%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software %%(www.radicaleye.com) %%Title: paolo.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips paolo.dvi -o paolo.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.04.04:1640 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 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y(tions)j(on)g(\(2.9\))g (inspired)g(b)m(y)h(a)f(di\013eren)m(t)g(pro)s(of)g(of)g(con)m(v)m (ergence)j(to)d(motion)e(b)m(y)j(mean)f(curv)-5 b(ature,)0 3586 y(based)30 b(on)g(the)g(analysis)f(in)g([23)o(])h(of)f(the)h (isotropic)e(case.)43 b(The)31 b(adv)-5 b(an)m(tage)29 b(of)h(this)f(approac)m(h)h(is)f(a)g(new)0 3702 y(form)m(ula)e(for)h Fn(h)p Fp(u;)17 b(A)p Fr(\()p Fp(e)p Fr(\))p Fp(u)p Fn(i)27 b Fr(whic)m(h)i(is)f(at)g(\014rst)h(sigh)m(t)f(non)h(negativ)m(e)g(and) f(for)g(whic)m(h)h(it)f(is)g(easier)g(to)h(pro)m(v)m(e)0 3818 y(strict)35 b(p)s(ositivit)m(y)-8 b(.)51 b(Less)37 b(ob)m(vious)f(is)f(c)m(hec)m(king)i(directly)e(that)g(the)h(new)h (expression)g(is)e(equal)g(to)h(the)0 3934 y(one)27 b(giv)m(en)g(b)m(y) i(\(2.9\):)40 b(it)26 b(m)m(ust)h(b)s(e)g(b)m(y)h(construction,)g (since)f(they)h(b)s(oth)f(lead)f(to)g(the)i(same)e(anisotropic)0 4051 y(motion)33 b(b)m(y)j(mean)e(curv)-5 b(ature.)51 b(W)-8 b(e)36 b(do)e(not)h(w)m(an)m(t,)i(though,)e(to)g(go)f(through)h 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1848 3323 279 4 v 1867 3414 a(\026)-68 b Fp(m)1933 3385 y Fm(0)1933 3439 y Fj(e)1971 3414 y Fr(\()p Fp(r)2053 3429 y Fj(e)2089 3414 y Fr(\))2137 3346 y Fp(J)9 b Fr(\()p Fp(r)2285 3305 y Fm(0)2331 3346 y Fn(\000)22 b Fp(r)s Fr(\))c(\026)-67 b Fp(m)2600 3305 y Fm(0)2600 3371 y Fj(e)2637 3346 y Fr(\()p Fp(r)2722 3305 y Fm(0)2719 3371 y Fj(e)2756 3346 y Fr(\))2794 3205 y Fo(\033)3639 3346 y Fr(\(5.6\))917 3626 y Fp(c)959 3641 y Fj(e)995 3626 y Fr(\()p Fp(\030)5 b Fr(\))27 b(:=)h(\010)1347 3585 y Fm(0)q(0)1390 3626 y Fr(\()18 b(\026)-67 b Fp(m)1513 3641 y Fj(e)1550 3626 y Fr(\()p Fp(\030)5 b Fr(\)\))p Fp(;)114 b(c)1895 3641 y Fj(e)1931 3626 y Fr(\()p Fp(r)m(;)17 b(r)2101 3585 y Fm(0)2124 3626 y Fr(\))28 b(:=)g Fp(c)2363 3641 y Fj(e)2399 3626 y Fr(\()p Fp(r)2481 3641 y Fj(e)2518 3626 y Fr(\))17 b Fp(K)2656 3641 y Fj(e)2693 3626 y Fr(\()p Fp(r)m(;)g(r)2863 3585 y Fm(0)2885 3626 y Fr(\))716 b(\(5.7\))0 3962 y Fu(Lemma)37 b(5.1.)49 b Fp(K)703 3977 y Fj(e)740 3962 y Fr(\()p Fp(r)m(;)17 b(r)910 3926 y Fm(0)933 3962 y Fr(\))35 b Fg(is)f(a)h(tr)-5 b(ansition)35 b(pr)-5 b(ob)g(ability)34 b(density)h(and)1119 4181 y Fr(\()p Fn(L)1226 4196 y Fj(e)1262 4181 y Fp(f)11 b Fr(\)\()p Fp(r)s Fr(\))27 b(=)1613 4045 y Fo(Z)1712 4181 y Fp(dr)1810 4140 y Fm(0)1850 4181 y Fp(c)1892 4196 y Fj(e)1929 4181 y Fr(\()p Fp(r)m(;)17 b(r)2099 4140 y Fm(0)2121 4181 y Fr(\)[)p Fp(f)11 b Fr(\()p Fp(r)2330 4140 y Fm(0)2353 4181 y Fr(\))22 b Fn(\000)h Fp(f)11 b Fr(\()p Fp(r)s Fr(\)])917 b(\(5.8\))0 4399 y Fn(L)69 4414 y Fj(e)142 4399 y Fg(is)36 b(then)h(the)f(gener)-5 b(ator)37 b(of)f(a)g(jump)h (Markov)f(pr)-5 b(o)g(c)g(ess)36 b(with)g(jump)g(intensities)g Fp(c)3176 4414 y Fj(e)3213 4399 y Fr(\()p Fp(r)m(;)17 b(r)3383 4363 y Fm(0)3406 4399 y Fr(\))36 b Fg(and)g(it)h(is)0 4515 y(self-adjoint)f(w.r.t.)h(the)g