We consider the SPDE:dZ=(AZ+b(Z))dt+dW(t),Z0=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$:dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x$$\end{document}, on a separable Hilbert space H, where A:H -> H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:H\rightarrow H$$\end{document} is self-adjoint b:H -> H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b:H\rightarrow H$$\end{document} is Lipschitz continuous and W is a cylindrical Wiener process on H. We determine, with the help of a well-known formula for nonlinear transformations of Gaussian integrals due to R. Ramer [16], an explicit representation for the law of Zx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_x$$\end{document} in C([0, T]; H), see Theorem 3.2 below. When b is, in addition, dissipative, we determine the invariant measure nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} of the semigroup Pt phi(x)=E[phi(Zx(t))]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_t\varphi (x)=\mathbb {E}[\varphi (Z_x(t))]$$\end{document}, the corresponding stationary process ZR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{\mathbb {R}}$$\end{document}. The final Section 5 is devoted to colored noise.
A mild Girsanov formula
Priola E.Membro del Collaboration Group
;
2025-01-01
Abstract
We consider the SPDE:dZ=(AZ+b(Z))dt+dW(t),Z0=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$:dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x$$\end{document}, on a separable Hilbert space H, where A:H -> H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:H\rightarrow H$$\end{document} is self-adjoint b:H -> H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b:H\rightarrow H$$\end{document} is Lipschitz continuous and W is a cylindrical Wiener process on H. We determine, with the help of a well-known formula for nonlinear transformations of Gaussian integrals due to R. Ramer [16], an explicit representation for the law of Zx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_x$$\end{document} in C([0, T]; H), see Theorem 3.2 below. When b is, in addition, dissipative, we determine the invariant measure nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} of the semigroup Pt phi(x)=E[phi(Zx(t))]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_t\varphi (x)=\mathbb {E}[\varphi (Z_x(t))]$$\end{document}, the corresponding stationary process ZR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{\mathbb {R}}$$\end{document}. The final Section 5 is devoted to colored noise.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
