The stable manifold theorem for non-linear stochastic systems with memory. I: Existence of the semiflow.
Let \(M_2= \mathbb{R}^d\times L^2([-r, 0],\mathbb{R}^d)\) be the Hilbert space with the norm \[ \|(\nu,\eta)\|_{M_2}= (| \nu|^2+ \|\eta\|^2_{L^2})^{1/2},\quad \nu\in\mathbb{R}^d,\;\eta\in L^2([-r, 0],\mathbb{R}^d). \] The authors consider the stochastic functional equation \[ \begin{gathered} dx(t)= H(t,x(t),x_t)\,dt+ G(t,x(t))\,dW(t),\quad t\geq t_0\geq 0,\\ x(t_0)= \nu\in\mathbb{R}^d,\quad x_{t_0}= \eta\in L^2([-r,0],\mathbb{R}^d),\end{gathered}\tag{1} \] where the drift is a nonlinear functional \(H: \mathbb{R}^+\times M_2\to \mathbb{R}^d\), the noise coefficient is a mapping \(G: \mathbb{R}^+\times \mathbb{R}^d\to \mathbb{R}^{d\times p}\) and \(W\) is \(p\)-dimensional Brownian motion on some probability space \((\Omega,{\mathfrak I},P)\). A soluton of (1) is a process \(x:[t_0- r,\infty)\times \Omega\to \mathbb{R}^d\), where \[ x_t(\cdot,\omega)(s)= x(t+ s,\omega),\quad s\in [-r,0],\;\omega\in\Omega,\;t\geq t_0, \] and (1) holds a.s. It is show that under sufficient regularity conditions on the coefficients (1) admits a stochastic semiflow \(X:\Delta\times M_2\times \Omega\to M_2\) satisfying \(X(t_0, t,(\nu,\eta),\cdot)= (x(t), x_t)\) for all \((\nu,\eta)\in M_2\) and \(t\geq t_0\) a.s., where \(x\) is the solution of (1) and \(\Delta= \{(s,t)\in \mathbb{R}^2: 0\leq s\leq t\}\). It is also established a global locally compact semiflow for (1) and it is show that, if the coefficients in (1) are time-independent, then the global semiflow defines a cocycle on the space \(M_2\). The authors also give some extensions and generalizations of the equation (1).
stochastic functional differential equation
scientific article; zbMATH DE number 2037130
scientific article