Gaussian estimates on networks with applications to optimal control

Abstract

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.

Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions

“…We would like to recall that in [11,13,19], a diffusion problem has been considered, stated on a finite graph, where the boundary conditions exhibit non-local behaviour, namely what happens on a given node also depends on the state of the remaining nodes, even without a direct connection. In the present work, we will consider a different type of non-local condition, studying a diffusion on a finite graph where the boundary conditions, at a given time, are affected by the present value of the state equation on each nodes, as well as by the past values of the underlying dynamic.…”

“…We underline that the operator (A, D(A)) just defined, generates a C 0 −semigroup on the space X 2 , see, e.g., [8,19,32]. Moreover, in writing system (3), we also made use of the so-called feedback operator C : D(A) → R n , which is defined as follows…”

“…[36], the books [24,31] and references therein; in neurobiology, as an example concerning the study of stochastic system of the FitzHugh-Nagumo type, see, e.g., [1,2,3,8,10]; or in finance, see, e.g., [6,14,15,26] and references therein, particularly in the light of numerical applications, see, e.g., [18] Concerning the aforementioned ambit, a possible approach which has shown to be particularly useful, is to introduce a suitable infinite dimensional space of functions that takes into account the underlying graph domain and then tackle the diffusion problem exploiting both functional analytic tools and infinite dimensional analysis. This technique had led to a systematic study of Stochastic Partial Differential Equations (SPDEs) on networks, showing that it is in general possible to rewrite a diffusion problem defined on a network in a general abstract form, see, e.g., [8,10,11,19], and the monograph [33] for a detailed introduction to the subject.…”

“…The typical conditions when one has to deal with diffusion problems governed by a second order differential operator are the so-call generalized Kirchhoff conditions, see, e.g., [32]. Nevertheless rather recently, different types of general BC has been proposed, such as non-local BC, allowing for non-local interaction of non-adjacent vertex of the graph, see, e.g., [10,19], or dynamic BC, see, e.g., [8,34], or also mixed type BC, allowing for both static and dynamic non-local boundary conditions, see, e.g., [13].…”

Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable

“…Latter problem is often overcome taking into account some additional regularity properties of the infinitesimal generator, namely the Laplacian ∆ appearing in eq. (1.1), such as the so-called m−dissipativity assumption, see, e.g., [2,3,21] and references therein, for details. We will not concern in the present paper with the existence and uniqueness result, since it is an already established result in literature, but on the existence of an optimal control for the aforementioned equation.…”