On the stochastic Allen-Cahn equation on networks with multiplicative noise

We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.


Introduction
We consider a finite connected network, represented by a finite graph G with m edges e 1 , . . . , e m and n vertices v 1 , . . . , v n . We normalize and parametrize the edges on the interval [0, 1]. We denote by Γ(v i ) the set of all the indices of the edges having an endpoint at v i , i.e., Denoting by Φ := (φ ij ) n×m the so-called incidence matrix of the graph G, see Subsection 2.1 for more details, we aim to analyse the existence, uniqueness and regularity of solutions of the problem t ∈ (0, T ], x ∈ (0, 1), j = 1, . . . , m, u j (t, v i ) = u ℓ (t, v i ) =: q i (t), t ∈ (0, T ], ∀j, ℓ ∈ Γ(v i ), i = 1, . . . , n, x ∈ [0, 1], j = 1, . . . , m, (1.1) where ∂w j ∂t are independent space-time white noises. The reaction terms in (1.1) are classical Allen-Cahn nonlinearities h j (η) = −η 3 + β 2 j η with β j > 0, j = 1, . . . , m. Note that h j = −H ′ j where H j (η) = 1 4 (η 2 − β 2 j ) 2 is a double well potential for each j with potential barrier height β 4 j /4. The diffusion coefficients g j are assumed to be locally Lipschitz continuous and of linear growth. The coefficients of the linear operator satisfy standard smoothness assumptions, see Subsection 2.1, the matrix M satisfies Assumptions 2.7 and µ j , j = 1, . . . , m, are positive constants. The classical Allen-Cahn equation belongs to the class of phase field models and is a classical tool to model processes involving thin interface layers between almost homogeneous regions, see [AC79]. It is a particular case of a reaction-diffusion equation of bistable type and it can be used to study front propagations as in [BBS92] . Effects due to, for example, thermal fluctuations of the system can be accounted for by adding a Wiener type noise in the equation, see [Co70]. While deterministic evolution equations on networks are well studied, see, [Al84, Al94, BFN16, BFN16b, Be85, Be88, Be88b, BN96, Ca97, CF03, EK19, Ka66, KMS07, KS05, Lu80, MS07, Mu07, MR07,Mu14] which is, admittedly, a rather incomplete list, the study of their stochastic counterparts is surprisingly scarce despite their strong link to applications, see e.g. [BMZ08,BM10,Tu06] and the references therein. In [BMZ08] additive Lévy noise is considered that is square integrable with drift being a cubic polynomial. In [BZ14] multiplicative square integrable Lévy noise is considered but with globally Lipschitz drifts f j and diffusion coefficients and with a small time dependent perturbation of the linear operator. Paper [BM10] treats the case when the noise is an additive fractional Brownian motion and the drift is zero. In [CP17a] multiplicative Wiener perturbation is considered both on the edges and vertices with globally Lipschitz diffusion coefficient and zero drift and timedelayed boundary condition. Finally, in [CP17], the case of multiplicative Wiener noise is treated with bounded and globally Lipschitz continuous drift and diffusion coefficients and noise both on the edges and vertices.
In all these papers the semigroup approach is utilized in a Hilbert space setting and the only work that treats non-globally Lipschitz continuous drifts on the edges, similar to the ones considered here, is [BMZ08] but the noise is there additive and square-integrable. In this case, energy arguments are possible using the additive nature of the equation which does not carry over to the multiplicative case. Therefore, we use an entirely different tool set based on the semigroup approach for stochastic evolution equations in Banach spaces [vNVW08], or for the classical stochastic reaction-diffusion setting [KvN12,KvN19], see also, [BG99,BP99,Ce03,Pe95]. We are able to rewrite (1.1) in a form that fits into this framework. After establishing various embedding and isomorphy results of function spaces and interpolation spaces, we may use [KvN19, Theorem 4.9] to prove our main existence and uniqueness result, Theorem 3.15, which guarantees existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph, denoted by B in the paper (see Definition 3.4); that is, in the space of continuous functions that are continuous on the edges and also across the vertices. When the initial data is sufficiently regular, then Theorem 3.15 also yields certain space-time regularity of the solution.
