Existence Results For Some Second-Order Stochastic Differential Equations

Abstract

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 $0 < \lambda_1 < \lambda_2 < \ldots < \lambda_n \to \infty$ with each eigenvalue having a finite multiplicity $\gamma_j$ equals to the multiplicity of the corresponding eigenspace, the functions $a, b : \mathbb{R} \to (0, \infty)$ are almost periodic functions, and the function $f_i(i = 1, 2) : \mathbb{R} \times L^2(\Omega, \mathbb{H}) \to L^2(\Omega, \mathbb{H})$ are jointly continuous functions satisfying some additional conditions and $\mathbb{W}$ is a one dimensional Brownian motion.