Abstract
In this paper, we study the questions of the existence of global weak solutions and local strong solutions of paired stochastic functional differential equations in a Hilbert space, one of which is an equation with an unbounded operator, and the other is an ordinary differential equation. We proved the existence and uniqueness theorems in the case of coefficients with polynomial growth.
Highlights
We are concerned with the existence and uniqueness of weak and strong solutions to stochastic functional differential equations in a Hilbert space of the form
Ut = u(t + θ), yt = y(t + θ), θ ∈ [−h, 0], A is an infinitesimal generator of an analytic semigroup of bounded linear operators {S(t) : t 0} in a separable Hilbert space H, W (t) is a Q-Wiener process on a separable Hilbert space K, u(t) is a state process, the functionals f and g map the space of functions continuous on [−h, 0] into H, σ maps the same space into a special space of Hilbert-Schmidt operators
Functional differential equations are mathematical models of processes whose evolution depends on their previous states
Summary
We are concerned with the existence and uniqueness of weak and strong solutions to stochastic functional differential equations in a Hilbert space of the form. The paired stochastic equations of type (1.1) arise in various applications; for instance, the bidomain equation (defibrillator model) [14], the Hodgkin-Huxley equation for nerve axons [15], the nuclear reactor dynamics equation, etc These equations are characterized by the fact that one of them is a partial differential equation (infinite-dimensional), and the other is an ordinary one (finite-dimensional). As a rule, the right-hand sides of these equations satisfy some monotonicity conditions, which makes it possible to apply Galerkin approximations This method is the main technique for obtaining the existence and uniqueness of weak solutions in this paper.
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