By S. R. S. Varadhan

Many events exist during which ideas to difficulties are represented as functionality house integrals. Such representations can be utilized to check the qualitative houses of the ideas and to guage them numerically utilizing Monte Carlo tools. The emphasis during this publication is at the habit of strategies in exact events while yes parameters get huge or small.

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In Section 10 we define, for any Q&Ms(fy, the entropy, H(Q), of the stationary process O with respect to the Markov process P0>x. We will then show that the large deviation principle holds for F tx with a rate function given by H(Q). , LtfcXA) is the proportion of time up to t that the particular path o>(-) occupies the set A <=X. For fixed t>0 and w Gft, L,>a)(-) is then a probability measure on X. We denote the space of probability measures on X by M(X). Now, the relation between L^ in M(X) and Rt>ta in ^s(^) isa made clear byoccupies the set A <=X.

Consider, then, the equation for u(x, t): By the Feynman-Kac formula, the solution is given by but, from hypothesis (3), -(Lun/un)(x) = V n (x), so that the solution u(x, t) = Un(x), and thus 46 SECTION 11 From hypothesis (2), for any compact set K there exists a constant CK such that supxeK supn Un(x)^ CK. Also, from hypothesis (1), there exists a constant c such that Un(x)^c>0 for all x and n. 22) we conclude that, for any compact set K, Using hypothesis (4) we get from Fatou's lemma that, for all t, By the definition of I\x measure we have so, in particular, if then F(Q) = JX V(y)/x(dy) where ^ is the one-dimensional marginal of Q.

Let Qedts(£i) be such thatH(Q)«*> and let /ut be the margina distribution of Q. Then /x « a. Proof. Let A c X be such that a (A) = 0. We want to show that if Q  Ms(fl such that H(Q)(o)f°r almost all o> (Q measure). 1) we conclude that for almost all o> (Q measure) Qo«{ft>':«'(l)eA} = 0. , ti(A) = 0. 2. The reference measure a on X is absolutely continuous with respect to p(l, x, •) uniformly for x in compact subsets of X.

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