On the small time asymptotics of diffusion processes on Hilbert spaces. (English) Zbl 1044.60071
Summary: We establish a small time large deviation principle and obtain the following small time asymptotics:
\[
\lim_{t\to 0}2t\log P(X_0 \in B,\;X_t\in C)= -d^2(B,C),
\]
for diffusion processes on Hilbert spaces, where \(d(B,C)\) is the intrinsic metric between two subsets \(B\) and \(C\) associated with the diffusions. The case of perturbed Ornstein-Uhlenbeck processes is treated separately at the end of the paper.
MSC:
| 60J60 | Diffusion processes |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 31C25 | Dirichlet forms |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60F10 | Large deviations |
Keywords:
Dirichlet form; intrinsic metric; large deviation; stochastic evolution equation; Girsanov transformReferences:
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