Fp(\033)t Fg(-\014nite)g(me)-5 b(asur)g(e)55 b Fr(\026)-67 b Fp(m)1750 4479 y Fm(0)1750 4540 y Fj(e)1787 4515 y Fr(\()p Fp(r)1869 4530 y Fj(e)1906 4515 y Fr(\))1944 4479 y Fl(2)1983 4515 y Fp(dr)s Fg(.)51 b(Mor)-5 b(e)g(over)37 b Fn(L)2662 4530 y Fj(e)2736 4515 y Fg(le)-5 b(aves)36 b(invariant)h(the)g(set)g(of)0 4631 y(functions)d(which)g(dep)-5 b(end)34 b(only)h(on)f Fp(r)1416 4646 y Fj(e)1453 4631 y Fg(.)0 4882 y(Pr)-5 b(o)g(of.)51 b Fp(K)391 4897 y Fj(e)428 4882 y Fr(\()p Fp(r)m(;)17 b(r)598 4845 y Fm(0)621 4882 y Fr(\))35 b(is)g(eviden)m(tly)h(non)f (negativ)m(e)h(and,)g(b)m(y)i(\(4.7\))c(and)i(\(4.5\))o(,)2850 4801 y Fo(R)2916 4882 y Fp(dr)3014 4845 y Fm(0)3053 4882 y Fp(K)3136 4897 y Fj(e)3173 4882 y Fr(\()p Fp(r)m(;)17 b(r)3343 4845 y Fm(0)3366 4882 y Fr(\))32 b(=)g(1.)52 b(The)0 4998 y(other)33 b(statemen)m(ts)g(are)g(trivial.)p 3769 4998 4 66 v 3773 4935 59 4 v 3773 4998 V 3831 4998 4 66 v 100 5279 a(By)h(Lemma)e(5.1)h(the)h(action)e(of)i Fn(L)1405 5294 y Fj(e)1475 5279 y Fr(on)f(functions)h(whic)m(h)g(dep)s (end)g(only)f(on)h Fp(r)3048 5294 y Fj(e)3118 5279 y Fr(is)f(still)e(a)i(function)0 5408 y(of)e(only)g Fp(r)367 5423 y Fj(e)404 5408 y Fr(,)g(so)h(that)f(w)m(e)i(can)e(de\014ne)i(a)e (generator)g Fn(L)1973 5357 y Fl(\(1\))1973 5418 y Fj(e)2099 5408 y Fr(on)g Fp(L)2299 5372 y Fm(1)2374 5408 y Fr(\()p Fk(R)5 b Fr(\))37 b(b)m(y)c(setting,)e(for)g Fp(f)38 b Fn(2)29 b Fp(L)3429 5372 y Fm(1)3504 5408 y Fr(\()p Fk(R)5 b Fr(\))37 b(and)1882 5507 y Fi(12)p eop %%Page: 13 13 13 12 bop 0 253 a Fr(denoting)422 227 y(^)401 253 y Fp(f)11 b Fr(\()p Fp(r)s Fr(\))27 b(=)h Fp(f)11 b Fr(\()p Fp(r)855 268 y Fj(e)891 253 y Fr(\),)1411 368 y Fo(\000)1456 448 y Fn(L)1525 407 y Fl(\(1\))1525 473 y Fj(e)1619 448 y Fp(f)1678 368 y Fo(\001)1724 448 y Fr(\()p Fp(\030)5 b Fr(\))27 b(:=)2005 368 y Fo(\000)2051 448 y Fn(L)2120 463 y Fj(e)2178 422 y Fr(^)2157 448 y Fp(f)2215 368 y Fo(\001)2261 448 y Fr(\()p Fp(\030)5 b(e)p Fr(\))1209 b(\(5.9\))0 663 y(The)34 b(jump)e(in)m(tensit)m(y)g(of)g Fn(L)1031 612 y Fl(\(1\))1031 673 y Fj(e)1158 663 y Fr(is)g Fp(c)1298 678 y Fj(e)1335 663 y Fr(\()p Fp(\030)5 b Fr(\))p Fp(K)1549 612 y Fl(\(1\))1542 673 y Fj(e)1643 663 y Fr(\()p Fp(\030)g(;)17 b(\030)1821 626 y Fm(0)1843 663 y Fr(\),)32 b(where)1037 894 y Fp(K)1127 853 y Fl(\(1\))1120 918 y Fj(e)1221 894 y Fr(\()p Fp(\030)5 b(;)17 b(\030)1399 853 y Fm(0)1421 894 y Fr(\))28 b(=)1677 826 y(1)p 1600 871 203 4 v 1600 962 a Fp(c)1642 977 y Fj(e)1679 962 y Fr(\()p Fp(\030)5 b Fr(\))1937 826 y(1)p 1839 871 246 4 v 1857 962 a(\026)-67 b Fp(m)1924 933 y Fm(0)1924 987 y Fj(e)1961 962 y Fr(\()p Fp(\030)5 b Fr(\))2111 894 y Fp(J)2165 909 y Fj(e)2202 894 y Fr(\()p Fp(\030)2288 853 y Fm(0)2333 894 y Fn(\000)22 b Fp(\030)5 b Fr(\))35 b(\026)-67 b Fp(m)2620 853 y Fm(0)2620 918 y Fj(e)2656 894 y Fr(\()p Fp(\030)2742 853 y Fm(0)2765 894 y Fr(\))788 b(\(5.10\))0 1131 y(Ob)m(viously)-8 b(,)32 b(for)g(an)m(y)i Fp(t)28 b Fn(\025)g Fr(0,)1352 1258 y Fo(\010)1410 1339 y Fp(e)1455 1297 y Fm(L)1504 1262 y Ff(\(1\))1504 1308 y Fe(e)1587 1297 y Fj(t)1616 1339 y Fp(f)1675 1258 y Fo(\011)1733 1339 y Fr(\()p Fp(\030)5 b Fr(\))27 b(=)1988 1258 y Fo(\010)2046 1339 