Existence Results For Some Second-Order Stochastic Differential Equations

The impetus of the work done in this chapter comes from two main sources from the deterministic setting. The first one is the work of Mawhin [139], in which the dissipativeness and the existence of bounded solutions on the whole real number line to the second-order differential equations given by $$u^{\prime \prime}(t) + cu^{\prime} + Au + g(t, u) = 0, \ \ t \in \mathbb{R},$$ where $$A : D(A) \subset \mathbb{H} \to \mathbb{H}$$ is a self-adjoint operator on a Hilbert space $$\mathbb{H}$$ , which is semipositive definite and has a compact resolvent, $$c>0, \ {\rm and} \ g : \mathbb{R} \times \mathbb{H} \to \mathbb{H}$$ is bounded, sufficiently regular, and satisfies some semi-coercivity condition, was established. The abstract results in [139] were subsequently utilized to study the existence of bounded solutions to the so-called nonlinear telegraph equation subject to some Neumann boundary conditions. Unfortunately, the main result of this chapter does not apply to the telegraph equation as the linear operator presented in [139], which involves Neumann boundary boundary conditions, lacks exponential dichotomy. The second source is the work by Leiva [118], in which the existence of (exponentially stable) bounded solutions and almost periodic solutions to the second-order systems of differential equations given by $$u^{\prime\prime}(t) + cu^{\prime}(t) + dAu +kH(u)=P(t),\quad u\in \mathbb{R}^n, \quad t\in \mathbb{R},$$ where $$A \ {\rm is \ an} \ n \times n$$ -matrix whose eigenvalues are positive, c, d, k are positive constants, $$H : \mathbb{R}^n \to \mathbb{R}^n$$ is a locally Lipschitz function, $$P : \mathbb{R} \to \mathbb{R}^n$$ is a bounded continuous function, was established. In this chapter, using slightly different techniques as in [118, 139], we study and obtain some reasonable sufficient conditions, which do guarantee the existence of square-mean almost periodic solutions to the classes of nonautonomous second-order stochastic differential equations $$\begin{array}{lll}dX^{\prime}(\omega, t) + a(t) dX(\omega, t) & = & \left[ -b(t) \mathcal{A}X(\omega, t) + f_1(t, X(\omega, t))\right]dt \\ {} & {} & +f_2(t, X(\omega, t)) d\mathbb{W}(\omega, t), \end{array}$$ for all $$\omega \in \Omega \ {\rm and} \ t\in \mathbb{R}, \ {\rm where} \ \mathcal{A} : D(\mathcal{A}) \subset \mathbb{H} \to \mathbb{H}$$ is a self-adjoint linear operator whose spectrum consists of isolated eigenvalues $$0