On the Bidomain equations driven by stochastic forces

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

1. Introduction. The bidomain equation arises in various models describing the propagation of impulses in electrophysiology. These models have a long tradition, starting with the celebrated classical model by Hodgkin and Huxley in the 1950s. The bidomain equations are described in detail e.g. in the monographs by Keener and Sneyd [14] and by Colli Franzone, Pavarino and Scacchi [5]. The system is given by ∂ t u + f (u, w) + ∇ · (a e ∇u e ) = −I e in (0, ∞) × Q, subject to the boundary conditions a i ∇u i · ν = 0, a e ∇u e · ν = 0 on (0, ∞) × ∂Q, and the initial data u(0) = u 0 and w(0) = w 0 .
Here Q ⊂ R d denotes a domain in R d , the functions u i and u e model the intraand extracellular electric potentials, u the transmembrane potential, and ν the outward unit normal vector to ∂Q. The anisotropic properties of this system are described by the conductivity matrices a i (x) and a e (x). Furthermore, I i and I e stand for the intra-and extracellular stimulation currents, respectively. There exist various models describing the ionic transport. The most classical one is the one by FitzHugh-Nagumo given by f (u, w) = u(u − a)(u − 1) + w = u 3 − (a + 1)u 2 + au + w, where 0 < a < 1 and b, c > 0 are constants. We will also consider the models due to Allen-Cahn, Aliev-Panfilov [1] and Rogers-McCulloch [20] described in detail in Section 2.
Despite its importance, not many results on the bidomain equations are known until today. One reason for this might be the fact that the associated bidomain operator is a highly non local operator, which makes the analysis of this set of equations seriously more complicated compared to standard reaction-diffusion systems.
This article is devoted to the study of bidomain equations driven by stochastic forces. More precisely, denoting by W white in time and colored in space cylindrical Q-Wiener process on some probability space (Ω, F, P ), we consider the equation du + [f (u, w) + ∇ · (a e ∇u e ) + I e ]dt = −dW in (0, ∞) × Q, ∂ t w + g(u, w) = 0 in (0, ∞) × Q, subject to the above boundary and initial conditions. The rigorous analysis of the system (BDE) was pioneered by Colli-Franzone and Savaré [6]. They introduced a variational formulation of the problem and showed global existence and uniqueness of weak and strong solutions for the FitzHugh-Nagumo ionic transport for data in H 1 in space dimension 3. Bourgault, Cordière, and Pierre presented in [2] a new approach to this system by introducing for the first time the so-called bidomain operator within the L 2 -setting. They proved existence and uniqueness of a local strong solution and existence of a global weak solution to the system (BDE), for various classes of ionic models including the one by FitzHugh-Nagumo.
Recently, Giga and Kajiwara [10] considered the bidomain equations within the L q -setting for 1 < q ≤ ∞. They showed that the bidomain operator generates an analytic semigroup on L q (Q) for q ∈ (1, ∞] and constructed a local, strong solution to the bidomain system within this setting. A maximal L q regularity approach to the bidomain equation was developed in [12] and [13] yielding the existence of a unique, global, strong solution for initial data lying even in critical spaces. To describe this approach denote by L q,0 (Q) the space of all L q -functions having mean zero. Consider the operators Under suitable conditions given precisely in Section 2 we have D(A i ) = D(A e ). This allows us to define the bidomain operator A as It was then shown in [13] that A admits a bounded H ∞ -calculus on L q (Q). Modern theory of parabolic evolution equations allowed then to prove global strong wellposedness of the bidomain equations.
In this article we consider for the first time the bidomain equations driven by white noise. Using the above bidomain operator A, equation (BDES) may be reformulated as Note that if a i is proportional to a e , the operator (1.1) reduces to monodomain operator, which is local. In this case, we may use the techniques from, e.g., [15,16,17] and references therein, to study the stochastic monodomain equation.
The aim of this article is twofold. We ask first whether equation (1.2) is globally well-posedness in the weak sense. Secondly, we study the existence and stability of stationary solutions. We show in particular that the bidomain equations driven by white noise admit a global weak solution. Furthermore, we consider the long time behavior of the bidomain system subject to Fitzhugh-Nagumo or Allen-Cahn nonlinearities, and show that the stochastic bidomain equation admits a unique stable stationary solution.
This paper is organized as follows. In Section 2 we introduce the setting, state preliminary results and establish a property of stochastic convolution, see Lemma 2.1. Section 3 establishes the existence and uniqueness of a solution in the case when the right hand side satisfies a monotonicity assumption. This assumption is satisfied in particular for the Allen -Cahn and Fitz-Nagumo models. These results are obtained without any extra assumption on the coloring of the noise. Section 4 discusses the existence and uniqueness of weak solutions in the general case, when no monotonicity assumptions are made, however, under an additional condition on the coloring of the noise. Finally, in Section 5 we address the long time behavior of the bidomain system. We show that the stochastic bidomain equation admits a stationary solution.
2. Preliminaries. In this section we first fix our notation and introduce the bidomain operator in the weak and strong setting. Throughout this article, Q ⊂ R 3 denotes a bounded domain with smooth boundary ∂Q. We set The canonical pairing in H is denoted by (u, v) and the one in V by u, v . We assume that the conductivities σ i and σ e satisfiy the following assumptions.

