Stochastic evolution systems with constant coefficients (Q483629)

The authors consider the equation \[ u(t)=u_0+\int_0^t(Au(s)+f(s))\,ds+\int_0^t(Bu(s)+g(s))\,dw(s) \] on \([0,T]\), where \(w\) is a standard Brownian motion and \(A\) and \(B\) are pseudo-differential operators on \(H_\infty=\bigcup_{\gamma<0}H^\gamma(\mathbb R^d;\mathbb R^N)\) with constant coefficients whose symbols are continuous and have at most a polynomial growth. The equation is called well-posed if, for every \(r\in\mathbb R\), \(u_0\in H^r(\mathbb R^d;\mathbb R^N)\), \(f,g\in L^2(0,T;H^r(\mathbb R^d;\mathbb R^N))\), there exists a unique solution \(u\in L^2(0,t;H^\gamma(\mathbb R^d;\mathbb R^N))\) for some \(\gamma\leq r\) and an apriori estimate \[ \mathbb E\,\|u(t)\|^2_\gamma\leq C\left(\|u_0\|^2_r+\int_0^t\|f(s)\|^2_r\,ds+\int_0^t\|g(s)\|^2_r\,ds\right),\qquad 0\leq t\leq T \] holds. An \((N^2\times N^2)\)-matrix \(M(y)\) is defined using the symbols of the operators \(A\) and \(B\) and it is proved that well-posedness holds if and only if \(\sup_{0\leq t\leq T}\|\exp\{t M(y)\}\|\) grows at most polynomially in \(y\in\mathbb R^d\) or, equivalently, if the spectral abscissa of \(M(y)\) (i.e., the largest real part of the eigenvalue of \(M(y)\)) is bounded by \(C_0\ln(2+|y|)\). Based on this result, the authors define several subclasses of the studied equation, introduce finer definitions of well-posedness and stability and determine the links between them.