A parabolic stochastic differential equation with fractional Brownian motion input
Let \((V,H,V^*)\) be a Gelfand triple of rigged Hilbert spaces, \(\langle\cdot,\cdot\rangle\) denotes the duality between \(V\) and \(V^*\). Assume \(f:H\to H\) is Lipschitz continuous and there is a constant \(C>0\) with \(\|f(\nu)\|_H\leq C(1+\|\nu\|_H)\) \(\forall \nu\in H\). Let \(A:V\to V^*\) be linear so that \(\langle A\nu,\nu\rangle+\alpha\|\nu\|^2\leq 0\) \(\forall \nu\in V\) for a fixed constant \(\alpha>0\). An \(H\)-valued continuous stochastic process \((X(t))_{t\in[0,T]}\) with \(X(t)\in V\) \((P\text{-a.s.})\) is a solution of \[ dX(t)=AX(t)dt+f\bigl(X(t)\bigr)dt+g(t)dB^h(t),\quad X(0)=X_0, \] if it holds for all \(\nu\) from a dense set \(M\subset V\) and all \(t\in[0,T]\), \[ (X(t),\nu)_H=(X_0,\nu)_H+\int^t_0\langle\Lambda X(s),\nu\rangle ds+\int^t_0(f(X(s)),\nu)_Hds+\left(\int^t_0g(s)dB^h(s),\nu\right)_H,\;P\text{-a.s.} \] Here \(g\) is a measurable bounded function defined on \([0,T]\subset R\), \(B^h(t)\) is a Hilbert space-valued fractional Brownian motion, \(X_0\) is an \(H\)-valued random function with \(E\|X_0\|^2_H<\infty\). An existence and uniqueness theorem is proved for such quasilinear stochastic evolution equations.
scientific article; zbMATH DE number 1348294