MATTHIAS HIEBER, OLEKSANDR MISIATS AND OLEKSANDR STANZHYTSKYI
for all x ∈ Q and all ξ ∈ R n . Moreover, it is assumed that We now introduce the bidomain operator in the weak setting. To this end, we define the bilinear forms Due to assumption (2.1), these forms are symmetric, continuous and uniforly elliptic on V 0 × V 0 . The weak operators A i and A e are thus defined by Denote by P 0 the projection from V to V 0 and denote its transpose by P T 0 . Then the weak bidomain operator and its corresponding bilinear form are given by It was proved in [2] that a is a symmetric, continuous and coercive form on V and that for all u, v ∈ V and some constants M, α > 0. Moreover, there exists an increasing The strong bidomain operator is defined as follows. Let P 0 be the orthogonal projection from H onto H 0 . We then set (2.5) We may thus may define the strong bidomain operator A as Assuming that the currents are conserved, i.e.
the bidomain equation (BDE) may be equivalently rewritten as is the modifed source term. Given the sequence (λ n ) n∈N introduced above, we define the stochastic perturbation by We next describe the nonlinearities f and g. Following [2], we assume that the nonlinearities f (u, w) and g(u, w) are of the form where g 2 ∈ R and f 1 , f 2 as well as g 1 are continuous real functions, satisfying the conditions: In addition to the FitzHugh-Nagumo model, defined above, of particular interest are the Aliev-Panfilov model [1] given by the Rogers-McCulloch model [20] given by for 0 < a < 1 and b, c, d, k > 0 as well as the Allen-Cahn model given by Finally, let (Ω, F, P ) be a probability space and F t be a right-continuous filtration such that W (t, x) is adapted to F t and W (t) − W (s) is independent of F s for all s < t. We call an F t -adapted random process u(t, ·), for a.e. t > 0 and all v ∈ V and all z ∈ H. It was shown in [10] (and also in [12]) that −A generates an analytic semigroup S on L p (Q) for all p ∈ (1, ∞). By definition of λ i and ψ i we have S(t)ψ i = e −λit ψ i for all i ∈ N and all t > 0. The stochastic convolution of S against W is defined by We will make use of the following result on stochastic convolutions.
Proof. Observe first that Since a(ψ k , ψ k ) = λ k for all k ∈ N, assumption (2.3) yields α ψ k 2 V ≤ λ k + α for all k ∈ N and hence, by Sobolev embeddings, for some constant C > 0. By Hölder's inequality with 1/q + 1/p = 1, the term in the last line of (2.11) does not exceed (2.13) Here we used the factorization method of Da-Prato and Zabczyk [7] with a fixed β ∈ ( 1 p , 1 2 ). Using Hölder's inequality once more, and we obtain This implies the first assertion. In order to show the second one note that Proposition 4.15 of [8] yields W A (t, ·) ∈ D(A), which completes the proof.
3. Global existence of a weak solution under the monotonicity condition.
In this section we will make use of the results described in Chapter 6 of [3]. For the convenience of the reader, we briefly summarize these results here. Given a separable Hilbert space H, a dense subspace V ⊂ H, and V its dual, we consider for T > 0 thr equation and assume that the operator A satisfies the following conditions: In addition, we assume that the nonlinearities F and Σ satisfy the following conditions: B1) There exists a constant b > 0 such that and for any N > 0 there exists a constant C N > 0 such that Our aim is to apply the above result to the situation of the bidomain equations subject to various types of nonlinearities. To this end, we introduce a setting as follows: We setṼ let z := (u, w) T ∈ R 2 and define F by Let us assume that there exists a constant C ∈ R such that for all (u 1 , w 1 ), (u 2 , w 2 ) ∈ R 2 the function F satisfies the monotonicity condition Theorem 3.1. Assume that f and the g satisfy the conditions C1)-C3) as well as the monotonicity condition (3.2) and let T > 0. Then the system (1.2) admits a unique, weak solution z ∈ L 2 (Q; C[0, T ];Ṽ ). This is in particular true for the bidomain equation subject to Allen-Cahn or FitzHugh-Nagumo nonlinearities.
Proof. In H we introduce the operatorÃ bỹ and thus B3). In order to verify condition B4), we use of (2.3) and (2.4) to conclude that for any u 1 , u 2 ∈ V we have for some c > 0. Hence B4) holds true.