y Fp(e)2091 1297 y Fm(L)2140 1305 y Fe(e)2173 1297 y Fj(t)2224 1312 y Fr(^)2203 1339 y Fp(f)2261 1258 y Fo(\011)2319 1339 y Fr(\()p Fp(\030)5 b(e)p Fr(\))1103 b(\(5.11\))0 1553 y(Moreo)m(v)m(er,)44 b(b)m(y)e(Lemma)d(5.1,)k Fn(L)1229 1502 y Fl(\(1\))1229 1563 y Fj(e)1363 1553 y Fr(is)e(the)g(generator)f(of)h(a)f(jump)g(Mark) m(o)m(v)i(semigroup)e(on)h Fk(R)51 b Fr(with)0 1669 y(\014nite,)45 b(p)s(ositiv)m(e,)g(in)m(v)-5 b(arian)m(t)41 b(measure)61 b(\026)-67 b Fp(m)1584 1633 y Fm(0)1584 1694 y Fj(e)1621 1669 y Fr(\()p Fp(\030)5 b Fr(\))1745 1633 y Fl(2)1784 1669 y Fp(d\030)46 b Fr(with)d(resp)s(ect)g(to)g(whic)m(h)g(it)e(is)i (self-adjoin)m(t.)72 b(W)-8 b(e)0 1785 y(denote)33 b(b)m(y)h(\()p Fn(\001)p Fp(;)17 b Fn(\001)p Fr(\))626 1800 y Fj(e)694 1785 y Fr(the)33 b(scalar)f(pro)s(duct)h(in)f Fp(L)1685 1749 y Fl(2)1725 1785 y Fr(\()p Fk(R)t Fp(;)41 b Fr(\026)-73 b Fp(m)1957 1749 y Fm(0)1957 1810 y Fj(e)2000 1785 y Fr(\()p Fp(\030)5 b Fr(\))2124 1749 y Fl(2)2163 1785 y Fp(d\030)g Fr(\).)100 1902 y(With)32 b(reference)i(to)e(\(2.9\))g(w)m (e)i(de\014ne)1023 2137 y Fp(b)1064 2152 y Fj(e;u)1190 2137 y Fr(:=)1330 2069 y(1)p 1330 2114 49 4 v 1330 2205 a(2)1406 2001 y Fo(Z)1505 2137 y Fp(d\030)1620 2001 y Fo(Z)1720 2137 y Fp(dr)19 b(J)9 b Fr(\()p Fp(r)s Fr(\))18 b(\026)-67 b Fp(m)2105 2096 y Fm(0)2105 2161 y Fj(e)2141 2137 y Fr(\()p Fp(\030)5 b Fr(\))18 b(\026)-67 b Fp(m)2350 2096 y Fm(0)2350 2161 y Fj(e)2387 2137 y Fr(\()p Fp(\030)26 b Fr(+)c Fp(r)2636 2152 y Fj(e)2673 2137 y Fr(\))17 b Fp(r)2775 2096 y Fl(2)2772 2161 y Fj(u)3591 2137 y Fr(\(5.12\))0 2366 y(so)33 b(that)697 2577 y Fn(h)p Fp(u;)17 b(A)p Fr(\()p Fp(e)p Fr(\))p Fp(u)p Fn(i)26 b Fr(=)h Fp(b)1295 2592 y Fj(e;u)1415 2577 y Fr(+)1513 2441 y Fo(Z)1613 2577 y Fp(d\030)1728 2441 y Fo(Z)1827 2577 y Fp(dr)1925 2536 y Fm(0)1964 2577 y Fp(J)9 b Fr(\()p Fp(r)2112 2536 y Fm(0)2135 2577 y Fr(\))19 b(\026)-68 b Fp(m)2258 2536 y Fm(0)2258 2601 y Fj(e)2296 2577 y Fr(\()p Fp(\030)5 b Fr(\)\()p Fp(@)2509 2592 y Fj(u)2572 2577 y Fr(\026)-68 b Fp(m)2638 2592 y Fj(e)2676 2577 y Fr(\)\()p Fp(\030)26 b Fr(+)c Fp(r)2966 2536 y Fm(0)2963 2601 y Fj(e)3000 2577 y Fr(\))17 b Fp(r)3102 2536 y Fm(0)3099 2601 y Fj(u)3591 2577 y Fr(\(5.13\))0 2971 y Fu(Lemma)37 b(5.2.)49 b Fg(L)-5 b(et)1524 3153 y Fp(w)s Fr(\()p Fp(\030)5 b Fr(\))26 b(:=)i(\()p Fn(L)1985 3168 y Fj(e)2021 3153 y Fp(r)2065 3168 y Fj(u)2110 3153 y Fr(\)\()p Fp(\030)5 b(e)p Fr(\))1274 b(\(5.14\))0 3336 y Fg(Then)1043 3561 y Fp(a)1094 3576 y Fj(e;u)1191 3561 y Fr(\()p Fp(t)p Fr(\))28 b(=)g Fp(tb)1510 3576 y Fj(e;u)1630 3561 y Fn(\000)1730 3425 y Fo(Z)1829 3452 y Fj(t)1785 3651 y Fl(0)1859 3561 y Fp(ds)1973 3425 y Fo(Z)2072 3452 y Fj(s)2027 3651 y Fl(0)2108 3561 y Fp(ds)2205 3520 y Fm(0)2245 3480 y Fo(\000)2291 3561 y Fp(w)s(;)17 b(e)2453 3520 y Fm(L)2502 3484 y Ff(\(1\))2502 3530 y Fe(e)2583 3520 y Fj(s)2616 3496 y Fh(0)2642 3561 y Fp(w)2715 3480 y Fo(\001)2760 3600 y Fj(e)3591 3561 y Fr(\(5.15\))0 3878 y Fg(Pr)-5 b(o)g(of.)43 b Fr(By)34 b(\(5.4\),)964 4127 y Fp(a)1015 4142 y Fj(e;u)1113 4127 y Fr(\()p Fp(t)p Fr(\))28 b(=)1365 4060 y(1)p 1365 4104 V 1365 4196 a(2)1441 3992 y Fo(Z)1540 4127 y Fp(d\030)39 b Fr(\026)-67 b Fp(m)1740 