Abstract:
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(
Rd))), where \varepsilon(f, g)=\frac12 \int_g^R\langle\nabla f, a \cdot \nabla g\rangle(x) d x and
Px,yσ be the law of the process
Xε(
t)=
X(
εt) conditional on
Xε(0) =
x and
Xε(l)=
y. Then wo show that (
Xx,yσ) 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).