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

$$0 < \lambda_1 < \lambda_2 < \ldots < \lambda_n \to \infty$$

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.