Asymptotic Behavior of the Invariant Density of a Diffusion Markov Process with Small Diffusion

Abstract

Let $X(t)$ be the n-dimensional diffusion process governed by the following equation: $dx(t) = b(x(t))dt + \sqrt \varepsilon dw(t)$. Assume $b(x) = \nabla I(x) + l(x)$, $\nabla I \cdot l = 0$. Then under some growth conditions on I and l, we show $x( \cdot )$ has unique invariant measure with density $p^\varepsilon (x)$. And we establish the following asymptotic behavior for $p^\varepsilon :p^\varepsilon (x) = \varepsilon ^{{{ - n} / 2}} \exp ({{ - 2I(x)} / \varepsilon })(R_0 (x) + O(\varepsilon ))$.