalization concerns nonexponential weights. For instance, we may define Choosing again µ i t = 0 and σ t = diag(σ 1 , . . . , σ n ), we find that This justifies the assumption µ i ∈ R in (1) but there seems to be no financial interpretation of the trigonometric weight functions.

Some auxiliary lemmas
In this section, we prove some algebraic properties of the matrices h ′′ (u) and A(u) and some estimates related to the entropy density h(u) and the components of A(u). Recall that h(u) is defined in (6) and A(u) in (8).
where we recall that Proof. We proceed in several steps.
Step 2: h ′ is invertible. Since the Hessian h ′′ is positive definite on (0, ∞) 2 , h ′ is oneto-one and the image R(h ′ ) is open. If R(h ′ ) is also closed, it follows that R(h ′ ) = R 2 which means that h ′ is surjective. For this, let (w n ) ∈ R(h ′ ) for n ∈ N such that w n → w as n → ∞. We show that w ∈ R(h ′ ). By definition, there exists u n > 0 such that w n = h ′ (u n ) for n ∈ N. The idea is to prove that (u n ) = (u 1,n , u 2,n ) is a bounded and strictly positive sequence. This implies that, up to a subsequence, u n → u ∈ (0, ∞) 2 as n → ∞. By continuity of h ′ , we infer that h ′ (u n ) → h ′ (u) as n → ∞. We already know that h ′ (u n ) = w n → w which shows that w = h ′ (u) ∈ R(h ′ ), and R(h ′ ) is closed.
Expanding these expressions yields and the product also converges, u 2 1,n u 2 2,n → 1/4. This is absurd since (u n ) converges to zero. Therefore, u 1,n is strictly positive. With an analogous argument, we conclude that u 2,n is strictly positive too.
Step 3: proof of (14). Observing that x − log x ≥ 1 for all x > 0, it follows that The elementary inequality x α + (1/x) α+2 ≥ 1 for x > 0 shows the first inequality in (14): For the second inequality in (14), we employ definition (6) of h and the elementary in- This finishes the proof.

Next, we prove that h ′′ (u)A)(u) is positive semidefinite. Then
there exists a constant κ = κ(α) > 0 such that for all u = (u 1 , u 2 ) ∈ (0, ∞) 2 and z ∈ R 2 , Proof. Let α(α + 2) > 1 and let By the first condition in (2) where k(α) = (α(α + 2) − 1)/(α + 2) 2 . In a similar way, we find that Since det M/ tr M is a lower bound for the eigenvalues of any symmetric positive definite matrix M ∈ R 2×2 (and taking into account that M (3) is positive definite), we deduce that for z ∈ R 2 , In the last inequality, we have employed the elementary inequality ( ≥ 0, and this holds true for all x > 0. By the second condition in (2), which concludes the proof with κ = a 0 k(α)/4.
The following two lemmas concern elementary estimates for a(r).
Proof. The first inequality in (2) implies that r → a(r)/r is nonincreasing, while r → a(r)r is nondecreasing. Writing these monotonicity properties in an explicit way gives the result.
where C a = a(1) 2 and ξ α > 0 is a suitable constant which only depends on α.
Proof. The first inequality follows from an application of Lemma 6 with r 0 = 1. Indeed, if u 1 /u 2 ≥ 1, we obtain These inequalities show the claim with C a = a(1) 2 . The second inequality follows from where ξ α > 0 is a suitable constant, which depends only on α. This finishes the proof.

Lemma 8.
Recall that A(u) = (A ij (u)) is given by (8). Then there exists C A > 0, only depending max 0≤r≤1 a(r), such that for all u 1 , u 2 > 0, Proof. We apply the first condition in (2) and Lemma 6 with r 0 = 1 to find that Then Lemma 6 with r 0 = 1 implies that This estimate and Young's inequality conclude the proof.

Proof of Theorem 1
Let T > 0, N ∈ N, τ = T /N, and m ∈ N with m > d/2. Then the embedding a partial derivative of order |α|.
Step 1: solution of (18). Let w = ( w 1 , w 2 ) ∈ L ∞ (T d ) 2 and η ∈ [0, 1] be given. Set u = ( u 1 , u 2 ) := (h ′ ) −1 ( w). We solve first the linear problem It holds clearly S(w, 0) = 0. Standard arguments show that S is continuous (see, e.g., the proof of Lemma 5 in [8]). Because of the compact embedding H m (T d ) ֒→ L ∞ (T d ), the mapping S is even compact. In order to apply the Leray-Schauder fixed-point theorem, it remains to prove a uniform bound for all fixed points of S(·, η) in L ∞ (T d ) 2 .
Let w ∈ L ∞ (T d ) 2 be such a fixed point, i.e. a solution to (19) with u replaced by u := (h ′ ) −1 (w). The uniform bound will be a consequence of the entropy inequality. For this, we employ the test function w in (19): By the convexity of h, it follows that Moreover, by (9) and Lemma 5, we have Taking into account the second estimate in (14), we infer that Therefore, (20) becomes Choosing τ < 1/C h , this shows that w is uniformly bounded in H m (T d ). Thus, we can apply the fixed-point theorem of Leray-Schauder to conclude the existence of a weak solution w k := w with u k = h ′ (w k ) to (18) with η = 1.
Step 2: a priori estimates. Inequality (21) shows, for w = w k , u = u k , and η = 1, that We sum (21) for k = 1, . . . , j and divide the resulting inequality by 1 − C h τ (recall that we have chosen τ < 1/C h ): We apply the discrete Gronwall inequality [4] to obtain for jτ ≤ T , where C > 0 denotes a constant which is independent of τ (and independent of T if µ i ≤ 0) but dependent on the initial entropy H[u 0 ].

Proof of Theorem 2
Theorem 1 allows us to employ the test equations u 1 −ū 1 , u 2 −ū 2 in (1), respectively: ∇u i · ∇(a(u)u i )dx, which, together with Theorem 1, implies that (d/dt) and Theorem 2 shows that u * i (t) → u i in L 2 (T d ) as t → ∞, which translates to e −µt u i (t) − u i L 2 (T d ) → 0.