LARGE DEVIATIONS FOR SYMMETRIC DIFFUSION PROCESSES

Let a(x)=(a_(ij)(x)) be a uniformly continuous, symmetric and matrix-valued function satisfying uniformly elliptic condition, p(t, x, y) be the transition density function of the diffusion process associated with the Diriehlet space (, H_0~1 (R~d)), where(u, v)=1/2 integral from n=R~d sum from i=j to d(u(x)/x_i v(x)/x_ja_(ij)(x)dx).Then by using the sharpened Arouson's estimates established by D. W. Stroock, it is shown that2t ln p(t, x, y)=-d~2(x, y).Moreover, it is proved that P_y~6 has large deviation property with rate functionI(ω)=1/2 integral from n=0 to 1<(t), α~(-1)(ω(t)),(t)>dtas s→0 and y→x, where P_y~6 denotes the diffusion measure family associated with the Dirichlet form (ε, H_0~1(R~d)).

LARGE DEVIATIONS FOR SYMMETRIC DIFFUSION PROCESSES

Let a(x)=(a_ij(x)) be a uniformly continuous,symmetric and matrix-valued function satisfying uniformly elliptic condition,p(t,x,y) be the transition density function of the diffusion process assciated with the Dirichlet space $(\varepsilon,H_0^1(R^d))$,where $\\varepsilon (u,v) = \frac{1}{2}\int\limits_{{R^d}} {\sum\limits_{i,j}^d {\frac{{\partial u(x)}}{{\partial {x_i}}}} } \frac{{\partial v(x)}}{{\partial {x_j}}}{a_{ij}}(x)dx\]$.Then by using the sharpened Aronson's estimates established by D.W.Stroock,it is shown that $\\mathop {\lim }\limits_{t \to 0} 2t\ln p(t,x,y) = - {d^2}(x,y)\] $ .Moreover,it is proved that P_y^s has large deviation property with rate function $\I(w) = \frac{1}{2}\int\limits_0^1 { < \dot w(t),{a^{ - 1}}} (w(t)),\dot w(t) > dt\] $ as $s\rightarrow 0$ and $y\rightarrow x$,where P_y^s denotes the diffusion measure family associated with the Dirichlet form $\(\varepsilon ,H_0^1({R^d}))\] $.