The paper is organized as follows. In Section 2 we collect partially known semigroup results for the linear deterministic version of (1.1). In Subsection 3.1 we first recall an abstract result from [KvN12,KvN19] regarding abstract stochastic Cauchy problems in Banach spaces. In order to utilize the abstract framework in our setting we prove various preparatory results in Subsection 3.2: embedding and isometry results are contained in Lemma 3.5, Lemma 3.6 and Corollary 3.7, and a semigroup result in Proposition 3.8. Subsection 3.3 contain our main results where we first consider the abstract stochastic Itô equation corresponding to a slightly more general version of (1.1). An existence and uniqueness result for the abstract stochastic Itô problem is contained in Theorem 3.13 followed by a space-time regularity result in Theorem 3.14. These are then applied to the Itô equation corresponding (1.1) to yield the main result of the paper, Theorem 3.15, concerning the existence, uniqueness and space-time regularity of the solution of (1.1).

Heat equation on a network
2.1. The system of equations. We consider a finite connected network, represented by a finite graph G with m edges e 1 , . . . , e m and n vertices v 1 , . . . , v n . We normalize and parametrize the edges on the interval [0, 1].
The structure of the network is given by the n × m matrices Φ + := (φ + ij ) and for i = 1, . . . , n and j = 1, . . . m. We denote by e j (0) and e j (1) the 0 and the 1 endpoint of the edge e j , respectively. We refer to [KS05] for terminology. The n × m is known in graph theory as incidence matrix of the graph G. Further, let Γ(v i ) be the set of all the indices of the edges having an endpoint at v i , i.e., For the sake of simplicity, we will denote the values of a continuous function defined on the (parameterized) edges of the graph, that is of We start with the problem x ∈ [0, 1], j = 1, . . . , m (d) (2.1) on the network. Note that c j (·), p j (·) and u j (t, ·) are functions on the edge e j of the network, so that the right-hand side of (2.1a) reads in fact as The functions c 1 , . . . , c m are (variable) diffusion coefficients or conductances, and we assume that 0 < c j ∈ C 1 [0, 1], j = 1, . . . , m. Equation (2.1b) represents the continuity of the values attained by the system at the vertices in each time instant, and we denote by q i (t) the common functions values in the vertice i, for i = 1, . . . , n and t > 0.
In (2.1c), M := (b ij ) n×n is a matrix satisfying the following Assumption 2.1. The matrix M = (b ij ) n×n is real, symmetric and negative semidefinite, M ≡ 0.
On the left-hand-side, [M q(t)] i denotes the ith coordinate of the vector M q(t). On the right-hand-side, the coefficients 0 < µ j , j = 1, . . . , m are strictly positive constants that influence the distribution of impulse happening in the ramification nodes according to the Kirchhoff-type law (2.1c).
We now introduce the n × m weighted incidence matrices With these notations, the Kirchhoff law (2.1c) becomes In equation (2.1d) we pose the initial conditions on the edges.

Spaces and operators.
We are now in the position to rewrite our system in form of an abstract Cauchy problem, following the concept of [KMS07]. First we consider the (real) Hilbert space as the state space of the edges, endowed with the natural inner product Observe that E 2 is isomorphic to L 2 (0, 1) m with equivalence of norms.
We further need the boundary space R n of the vertices. According to (2.1b) we will consider such functions on the edges of the graph those values coincide in each vertex. Therefore we introduce the boundary value operator The condition u(t, ·) ∈ D(L) for each t > 0 means that (2.1b) is for the function u(·, ·) satisfied.
On E 2 we define the operator This operator can be regarded as maximal since no other boundary condition except continuity is supposed for the functions in its domain.
We further define the so called feedback operator acting on D(A max ) and having values in the boundary space R n as With these notations, we can finally rewrite (2.1) in form of an abstract Cauchy problem. Define (2.10) where Lu = q and Lv = r.
The next definition can be found e.g. in [Ou04, Section 1.2.3].