4086 y Fm(0)1740 4152 y Fj(e)1777 4127 y Fr(\()p Fp(\030)5 b Fr(\))1901 4086 y Fl(2)1957 3992 y Fo(Z)2056 4018 y Fj(t)2012 4217 y Fl(0)2086 4127 y Fp(ds)2216 4047 y Fo(\010)2274 4127 y Fp(e)2319 4086 y Fm(L)2368 4094 y Fe(e)2401 4086 y Fj(s)2438 4127 y Fn(L)2507 4142 y Fj(e)2543 4127 y Fp(r)2590 4086 y Fl(2)2587 4152 y Fj(u)2632 4047 y Fo(\011)2707 4127 y Fr(\()p Fp(\030)g(e)p Fr(\))715 b(\(5.16\))0 4361 y(W)-8 b(e)33 b(ha)m(v)m(e)794 4571 y Fn(L)863 4586 y Fj(e)900 4571 y Fp(r)947 4530 y Fl(2)944 4596 y Fj(u)1017 4571 y Fr(=)1245 4504 y(1)p 1130 4548 279 4 v 1149 4640 a(\026)-68 b Fp(m)1215 4611 y Fm(0)1215 4664 y Fj(e)1252 4640 y Fr(\()p Fp(r)1334 4655 y Fj(e)1371 4640 y Fr(\))1436 4436 y Fo(Z)1535 4571 y Fp(dr)1633 4530 y Fm(0)1673 4571 y Fp(J)9 b Fr(\()p Fp(r)1821 4530 y Fm(0)1844 4571 y Fr(\))18 b(\026)-67 b Fp(m)1967 4530 y Fm(0)1967 4596 y Fj(e)2004 4571 y Fr(\()p Fp(r)2089 4530 y Fm(0)2086 4596 y Fj(e)2145 4571 y Fr(+)22 b Fp(r)2287 4586 y Fj(e)2324 4571 y Fr(\))2395 4491 y Fo(\002)2436 4571 y Fr(\()p Fp(r)2521 4530 y Fm(0)2518 4596 y Fj(u)2563 4571 y Fr(\))2601 4530 y Fl(2)2663 4571 y Fr(+)g(2)p Fp(r)2854 4586 y Fj(u)2898 4571 y Fp(r)2945 4530 y Fm(0)2942 4596 y Fj(u)2987 4491 y Fo(\003)3591 4571 y Fr(\(5.17\))0 4814 y(Recalling)31 b(\(5.14\),)1117 5049 y Fp(w)s Fr(\()p Fp(\030)5 b Fr(\))26 b(=)1552 4982 y(1)p 1453 5026 246 4 v 1472 5117 a(\026)-68 b Fp(m)1538 5088 y Fm(0)1538 5142 y Fj(e)1576 5117 y Fr(\()p Fp(\030)5 b Fr(\))1726 4913 y Fo(Z)1825 5049 y Fp(dr)1923 5008 y Fm(0)1962 5049 y Fp(J)k Fr(\()p Fp(r)2110 5008 y Fm(0)2133 5049 y Fr(\))19 b(\026)-68 b Fp(m)2256 5008 y Fm(0)2256 5074 y Fj(e)2294 5049 y Fr(\()p Fp(\030)26 b Fr(+)c Fp(r)2546 5008 y Fm(0)2543 5074 y Fj(e)2580 5049 y Fr(\))17 b Fp(r)2682 5008 y Fm(0)2679 5074 y Fj(u)3591 5049 y Fr(\(5.18\))0 5292 y(so)39 b(that)g(the)h(second)g(term)f(on)g(the)g(r.h.s.)h(of)46 b(\(5.17\))38 b(reduces)j(to)e(2)p Fp(r)2633 5307 y Fj(u)2677 5292 y Fp(w)s Fr(\()p Fp(r)2832 5307 y Fj(e)2869 5292 y Fr(\).)63 b(On)39 b(the)g(other)h(hand,)0 5408 y(the)35 b(\014rst)g(term)f(on)h(the)g(r.h.s.)h(is)e(a)g(function)g(whic)m(h)i (dep)s(ends)g(only)e(on)h Fp(r)2789 5423 y Fj(e)2826 5408 y Fr(.)49 b(Hence)36 b(b)m(y)g(using)g(\(5.11\))1882 5507 y Fi(13)p eop %%Page: 14 14 14 13 bop 0 258 a Fr(and)29 b(self-adjoin)m(tness)f(of)g Fn(L)1034 208 y Fl(\(1\))1034 269 y Fj(e)1128 258 y Fr(,)h(its)f(con)m (tribution)f(in)h(\(5.16\))g(is)g(equal)g(to)g Fp(tb)2793 273 y Fj(e;u)2891 258 y Fr(.)42 b(W)-8 b(e)29 b(ha)m(v)m(e)h(th)m(us)f (pro)m(v)m(ed)0 375 y(that)763 582 y Fp(a)814 597 y Fj(e;u)911 582 y Fr(\()p Fp(t)p Fr(\))f(=)g Fp(tb)1230 597 y Fj(e;u)1350 582 y Fr(+)1448 447 y Fo(Z)1548 582 y Fp(d\030)39 b Fr(\026)-68 b Fp(m)1747 541 y Fm(0)1747 607 y Fj(e)1784 582 y Fr(\()p Fp(\030)5 b Fr(\))1908 541 y Fl(2)1964 447 y Fo(Z)2063 473 y Fj(t)2019 672 y Fl(0)2093 582 y Fp(ds)2223 501 y Fo(\010)2281 582 y Fp(e)2326 541 y Fm(L)2375 549 y Fe(e)2408 541 y Fj(s)2462 582 y Fr([)p Fp(r)2533 597 y Fj(u)2578 582 y Fp(w)s Fr(\()p Fp(r)2733 597 y Fj(e)2769 582 y Fr(\)])2834 501 y Fo(\011)2909 582 y Fr(\()p Fp(\030)g(e)p Fr(\))513 b(\(5.19\))0 799 y(W)-8 b(e)33 b(p)s(ostp)s(one)g(the)g(pro)s (of)f(that)365 875 y Fo(Z)464 1011 y Fp(d\030)39 b Fr(\026)-67 b Fp(m)664 970 y Fm(0)664 1035 y Fj(e)701 1011 y Fr(\()p Fp(\030)5 b Fr(\))825 970 y Fl(2)881 930 y Fo(\010)939 1011 y Fp(e)984 970 y Fm(L)1033 978 y Fe(e)1066 970 y Fj(s)1119 1011 y Fr([)p Fp(r)1190 1026 y Fj(u)1235 1011 y Fp(w)s Fr(\()p Fp(r)1390 1026 y Fj(e)1427 1011 y Fr(\)])1492 930 y Fo(\011)1566 1011 y Fr(\()p Fp(\030)g(e)p Fr(\))27 b(=)h Fn(\000)1960 875 y Fo(Z)2060 1011 y Fp(d\030)39 b Fr(\026)-68 b Fp(m)2259 970 y Fm(0)2259 1035 y Fj(e)2297 1011 y Fr(\()p Fp(\030)5 b Fr(\))2421 970 y Fl(2)2459 1011 y Fp(w)s Fr(\()p Fp(\030)g Fr(\))2673 930 y Fo(\010)2730 1011 y Fp(e)2775 970 y Fm(L)2824 978 y Fe(e)2857 970 y Fj(s)2894 1011 y Fp(r)2938 1026 y Fj(u)2983 930 y Fo(\011)3058 1011 y Fr(\()p Fp(\030)g(e)p Fr(\))364 b(\(5.20\))0 1229 y(By)33 b(arguing)e(as)i(in)f(\(5.17\))g(and)g(using)i(\(5.14\))e(w)m (e)h(\014nd)h(\()p Fn(L)2195 1244 y Fj(e)2231 1229 y Fp(r)2275 1244 y Fj(u)2320 1229 y Fr(\)\()p Fp(r)s Fr(\))27 b(=)h Fp(w)s Fr(\()p Fp(r)2767 1244 y Fj(e)2803 1229 y Fr(\))k(for)g(all)f Fp(r)f Fn(2)e Fk(R)3392 1192 y Fj(n)3445 1229 y Fr(.)44 b(Then)1091 1368 y Fo(\010)1149 1449 y Fp(e)1194 1408 y Fm(L)1243 1416 y Fe(e)1276 1408 y Fj(s)1313 1449 y Fp(r)1357 1464 y Fj(u)1402 1368 y Fo(\011)1476 1449 y Fr(\()p Fp(\030)5 b(e)p Fr(\))28 b(=)1776 1313 y Fo(Z)1876 1339 y Fj(s)1832 1539 y Fl(0)1913 1449 y Fp(ds)2010 1408 y Fm(0)2049 1368 y Fo(\010)2107 1449 y Fp(e)2152 1408 y Fm(L)2201 1416 y Fe(e)2234 1408 y Fj(s)2267 1384 y Fh(0)2294 1449 y Fp(w)s Fr(\()p Fp(r)2449 1464 y Fj(e)2485 1449 y Fr(\))2523 1368 y Fo(\011)2581 1449 y Fr(\()p Fp(\030)5 b(e)p 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Fp(w)s Fr(\()p Fp(\030)g Fr(\))0 5111 y(Hence)1380 5173 y Fo(\022)1453 5313 y Fn(L)1522 5328 y Fj(e)1569 5246 y Fp(@)1620 5261 y Fj(u)1683 5246 y Fr(\026)-67 b Fp(m)1750 5261 y Fj(e)p 1569 5290 219 4 v 1635 5381 a Fr(\026)g Fp(m)1702 5353 y Fm(0)1702 5406 y Fj(e)1797 5173 y Fo(\023)1887 5313 y Fr(\()p Fp(\030)5 b(e)p Fr(\))27 b(=)h Fn(\000)p Fp(w)s Fr(\()p Fp(\030)5 b Fr(\))1882 5507 y Fi(14)p eop %%Page: 15 15 15 14 bop 0 241 a Fr(whic)m(h)33 b(w)m(e)h(shorthand)f(as)1335 437 y Fn(L)1404 396 y Fl(\(1\))1404 462 y Fj(e)1498 437 y Fp(f)38 b Fr(=)28 b Fp(w)s(;)211 b(f)38 b Fr(=)28 b Fn(\000)2276 370 y Fp(@)2327 385 y Fj(u)2391 370 y Fr(\026)-67 b Fp(m)2458 385 y Fj(e)p 2276 414 219 4 v 2343 506 a Fr(\026)g Fp(m)2410 477 y Fm(0)2410 530 y Fj(e)3591 437 y Fr(\(5.21\))100 659 y(W)-8 b(e)33 b(ha)m(v)m(e)294 722 y Fo(Z)393 748 y Fj(s)349 947 y Fl(0)430 857 y Fp(ds)527 816 y Fm(0)567 777 y Fo(\000)612 857 y Fp(w)s(;)17 b(e)774 816 y Fm(L)823 781 y Ff(\(1\))823 827 y Fe(e)905 816 y Fj(s)938 793 y Fh(0)964 857 y Fp(w)1037 777 y Fo(\001)1082 896 y Fj(e)1147 857 y Fr(=)1250 722 y Fo(Z)1350 748 y Fj(s)1306 947 y Fl(0)1387 857 y Fp(ds)1484 816 y Fm(0)1523 777 y Fo(\000)1569 857 y Fp(w)s(;)g(e)1731 816 y Fm(L)1780 781 y Ff(\(1\))1780 827 y Fe(e)1861 816 y Fj(s)1894 793 y Fh(0)1920 857 y Fn(L)1989 816 y Fl(\(1\))1989 882 y Fj(e)2083 857 y Fp(f)2142 777 y Fo(\001)2188 896 y Fj(e)2252 857 y Fr(=)2356 777 y Fo(\000)2402 857 y Fp(w)s(;)g(e)2564 816 y Fm(L)2613 781 y