On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators |
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Authors: | Tomasz Klimsiak |
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Institution: | 1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87–100, Toruń, Poland
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Abstract: | We consider processes of the form s,T]?t?u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P s,x -Dirichlet process for quasi-every (s,x)∈0,T)×? d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale. |
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