Abstract
Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e i } ⊂ H be an orthonormal basis of H. Suppose that μ n = ρ n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms \(\{\varepsilon_n\}\), where \(\varepsilon_n(f) = \int_{X}\|\nabla_H f\|^2_H \rho_n \,{\rm d} \gamma\) and \(\sqrt{\rho_n} \in W^{2,1}(\gamma)\). We prove some sufficient conditions for Mosco convergence \(\varepsilon_n \to \varepsilon,\) where \(\varepsilon(f) = \int_{X} \|\nabla_H f\|^2_H \rho \,{\rm d} \gamma\). In particular, if X is a Hilbert space, \(\sup_{n} \|\sqrt{\rho_n}\|_{W^{2,1}(\gamma)} < \infty\) and \({\partial_{e_i} \rho_n}/{\rho_n} \) can be uniformly approximated by finite dimensional conditional expectations \({\rm IE}^{\mathcal{F}_N}_{\mu_n} \big({\partial_{e_i} \rho_n}/{\rho_n}\big)\) for every fixed e i , then under broad assumptions \(\varepsilon_n \to \varepsilon\) Mosco and the distributions of the associated stochastic processes converge weakly.
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Kolesnikov, A.V. Weak convergence of diffusion processes on Wiener space. Probab. Theory Relat. Fields 140, 1–17 (2008). https://doi.org/10.1007/s00440-006-0023-4
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DOI: https://doi.org/10.1007/s00440-006-0023-4
Keywords
- Dirichlet forms
- Mosco convergence
- Convergence of stochastic processes
- Gaussian measures