CONDITIONAL DIFFUSION PROCESSES IN A SMALL TIME IOTERVERVAL

In this paper, we study the large deviation property of the conditional diffusion process-with uniformly continuous diffusion coefficents. Let X（t） be a diffusion process associated with the Dirichlet spaoe（δ, H₀¹（R^d））), where \varepsilon(f, g)=\frac12 \int_g^R\langle\nabla f, a \cdot \nabla g\rangle(x) d x and P_x,y^σ be the law of the process X_ε(t)=X（εt） conditional onX_ε（0） = x and X_ε（l）=y. Then wo show that （X_x,y^σ） has large deviation property as ε→0 with the rate functionJ_\sigma, y(\omega)=\frac12 \int_0^1\left\langle\dot\omega(t), a^-1(\omega(t)) \cdot \dot\omega(t)\right\rangle d t-\frac12 d^2(x, y).