Ff(\(1\))2613 827 y Fe(e)2694 816 y Fj(s)2731 857 y Fp(f)2790 777 y Fo(\001)2835 896 y Fj(e)2894 857 y Fn(\000)2994 777 y Fo(\000)3039 857 y Fp(w)s(;)g(f)3215 777 y Fo(\001)3260 896 y Fj(e)0 1076 y Fr(and)549 1151 y Fo(Z)649 1177 y Fj(t)605 1376 y Fl(0)679 1286 y Fp(ds)793 1151 y Fo(Z)892 1177 y Fj(s)847 1376 y Fl(0)928 1286 y Fp(ds)1025 1245 y Fm(0)1065 1206 y Fo(\000)1111 1286 y Fp(w)s(;)g(e)1273 1245 y Fm(L)1322 1210 y Ff(\(1\))1322 1256 y Fe(e)1403 1245 y Fj(s)1436 1222 y Fh(0)1462 1286 y Fp(w)1535 1206 y Fo(\001)1580 1325 y Fj(e)1700 1286 y Fr(=)83 b Fn(\000)p Fp(t)1971 1206 y Fo(\000)2017 1286 y Fp(w)s(;)17 b(f)2193 1206 y Fo(\001)2238 1325 y Fj(e)2297 1286 y Fr(+)2395 1151 y Fo(Z)2495 1177 y Fj(t)2450 1376 y Fl(0)2524 1286 y Fp(ds)2638 1206 y Fo(\000)2683 1286 y Fp(w)s(;)g(e)2845 1245 y Fm(L)2894 1210 y Ff(\(1\))2894 1256 y Fe(e)2976 1245 y Fj(s)3012 1286 y Fp(f)3071 1206 y Fo(\001)3117 1325 y Fj(e)1700 1528 y Fr(=)83 b Fn(\000)p Fp(t)1971 1447 y Fo(\000)2017 1528 y Fp(w)s(;)17 b(f)2193 1447 y Fo(\001)2238 1567 y Fj(e)2297 1528 y Fn(\000)2397 1447 y Fo(\000)2442 1528 y Fp(f)5 b(;)17 b(f)2598 1447 y Fo(\001)2644 1567 y Fj(e)2703 1528 y Fr(+)2801 1447 y Fo(\000)2846 1528 y Fp(f)5 b(;)17 b(e)2988 1486 y Fm(L)3037 1451 y Ff(\(1\))3037 1497 y Fe(e)3120 1486 y Fj(t)3149 1528 y Fp(f)3208 1447 y Fo(\001)3254 1567 y Fj(e)0 1695 y Fr(Since)950 1810 y Fo(\000)995 1891 y Fp(f)5 b(;)17 b(e)1137 1850 y Fm(L)1186 1814 y Ff(\(1\))1186 1860 y Fe(e)1269 1850 y Fj(t)1298 1891 y Fp(f)1357 1810 y Fo(\001)1403 1930 y Fj(e)1467 1891 y Fn(\024)1573 1810 y Fo(\000)1618 1891 y Fp(f)5 b(;)17 b(f)1774 1810 y Fo(\001)1820 1930 y Fj(e)1884 1891 y Fr(=)1988 1755 y Fo(Z)2087 1891 y Fp(d\030)k Fr(\()p Fp(@)2291 1906 y Fj(u)2355 1891 y Fr(\026)-68 b Fp(m)2421 1906 y Fj(e)2459 1891 y Fr(\)\()p Fp(\030)5 b Fr(\))2621 1850 y Fl(2)2687 1891 y Fp(<)28 b Fn(1)0 2116 y Fr(b)s(ecause)34 b Fp(@)412 2131 y Fj(u)476 2116 y Fr(\026)-68 b Fp(m)542 2131 y Fj(e)612 2116 y Fr(is)32 b(in)g Fp(L)890 2079 y Fl(2)930 2116 y Fr(\()p Fk(R)5 b Fp(;)17 b(d\030)5 b Fr(\))37 b(\(see)d(the)f(App)s(endix\))g (w)m(e)g(conclude)g(from)f(\(5.15\))f(that)1358 2343 y(lim)1342 2402 y Fj(t)p Fm(!1)1535 2275 y Fp(a)1586 2290 y Fj(e;u)1684 2275 y Fr(\()p Fp(t)p Fr(\))p 1535 2320 260 4 v 1647 2411 a Fp(t)1832 2343 y Fr(=)d Fp(b)1977 2358 y Fj(e;u)2097 2343 y Fr(+)2195 2262 y Fo(\000)2241 2343 y Fp(w)s(;)17 b(f)2417 2262 y Fo(\001)2461 2382 y Fj(e)3591 2343 y Fr(\(5.22\))0 2541 y(W)-8 b(e)33 b(\014nally)e (write)318 2685 y Fo(\000)363 2765 y Fp(w)s(;)17 b(f)539 2685 y Fo(\001)584 2804 y Fj(e)704 2765 y Fr(=)83 b Fn(\000)957 2630 y Fo(Z)1056 2765 y Fp(d\030)39 b Fr(\026)-67 b Fp(m)1256 2724 y Fm(0)1256 2790 y Fj(e)1293 2765 y Fr(\()p Fp(\030)5 b Fr(\))1417 2724 y Fl(2)1473 2625 y Fo(\032)1656 2698 y Fr(1)p 1557 2742 246 4 v 1575 2834 a(\026)-67 b Fp(m)1642 2805 y Fm(0)1642 2858 y Fj(e)1679 2834 y Fr(\()p Fp(\030)5 b Fr(\))1829 2630 y Fo(Z)1929 2765 y Fp(dr)2027 2724 y Fm(0)2066 2765 y Fp(J)k Fr(\()p Fp(r)2214 2724 y Fm(0)2237 2765 y Fr(\))18 b(\026)-67 b Fp(m)2360 2724 y Fm(0)2360 2790 y Fj(e)2397 2765 y Fr(\()p Fp(\030)27 b Fr(+)22 b Fp(r)2650 2724 y Fm(0)2647 2790 y Fj(e)2684 