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)×ℝd 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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Klimsiak, T. On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators. J Theor Probab 26, 437–473 (2013). https://doi.org/10.1007/s10959-011-0381-4
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DOI: https://doi.org/10.1007/s10959-011-0381-4