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**Extra info for Parabolic Anderson Problem and Intermittency**

**Sample text**

In other words, 72(«) is the supremum of the spectrum of the (deterministic) Schrodinger operator: H2 = 2«A + ri.

As above, we only argue the case of the single distribution of u(t, x) for t and x fixed. T h e desired conclusion follows from the fact that: Mt = u(i, x) - u0(x) - / [A + r i ] u ( s , • )(x) ds Jo is a martingale and t h a t : [ M , M ] t = 1^(0) f'lufaxtfds. e. a filtration with respect to which all of the processes {Ct{x)] t > 0} are Brownian motions. The proof is as follows. First one notices t h a t : Au(s,x)ds = lim (u^(t x v n — + 0 0 = lim(/ | Tt) rt+h + M ) - u ( n ) (*>x) - ft+h / Jt A t i ( n ) ( s , x ) d s I Tt) Zin\x)u(n\s,x)ds\Ft) and then one rewrites the integral between t and t + h as a sum of integrals over intervals of the form [k2~n, (k + l ) 2 - n ) .

6) does belong to some of these spaces. We postpone the definition of these spaces for we shall not use them now. But because of our special interest in nonnegative initial conditions UQ(X) we shall first consider the uniqueness problem in the class of nonnegative solutions u(t, x). We shall assume that the potential function £t{x) satisfies the following growth condition: for each T > 0 there exists a positive constant CT for which: Kt(*)l < c r ^ / i + iog+ |x|, zexd, o