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Lecture 11.
Scaling limits of Large Systems. Start with an example. Let cfw_xi (t) be a collection
of processes such that
dxi (t) = c[xi1 (t) 2xi (t) + xi+1 (t)]dt + i (t)
where c > 0 and i (t) = i,i+1 (t) i1,i (t). This is an example of a system of inter
Lecture 6.
More on Localization:
Suppose on the space C [0, T ], Rd], Ft , P ] we have a stopping time and a family
Q (), of measures on C [ (w), T ], Rd] such that
P : Q (), [y () : y ( ( ) = x( ( ), )] = 1 = 1
i.e. with probability 1 the Q-paths start w
Exit Problem. Consider
t
x ( t ) = x +
b(x (s)ds +
(t)
0
and let Qx, be the distribution of the solution x . As 0 the measure Qx, concentrates
on the trajectory which is the solution of
t
x( t ) = x +
b(x(s)ds
0
There is a large deviation principle for c
Ergodic Theorems: Suppose
L=
1
2
ai,j (x)Di,j +
i,j
bj (x)Dj
j
is the generator of a process. Assume that cfw_ai,j are continuous and strictly positive
denite, i.e. cfw_ai,j is positive denite for each x Rd . An invariant probability measure
is one for