Definition 2.2. From the form a -using the Riesz representation theorem -we can obtain a unique operator (B, D(B)) in the following way: We say that the operator (B, D(B)) is associated with the form a.
In the following, we will claim that the operator associated to the form a is (A, D(A)). Furthermore, we will state results regarding how the properties of a and the matrix M carry on the properties of the operator A, obtaining the wellposedness of the abstract Cauchy-problem (2.8) on E 2 and even on L p -spaces of the edges. The proofs of these statements combine techniques of [Mu07] (where no p j 's on the right-hand-side of (2.1b) are considered) and techniques of [MR07] (where p j 's are considered for the heat equation but the matrix M is diagonal). As a corollary we obtain well-posedness of (2.8). Proof. The claim follows from Proposition 2.4 and the fact that (A, D(A)) is resolvent compact. This is true since V is densely and compactly embedded in E 2 by the Rellich-Khondrakov Theorem, and we can use [Da89, Theorem 1.2.1].
In the following we will extend the semigroup (T 2 (t)) t≥0 on L p -spaces. To this end we define We can characterize features of the semigroup (T 2 (t)) t≥0 by those of (e tM ) t≥0 , the semigroup generated by the matrix M -hence, by properties of M . In particular, the following holds.
Proposition 2.6. The semigroup (T 2 (t)) t≥0 on E 2 associated with a enjoys the following properties: To obtain the desired extension of the semigroup on L p -spaces, we assume the following on the matrix M.
Assumption 2.7. For the matrix M = (b ij ) n×n we assume the following properties: (1) M satisfies Assumption 2.1; that is, the matrix is diagonally dominant.
Proposition 2.8. If M satisfies Assumptions 2.7 then the semigroup (T 2 (t)) t≥0 extends to a family of compact, contractive, positive one-parameter semigroups (T p (t)) t≥0 on E p , 1 ≤ p ≤ ∞. Such semigroups are strongly continuous if p ∈ [1, ∞), and analytic of angle π 2 − arctan |p−2| We also can prove that the generators of the semigroups in the spaces E p , 1 ≤ p ≤ ∞ have in fact the same form as in E 2 , with appropriate domain. (2.11) In particular, A p has compact resolvent for p ∈ [1, ∞].
As a summary we obtain the following theorem.
Theorem 2.10. The first order problem (2.1) is well-posed on E p , p ∈ [1, ∞), i.e., for all initial data u ∈ E p the problem (2.1) admits a unique mild solution that continuously depends on the initial data.

The stochastic Allen-Cahn equation on networks
3.1. An abstract stochastic Cauchy problem. Let (Ω, F , P) is a complete probability space endowed with a right continuous filtration To be able to handle the stochastic Allen-Cahn equation on networks, first we cite a result of M. Kunze and J. van Neerven, regarding the following abstract equation see [KvN12,Section 3]. If we assume that (A, D(A)) generates a strongly continuous, analytic semigroup S on the Banach space E with S(t) ≤ Ke ωt , t ≥ 0 for some K ≥ 1 and ω ∈ R, then for ω ′ > ω the fractional powers (ω ′ − A) α are well-defined for all α ∈ (0, 1). In particular, the fractional domain spaces are Banach spaces. It is well-known (see e.g. [EN00, §II.4-5.]), that up to equivalent norms, these spaces are independent of the choice of w ′ .
These spaces are independent of ω ′ > ω up to an equivalent norm.
To obtain the desired result for the solution of (SCP), one has to impose the following assumptions for the mappings in (SCP). These are -in the first 2 cases slightly simplified versions of -Assumptions (A1), (A4), (A5), (F'), (F") and (G") in [KvN12]. Let B be a Banach space, · will denote · B . For u ∈ B we define the subdifferential of the norm at u as the set which is not empty by the Hahn-Banach theorem. Furthermore, let E be a UMD Banach space of type 2.

Assumptions 3.2.