2765 y Fr(\))17 b Fp(r)2786 2724 y Fm(0)2783 2790 y Fj(u)2827 2625 y Fo(\033)g(\032)3003 2698 y Fr(\()p Fp(@)3092 2713 y Fj(u)3156 2698 y Fr(\026)-67 b Fp(m)3223 2713 y Fj(e)3260 2698 y Fr(\)\()p Fp(\030)5 b Fr(\))p 3003 2742 418 4 v 3108 2834 a(\026)-68 b Fp(m)3174 2805 y Fm(0)3174 2858 y Fj(e)3212 2834 y Fr(\()p Fp(\030)5 b Fr(\))3431 2625 y 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y(rescaled)h(energies.) h Fc(Eur)l(op)l(e)l(an)j(J.)f(Appl.)i(Math.)g Fb(9)d Ft(\(1998\))e(261{284.)21 2515 y([4])41 b(G.)26 b(Alb)r(erti,)g(G.)g (Bellettini,)h(M.)f(Cassandro,)e(E.)h(Presutti:)36 b(Surface)25 b(tension)g(in)h(Ising)f(systems)g(with)h(Kac)f(p)r(oten-)150 2615 y(tials.)i Fc(J.)j(Stat.)f(Phys.)j Fb(82)27 b Ft(\(1996\))f (743{796.)21 2785 y([5])41 b(L.)26 b(Am)n(brosio,)g(A.)h(Braides:)35 b(F)-7 b(unctionals)26 b(de\014ned)h(on)f(partitions)f(in)i(sets)f(of)h (\014nite)g(p)r(erimeter)f(I:)g(in)n(tegral)f(repre-)150 2885 y(sen)n(tation)i(and)g(\000)h(con)n(v)n(ergence.)d Fc(J.)30 b(Math.)h(Pur)l(es)e(Appl.)i(\(9\))f Fb(69)d Ft(\(1990\))g(285{305.)21 3056 y([6])41 b(G.)29 b(Barles,)f(P)-7 b(.E.)28 b(Souganidis:)38 b(A)29 b(new)g(approac)n(h)e(to)i(fron)n(t)f (propagation)f(problems:)38 b(theory)28 b(and)h(applications.)150 3155 y Fc(A)n(r)l(ch.)h(R)l(ational)g(Me)l(ch.)h(A)n(nal.)g Fb(141)26 b Ft(\(1998\))h(237{296.)21 3326 y([7])41 b(P)-7 b(.W.)38 b(Bates,)h(A.)f(Chma)5 b(j:)57 b(An)38 b(in)n(tegro)r (di\013eren)n(tial)f(mo)r(del)g(for)h(phase)f(transitions:)56 b(stationary)36 b(solutions)h(in)150 3425 y(higher)27 b(space)g(dimensions.)g Fc(J.)j(Stat.)f(Phys.)i Fb(95)c Ft(\(1999\))g(1119{1139.)21 3596 y([8])41 b(P)-7 b(.W.)24 b(Bates,)g(P)-7 b(.C.)23 b(Fife,)i(X.)g(Ren,)f(X.)h(W)-7 b(ang:)34 b(T)-7 b(ra)n(v)n(eling)22 b(w)n(a)n(v)n(es)g(in)i(a)f(con)n (v)n(olution)g(mo)r(del)g(for)h(phase)f(transitions.)150 3695 y Fc(A)n(r)l(ch.)30 b(R)l(ational)g(Me)l(ch.)h(A)n(nal.)g Fb(138)26 b Ft(\(1997\))h(105{136.)21 3866 y([9])41 b(G.)29 b(Bellettini,)h(M.)f(Cassandro,)e(E.)h(Presutti:)39 b(Constrained)28 b(minima)h(of)f(non)h(lo)r(cal)f(free)h(energy)e(functionals.)i Fc(J.)150 3966 y(Stat.)g(Phys.)j Fb(84)27 b Ft(\(1996\))f(1337{1349.)0 4136 y([10])41 b(G.)30 b(Bellettini,)h(M.)f(P)n(aolini:)40 b(An)n(ysotropic)28 b(motion)i(b)n(y)f(mean)h(curv)-5 b(ature)29 b(in)h(the)g(con)n(text)f(of)h(Finsler)f(geometry)150 4236 y Fc(Hokkaido)j(Math.)f(J.)f Fb(25)d 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y(Math.)c(Phys.)e Ft(to)f(app)r(ear.)1882 5507 y Fi(23)p eop %%Page: 24 24 24 23 bop 0 241 a Ft([15])41 b(P)-7 b(.)30 b(Butt\022)-42 b(a:)44 b(On)31 b(the)g(v)-5 b(alidit)n(y)31 b(of)g(an)g(Einstein)g (relation)f(in)h(mo)r(dels)g(of)g(in)n(terface)f(dynamics.)h Fc(J.)i(Stat.)f(Phys.)j Fb(72)150 340 y Ft(\(1993\))26 b(1401{1406.)0 511 y([16])41 b(P)-7 b(.)29 b(Butt\022)-42 b(a:)40 b(Motion)30 b(b)n(y)f(mean)g(curv)-5 b(ature)29 b(b)n(y)g(scaling)f(a)h(nonlo)r(cal)g(equation:)40 b(con)n(v)n(ergence) 27 b(at)i(all)g(times)h(in)g(the)150 611 y(t)n(w)n(o-dimensional)c (case.)h Fc(L)l(etters)i(in)g(Math.)j(Phys.)f Fb(31)c Ft(\(1994\))f(41{55.)0 781 y([17])41 b(R.)28 b(Cerf:)36 b(Large)26 