4.
Global existence of weak solutions in the general case. The aim of this section is to establish the existence of weak solutions to (1.2) without assuming the monoticity condition. Proof. We subdivide the proof into two steps.
Step 1. Construction of a local mild and weak solutions. We start by recalling that the semigroup S, generated by A, is an analytic strongly continous semigroup on L p (Q) for any p > 1 and that in particular S(t)u 0 (·) is measurable for all t > 0 and all x ∈ Q. Following [21], the operator in weak sense generates a semigroupS which is also analytic in L 2 (Q). Recall that W is given by We may now define the stochastic convolution W A in L 2 (Q) (in the weak sense) as However, since ψ i ∈ D(A), i ≥ 1, we haveS(t)ψ i ≡ S(t)ψ i , t ≥ 0, thus the stochastic convolutions in the weak and in the strong sense coincide, and will be denoted with If (u, w) a weak solution of (1.2), then the pair (U, w), where U := u − W A , is a weak solution of For ω ∈ Ω we may treat (4.1) as a coupled PDE-ODE system with a parameter. We proceed with defining the Galerkin approximations We now show that for any ω ∈ Ω Caratheodori's conditions are satisfied. To this end, note that is continuous in U im and w im in R 2m+2 . By assumption, f = f 1 (u) + f 2 (u)w, and hence since the U im are bounded (here C denotes some generic constant). Similarly, for bounded w im , the term f 2 is estimated as Note that if 1 p + 1 p = 1 and β = 2 for any i, j, k ∈ N. Similarly, we conclude that The dominated convergence theorem yields that for fixed t the integral term in the first equation in (4.2) is continuous in w im and Z im . Using g(u, w) = g 1 (u) + g 2 w, the integral term in the equation of (4.2) satisfies Appyling the dominated convergence theorem again, we conclude that for fixed t the integral term in the second equation in (4.2) is continuous with respect to w im and U im . Hence, Caratheodori's conditions are satisfied and (4.2) admits a local weak solution. Note that U im (t, ω) and w im (t, ω) are absolutely continuous functions of t on [0, t m (ω)).
Step 2. Existence of a global weak solution. We next show that t m (ω) = ∞, i.e. that the weak solution is defined globally. Note first that Due to (2.8) we have 3) we obtain for any ε > 0 which means that for some η > 0 we have Summing up, we showed that 1 2 for someη > 0. By Lemma 2.1 we see that for given T > 0 the right hand side of (4.6) is bounded on [0, T ]. Thus The bound above implies that the solutions U m and w m cannot "explode" in finite time. Hence, t m (ω) = ∞ and the system (4.2) admits a global solution. Similarly, since U m + W p ∈ L p , Lemma 25 of [2] yields In order to show the convergence of U m + W A , we will follow the main idea of the convergence proof in deterministic case, namely [2], section 5.2.3. Here we will highlight the main steps. The a priory bounds (4.7) imply that there exist weakly converging subsequences The difference with the determinimistic case is that the corresponding sequences depend on the realization ω. Thus, for fixed ω ∈ Ω, we conclude (the same way as in [2]) that U m → U converges strongly in L 2 (Q). This yields thatŨ = U ,w = w , and for every eigenfunction ψ i and every test function It remains to pass to the limit in the nonlinear terms. Note that by Lemma 2.1 we have Similarly, for 2 β + 2 p = 1 we have and finally These uniform bounds allow us to pass to the limit the same way, as it is done in the deterministic setting in [2], section 5.2.3. Hence, (4.1) admits a weak distributional solution, i.e., we have A similar identity holds also for w and thus (U, w) is a weak solution to equation (4.1). Consequently, (U (t), w(t)) is a mild solution of (4.1) and hence (u, w) is a mild solution of (1.2). This means In order to show that (u, w) is a weak solution of (1.2) we apply Proposition F.05 (ii) of [19].
We next define the mapping R : H → R by

MATTHIAS HIEBER, OLEKSANDR MISIATS AND OLEKSANDR STANZHYTSKYI
Then R 2 HS = ∞ k=1 |Rψ k | 2 . Indeed, setting z = ψ k and Bψ k = γ k ψ k , we see that the function u given by u(t) := S(t)(γ k ψ k ) satisfies On the other hand the problem (4.8) has the unique solution and (u, w) is a weak solution of (1.2).

5.
Existence of a stationary solution and invariant measure. This section concerns the existence and uniqueness of a stationary solution to the bidomain equation driven by white noise, and subject to Allen-Cahn and FitzHugh-Nagumo nonlinearities. Let C p be the constant in Poincare's inequality, i.e., u 2 L 2 (Q) ≤ C p ∇u 2 L 2 (Q) , u ∈ H 1 0 (Q). and α > 0 as in (2.3).
The main results in this section are the following theorems, which guarantee the existence, and, under slightly more restrictive conditions, uniqueness of the invariant measures.