(1) (A, D(A)) is densely defined, closed and sectorial on E; (2) Let S be the strongly continuous analytic semigroup generated by (A, D(A)).  F (t, ω, u) is strongly measurable and adapted. Finally, for suitable constants a, b ≥ 0 and N ≥ 1 we have Moreover, for all u ∈ B and h ∈ H the map (t, ω) → G(t, ω, u)h is strongly measurable and adapted. Finally, G is of linear growth, that is, for suitable constant c ′ , Recall that a mild solution of (SCP) is a solution of the following implicit equation denotes the "usual" convolution, and denotes the stochastic convolution with respect to W H .
The result of Kunze and van Neerven that will be useful for our setting is the following. We note that this was first proved in [KvN12, Theorem 4.9] but with a typo in the statement which was later corrected in the recent arXiv preprint [KvN19, Theorem 4.9].
Then for all ξ ∈ L q (Ω, F 0 , P; B) there exists a unique global mild solution of (SCP). Moreover, for some constant C > 0 we have

Preparatory results.
In order to apply the abstract result of Theorem 3.3 to the stochastic Allen-Cahn equation on a network we need to prove some preparatory results using the setting of Section 2.
On the edges of the graph G we will consider continuous functions that satisfy the continuity condition in the vertices, see Subsection 2.1. We will refer to such functions as continuous functions on the graph G and denote them by C(G). (3.4) We will show that for θ big enough the continuous, dense embeddings To do so, we first need a technical lemma, and define the maximal operator on E p as where W 2,p 0 (0, 1; µ j dx) = W 2,p (0, 1; µ j dx) ∩ W 1,p 0 (0, 1; µ j dx), j = 1, . . . m. That is, W 0 (G) contains such vectors of functions that are twice weakly differentiable on each edge and continuous on the graph with Dirichlet boundary conditions.
where the isomorphism is taken for D(A p,max ) equipped with the graph norm.
Proof. We will use the setting of [Gr87] for A = A p,max , X = E p and the boundary operator L : which is the operator (3.6) with Dirichlet boundary conditions. Hence, it is a generator on E p . Clearly holds. Now using (3.9) and that (3.10) is an isomorphism, the claim follows. Proof. Let u ∈ B arbitrary and r := Lu ∈ R n . We can define the unique v u ∈ B such that v u j is a first order polynomial for each j = 1, . . . , m taking values v u j (v i ) = r i , for e j ∈ Γ(v i ) j = 1, . . . , m, i = 1, . . . , n.
Then Lv u = r and By the construction of v u follows that since L : B → R n is onto, is a bijection. The operator L | B 1 is also bounded for the norm of B induced on B 1 . Hence, by the open mapping theorem, it is an isomorphism. Denoting its inverse by Proof. We know that (A p , D(A p )) is sectorial and maximal dissipative, hence it is injective and generates a contractive semigroup. By Remark 3.1 we have that holds with equivalence of norms. Defining (A p,max , D(A p,max )) as in (3.6), (3.7) we have that (3.12) By Lemma 3.5, Thus, by (3.12) and (3.13) holds. Hence, is true. Applying Lemma 3.6 we obtain that for θ > 1 is satisfied. Using Lemma 3.6 again, we have B ֒→ E p , and the claim follows.
In the following we will prove that the part of the operator (A p , D(A p )) in B is the generator of a strongly continuous semigroup on B. First notice that by the form (2.11) of D(A p ) and by (3.11) holds.
Proposition 3.8. The part of (A p , D(A p )) in B generates a positive strongly continuous semigroup of contractions on B.
Proof. 1. We first prove that the semigroup (T p (t)) t≥0 leaves B invariant. We take u ∈ B and use that (T p (t)) t≥0 is analytic on E p (see Proposition 2.8). Hence, T p (t)u ∈ D(A p ). By (3.17) also T p (t)u ∈ B holds.
