Convergence of time changed skew product diffusion processes
This paper treats a limit theorem for the time changed skew product diffusion processes. Roughly speaking, skew product diffusion processes \(X_n\) are given by one-dimensional diffusion processes \(R_n\) and the spherical Brownian motion \(\Theta\), and the time change \(\Phi_n(t)\) is based on a positive continuous additive functional, related to the local time \(\ell(t)\) of \(R_n\). In this paper, the author considers the time changed skew product diffusion processes \(X_n(t)\) \(( n \in {\mathbb N})\) which are associated with the following Dirichlet forms \({\mathcal E}_n\) on \(L^2( I \times S^{d-1}, \psi_n)\): \[ {\mathcal E}_n (u,v) = \int_{ S^{d-1}} {\mathcal E}^{ R_n } ( u( \cdot, \theta), v( \cdot, \theta) ) d m^{ \Theta}(\theta) + \int_I {\mathcal E}^{ \Theta} ( u(r, \cdot), v(r, \cdot)) d \nu_n(r). \tag{1} \] Here \({\mathcal E}^{R_n}\) is the Dirichlet form corresponding to the one-dimensional diffusion process \(R_n(t)\) on an interval \(I\) \(=\) \(( \ell_1, \ell_2)\) with the scale function \(s_n\) and the speed measure \(m_n\), and it is actually given by \[ {\mathcal E}^{R_n}(u,v) = \int_I \frac{ du}{d s_n} \frac{dv}{ d s_n} d s_n. \tag{2} \] \({\mathcal E}^{\Theta}\) is the Dirichlet form correponding to the spherical Brownian motion \(\Theta(t)\) on \(S^{d-1}\), and \(m^{\Theta}\) stands for the spherical element of \(S^{d-1}\). \(\psi_n\) denotes a bounded measure on \(I \times S^{d-1}\) and \(\nu_n\) is a Borel measure on \(I\). Assume that \newline (A.1) \(s_n(x) \to s(x)\), \(x \in I\); \newline (A.2) \(m_n(x) \to m(x)\), \(x \in {\mathcal C}(m)\) (= the set of all continuity points of \(m\)); \newline (A.3) \(\nu_n \to \nu\), vaguely; \newline (A.4) \(\psi_n \to \mu \otimes m^{ \Theta}\), vaguely, \newline where \(s\) is an increasing continuous function, \(m\) is an increasing right continuous function, \(\nu\) is a Borel measure satisfying \(\text{supp} [ \nu ] = I\), and \(\mu\) is a Borel measure on \(I\). The author is interested in the convergence problem of such a sequence of diffusion processes toward a stochastic process whose sample paths are not necessarily continuous. In other words, the above problem is nothing but a convergence problem of Dirichlet forms of local type towards that of nonlocal type. Let us denote by \(p_t^{X_n}\) the semigroup of \[ X_n = \{ R_n( \Phi_n(t)), \Theta( f_n( \Phi_n(t))), P_{(r,\theta)}^{ X_n} = P_r^{ R_n} \otimes P_{\theta}^{\Theta} \} \tag{3} \] with \(f_n(t) = \int_I \ell^{R_n} (t, \xi) d \nu_n( \xi)\) and the local time \(\ell^{R_n}\) of \(R_n\). Note that the process \(X_n\) satisfies the Feller property. Let \(R(t)\) be a one-dimensional diffusion process with the scale function \(s(x)\), the speed measure \(m(x)\), and no killing measure. Assume that the end point \(\ell_i\) is \((s,m)\)-entrance or natural for \(i=1,2\). \(\ell^R(t, \xi)\) denotes the local time of \(R\), and suppose that \(R\) is independent of \(\Theta\). Then the time changed skew product process \(X\) is defined by \[ X = \{ X(t) = ( R( \Phi(t)), \Theta( f( \Phi(t))) ), P_{(r,\theta)}^X = P_r^R \otimes P_{\theta}^{\Theta} \}, \tag{4} \] where \(f(t) = \int_I \ell^R( t, \xi) d \nu( \xi)\), \(\Psi(t) = \int_I \ell^R(t, \xi) d \mu(\xi)\) and \(\Phi(t)\) is the right continuous inverse of \(\Psi(t)\). On this account, with an additional condition on \(\psi_n\), the author proves the following: Theorem. The time changed processes \(X_n\) converge to the time changed process \(X\) in the following sense: \[ \lim_{ n \to \infty} p_t^{ X_n} g(r, \theta) = p_t^X g(r, \theta), \tag{5} \] for \(t > 0\), \((r, \theta ) \in \Gamma\), and \(g \in C_b( \Gamma)\) with \(\Gamma = \text{supp} [\mu ] \times S^{d-1}\), where the semigroup \(p_t^X\) of \(X\) is given by \(p_t^X g(r, \theta)\) \(=\) \(\operatorname{E}^{ P_{(r, \theta)}^X} [ g( X_t)]\). Some examples of limit processes are given as well, which illustrate Dirichlet forms with diffusion term, jump term and killing term. As for other related works, see, e.g. [\textit{Y. Ogura} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 4, 507--556 (2002; Zbl 0997.60084); \textit{T. Takemura} and \textit{M. Tomisaki}, Osaka J. Math. 48, No. 1, 269--290 (2011; Zbl 1234.60078)].
scientific article; zbMATH DE number 6136705
scientific article