By Petrosyan A., Valdinoci E.

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**Additional info for Density estimates for a degeneratesingular phase-transition model**

**Example text**

If there exist A and B such as in (a), then we set A\ = A and A2 = AU B. It is clear that Ai £ J7, A2 £ J7, Ax c A C A2, and fi(A2\A1)=0. (b)=>(a). If there exist Ai and A 2 such as in (b), then we set A = A\ and B = A2\Ai. It is clear that A £ T, B £ J7, A\A C A2\Ai = B, and H(B) = 0. (b)=>(c). If there exist Ai and A2 such as in (b), then we put A' = A\ and B = A2\AX. It follows that A' £ J7, B £ J7, AAA' C B, and fi (B) = 0. (c)=*>(b). If there exist A' £ J7 and B £ J7 such as in (3), then we set Ax = A'\B and A2 = 4 ' U B.

It is said that the function / has at least k e-oscillations on the set / provided that there exists a finite subset J = {si < S2 < • • • < Sk < Sfc+i} of the set / such that p(f(si),f(si+i)) > e for all 1 < i < k. Denote by Zk,e the class of all functions having at least k e-oscillations on /. 17 For every subinterval I = [a,b] of the interval [T,T], the class of functions on I having a finite number of e-oscillations for every e > 0 coincides with the class of functions on I with no discontinuities of the second kind.

Our next goal is to show that Xt=Xt =1 (1-59) for all t G [T, T] and also to prove that Xt is a separable process. Indeed, if t G {U}, then lxt = Xt\ = ft. If t € {**}, then lxt ^ Xt\ C JV(i). 59). Now we are ready to prove the separability of the process Xt- If t = ti for some i > 1, then Xti = Xti, and hence for all UJ G ft, the corresponding sample path is minimally continuous with respect to {U}. lit $. {tk} and ui G -/V(i), then X t coincides with a {ij}-separable process. If t £ {t,} and w $.

### Density estimates for a degeneratesingular phase-transition model by Petrosyan A., Valdinoci E.

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