2. In the next step we prove that (T p (t)| B ) t≥0 is a strongly continuous semigroup. By [EN00,Proposition I.5.3], it is enough to prove that there exist K > 0 and δ > 0 and a dense subspace To verify (a), we obtain by Proposition 2.8 that for u ∈ B To prove (b) take 1 2p < θ < 1 2 arbitrary. By (3.11) we have that D := E θ p ֒→ B with dense, continuous embedding. Hence, there exists C > 0 such that for u ∈ D, Summarizing 1. and 2., and using (3.17), we can apply [EN00, Proposition in Section II.2.3] for (A p , D(A p )) and Y = B, and obtain that the part of (A p , D(A p )) in B generates a positive strongly continuous semigroup of contractions on B.
Corollary 3.9. The first order problem (2.1) is well-posed on B, i.e., for all initial data u ∈ B the problem (2.1) admits a unique mild solution that continuously depends on the initial data.

Main results.
In this subsection we first apply the above results to the following stochastic evolution equation, based on (2.1). This corresponds to a slightly more general version of (1.1), see (3.33) later. Let (Ω, F , P) be a complete probability space endowed with a right-continuous filtration F = (F t ) t∈[0,T ] for some T > 0 given. We consider the problem We further assume a technical assumption regarding the graph structure that will play and important role in our setting.  1,l (t, ω, ·), . . . , a m,l (t, ω, ·)) ⊤ ∈ B for all l = 1, . . . , 2k + 1, t ∈ [0, T ] and almost all ω ∈ Ω.
For the functions g j we assume (3.20) We further assume that the functions are jointly measurable and adapted in the sense that for each j and t and B R denote the sigma-algebras of the Borel sets on [0, 1] and R, respectively. The above assumptions on the coefficients on the edges, except for Assumption 3.10 which is specific for the graph setting, are analogous to those in [KvN12, Section 5] and [KvN19, Section 5].
To handle system (3.18), we rewrite it in the form of the abstract stochastic Cauchy-problem (SCP). To do so, we specify the functions appearing in (SCP) corresponding to (3.18).
The operator (A, D(A)) = (A p , D(A p )) will be the generator of the strongly continuous analytic semigroup S := (T p (t)) t≥0 on the Banach space E := E p for some large p ≥ 2, see Proposition 2.8 and Lemma 2.9. Hence, E is a UMD space of type 2.
For  Let (A 2 , D(A 2 )) be the generator on H = E 2 , see Proposition 2.5, and pick κ G ∈ ( 1 4 , 1 2 ). By (3.14) in the proof of Corollary 3.7 we have that there exists a continuous embedding where H is a Hilbert space. Applying the steps (3.15) and (3.16) of Corollary 3.7 we obtain that H ֒→ B holds, and by (3.11), there exists a continuous embedding and thus (W H (t)) t∈[0,T ] is a cylindrical Wiener process, defined on (Ω, F , P), in the Hilbert space H with respect to the filtration F.
We will state now the result regarding system (SCP) corresponding to (3.18). Proof. The condition q > 4 allows us to choose 2 ≤ p < ∞, θ ∈ [0, 1 2 ) and κ G ∈ ( 1 4 , 1 2 ) such that θ > 1 2p (3.25) We will apply Theorem 3.3 with θ and κ G having the properties above. To this end we have to check Assumptions 3.2 for the mappings in (SCP), taking A = A p and E = E p for the p chosen above. Assumption (1) is satisfied because of the generator property of A p , see Proposition 2.8. Assumption (2) is satisfied by the statement of Proposition 3.8. Assumption (3) is satisfied since (3.25) holds and we can use Corollary 3.7. Using that the functions f j are polynomials of the 4th variable of the same degree 2k + 1 (see (3.19 In the following theorem we will state a result regarding Hölder regularity of the mild solution of (SCP) corresponding to (3.18), see (3.3).

If the inequality
is fulfilled, then the mild solution X of (SCP) from Theorem 3.13 satisfies Proof. Using the continuous embedding (3.11), we have that holds. Since (2k + 1)q > 4, by Theorem 3.13 there exists a global mild solution This solution satisfies the following implicit equation (see (3.3)): where S denotes the semigroup generated by A p on E p , * denotes the usual convolution, ⋄ denotes the stochastic convolution with respect to W. In the following we have to estimate the L q (Ω; C λ ([0, T ], E η p ))-norm of X, and we will do this using the triangle-inequality in (3.27).
