Stochastic reaction–diffusion equations on networks

Abstract

We consider stochastic reaction–diffusion equations on a finite network represented by a finite graph. On each edge in the graph, a multiplicative cylindrical Gaussian noise-driven reaction–diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so, we generalize existing results on abstract stochastic reaction–diffusion equations in Banach spaces.

Reaction-diffusion equations on metric graphs with edge noise

“…A crucial point is to prove Proposition 3.9 claiming the existence of continuous, dense embeddings of the fractional domain spaces of the generators A p (see (2.8)) into the space B. In contrary to [15,Lemma 4.2], we do not have isometry here. Techniques and results from [15] make us possible to show in Theorem 3.14 that for any initial value from E c problem (1.4) admits a unique mild solution with trajectories in the space B which is a more general result than obtained in [13,Thm.…”

“…Our equations differ from those in [15] since we have no dynamics, and correspondingly no noise in the vertices. Hence, in contrary to [15], the state space of the problem will consist of functions on the edges and no E. SIKOLYA 5 boundary space is included.…”

“…Our equations differ from those in [15] since we have no dynamics, and correspondingly no noise in the vertices. Hence, in contrary to [15], the state space of the problem will consist of functions on the edges and no E. SIKOLYA 5 boundary space is included. To work in this new setting we have to introduce the product space of continuous functions on the edges E c , see (3.5) and verify results concerning this space.…”

“…In the literature stochastic (reaction-)diffusion equations on networks are treated e.g. in [5], [6], [8], [9] and [15]. In the first four papers the semigroup approach is utilized in a Hilbert space setting.…”

“…In the first four papers the semigroup approach is utilized in a Hilbert space setting. In the recent work [15] a much more general problem is treated with multiplicative Wiener-type noise on the edges as well as in the vertices and with locally Lipschitz continuous diffusion coefficients satisfying appropriate growths conditions. This paper uses an entirely different tool-set based on the semigroup approach for stochastic evolution equations in Banach spaces from [7], [16], [17] and [24].…”