For the qth power of first term we have (3.28) where · F p.λ denotes the Favard norm of order λ on E p , see [EN00, Definition II.5.10]. Furthermore, because of the continuous inclusion D((−A p ) λ ) ֒→ C λ p , we have that there exists c = c(λ) such that To estimate the qth power of the second term We will use [vNVW08, Lemma 3.6] with this θ, α = 1, and q instead of p, and obtain that there exist constants C ≥ 0 and ε > 0 such that We have to estimate the expectation of the qth power on the right-hand-side of (3.29). By Corollary 3.7 we obtain 2p holds and (ω ′ − A p ) −θ is an isomorphism between E −θ p and E p . Using this and Assumptions 3.2(5) with m = 2k + 1 (which holds by the proof of Theorem 3.13), we have where denotes that the expression on the left-hand-side is less or equal to a constant times the expression on the right-hand-side. This implies that for each T > 0 there exists C T > 0 such that T ],B)) , (3.30) and the right-hand-side is finite.
To estimate the stochastic convolution term in (3.27) we first fix 0 < α < 1 2 such that holds. We now choose κ G ∈ ( 1 4 , 1 2 ) such that is satisfied. Applying [vNVW08, Proposition 4.2] with θ = κ G and q instead of p, we have that there exist ε > 0 and C ≥ 0 such that In the following we proceed similarly as done in the proof of [KvN12,Theorem 4.3], with N = 1 and q instead of p. Since E −κ G p is a Banach space of type 2 (because E p is of that type), the continuous embedding holds. Using this, Young's inequality and the properties of G, respectively, we obtain the following estimates Hence, for each T > 0 there exists constant C ′ T > 0 such that (3.31) In summary, by (3.28), (3.30) and (3.31), we obtain that for the mild solution X ∈ L q (Ω; C λ ([0, T ], E η p )) holds, hence the proof is completed.
We are now in the position to finally consider (1.1). Let β := max 1≤j≤m β j .
We define the operator A p on E p as in (2.5) withp j 's instead of p j 's and with domain (2.11).
Proof. First note that the coefficientsp j stay nonnegative as the constants ̺ j are nonnegative. Furthermore, the nonlinear terms f j = f in (3.32) are of the form (3.19) with k = 1 and constant coefficients. Hence, Assumption 3.10 is fullfilled by Remark 3.11. The statement then follows from Theorems 3.13 and 3.14.
3.4. Concluding remarks. In equation (3.18a) we could have prescribed coloured noise instead of white noise on the edges of the graph. That is, we could seṫ u j (t, x) = (c j u ′ j ) ′ (t, x) − p j (x)u j (t, x) + f j (t, x, u j (t, x)) + g j (t, x, u j (t, x))R j ∂w j ∂t (t, x), t ∈ (0, T ], x ∈ (0, 1), j = 1, . . . , m, (3.34) with R j ∈ γ(L 2 (0, 1; µ j dx), L p (0, 1; µ j dx)). Then we define  Ep) where L (r) is the maximum of the Lipschitz-constants of the functions g j on the ball of radius r. If setting (3.34) instead of (3.18a), Theorem 3.13 remains true as stated; that is, for q > 4, but one may use a simpler Hilbert space machinery; that is, one may set p = 2 in the proof. However, in the coloured noise case, Theorem 3.13 is true also for q > 2. But this can only be shown by choosing p > 2 large enough in the proof and hence, in this case, the Banach space arguments are crucial.
In Theorem 3.14, if one takes p = 2 (Hilbert space) and q > 4, then the statement is true for λ + η > 1 4 with λ + η < 1 2 − 1 q (3.35) instead of (3.26). In this case R will be a Hilbert-Schmidt operator whence the covariance operator of the driving process is trace-class. However, the statement of the theorem remains true for q > 2 as well assuming (3.35) instead of (3.26), but only for the Banach space E p for p large enough so that λ + η > 1 2p . The statements of Theorem 3.15 could also be changed accordingly.