b(deviations)h(for)g(three)g(dimensional)h(sup)r(ercritical) e(p)r(ercolation.)h Fc(Pr)l(eprint)p Ft(.)0 951 y([18])41 b(R.)28 b(Cerf,)f(A.)h(Pisztora:)35 b(On)27 b(the)h(W)-7 b(ul\013)29 b(crystal)e(in)g(the)h(Ising)f(mo)r(del.)h Fc(Pr)l(eprint)p Ft(.)0 1122 y([19])41 b(G.)28 b(Dal)f(Maso:)36 b(An)28 b(In)n(tro)r(duction)f(to)h(\000-Con)n(v)n(ergence.)d Fc(Birkh\177)-42 b(auser)29 b Ft(\()f(1993\).)0 1292 y([20])41 b(E.)27 b(De)g(Giorgi,)f(T.)i(F)-7 b(ranzoni:)36 b(Su)27 b(un)h(tip)r(o)f(di)h(con)n(v)n(ergenza)c(v)-5 b(ariazionale.)26 b Fc(A)n(tti)i(A)l(c)l(c)l(ad.)i(Naz.)g(Linc)l(ei)g (R)l(end.)g(Cl.)150 1392 y(Sci.)g(Mat.)h(Natur.)e(\(8\))h Fb(58)e Ft(\(1975\))e(842{850.)0 1563 y([21])41 b(A.)33 b(De)g(Masi,)g(P)-7 b(.)33 b(F)-7 b(errari,)32 b(S.)h(Goldstein,)h(D.)f (Wic)n(k:)47 b(In)n(v)-5 b(ariance)31 b(principle)i(for)f(rev)n (ersible)f(Mark)n(o)n(v)g(pro)r(cesses)150 1662 y(with)d(application)f (to)g(di\013usion)h(in)g(the)g(p)r(ercolation)f(regime.)f Fc(J.)k(Stat.)g(Phys.)h Fb(44)c Ft(\(1989\))f(787{855.)0 1833 y([22])41 b(A.)24 b(De)g(Masi,)g(T.)g(Gobron,)g(E.)f(Presutti:)35 b(T)-7 b(ra)n(v)n(elling)22 b(fron)n(ts)h(in)h(non)g(lo)r(cal)f(ev)n (olution)g(equations.)g Fc(A)n(r)l(ch.)j(R)l(ational)150 1932 y(Me)l(ch.)31 b(A)n(nal.)f Fb(132)d Ft(\(1995\))f(143{205.)0 2103 y([23])41 b(A.)33 b(De)g(Masi,)g(E.)f(Orlandi,)h(E.)g(Presutti,)g (L.)g(T)-7 b(riolo:)45 b(Motion)32 b(b)n(y)h(curv)-5 b(ature)32 b(b)n(y)g(scaling)f(non)i(lo)r(cal)f(ev)n(olution)150 2202 y(equations.)27 b Fc(J.)i(Stat.)h(Phys.)h Fb(73)c Ft(\(1993\))g(543{570.)0 2373 y([24])41 b(A.)28 b(De)h(Masi,)f(E.)f (Orlandi,)h(E.)g(Presutti,)f(L.)i(T)-7 b(riolo:)36 b(Glaub)r(er)28 b(ev)n(olution)f(with)i(Kac)e(p)r(oten)n(tials.)h(I.)g(Mesoscopic)150 2473 y(and)f(macroscopic)f(limits,)i(in)n(terface)f(dynamics.)g Fc(Nonline)l(arity)k Fb(7)c Ft(\(1994\))f(1-67.)0 2643 y([25])41 b(A.)21 b(De)h(Masi,)g(E.)f(Orlandi,)h(E.)f(Presutti,)h(L.)g (T)-7 b(riolo:)32 b(Stabilit)n(y)22 b(of)f(the)h(in)n(terface)e(in)i(a) f(mo)r(del)g(of)h(phase)f(separation.)150 2743 y Fc(Pr)l(o)l(c.)31 b(R)l(oy.)f(So)l(c.)g(Edinbur)l(gh)h(Se)l(ct.)e(A)h Fb(124)d Ft(\(1994\))f(1013{1022.)0 2913 y([26])41 b(A.)24 b(De)g(Masi,)h(E.)e (Orlandi,)h(E.)g(Presutti,)g(L.)g(T)-7 b(riolo:)34 b(Uniqueness)24 b(and)g(global)e(stabilit)n(y)i(of)g(the)g(instan)n(ton)g(in)g(non)150 3013 y(lo)r(cal)j(ev)n(olution)g(equations.)g Fc(R)l(end.)i(Math.)i (Appl.)g(\(7\))f Fb(14)d Ft(\(1994\))g(693{723.)0 3183 y([27])41 b(A.)26 b(De)h(Masi,)f(E.)g(Olivieri,)f(E.)h(Presutti:)36 b(Sp)r(ectral)26 b(prop)r(erties)f(of)h(in)n(tegral)f(op)r(erators)f (in)i(problems)g(of)g(in)n(terface)150 3283 y(dynamics)h(and)g (metastabilit)n(y)-7 b(.)28 b Fc(Markov)j(Pr)l(o)l(c)l(ess.)g(R)l (elate)l(d)f(Fields)h Fb(4)c Ft(\(1998\))f(27{112.)0 3453 y([28])41 b(G.B.)32 b(Ermen)n(trout,)f(J.B.)g(McLeo)r(d:)45 b(Existence)31 b(and)h(uniqueness)f(of)h(tra)n(v)n(elling)e(w)n(a)n(v)n (es)g(for)h(a)g(neural)g(net)n(w)n(ork.)150 3553 y Fc(Pr)l(o)l(c.)g(R)l (oy.)f(So)l(c.)g(Edinbur)l(gh)h(Se)l(ct.)e(A)h Fb(123)d Ft(\(1993\))f(461{478.)0 3724 y([29])41 b(L.C.)27 b(Ev)-5 b(ans,)27 b(R.F.)h(Gariep)n(y:)36 b(Measure)26 b(Theory)h(and)g(Fine)h (Prop)r(erties)e(of)i(F)-7 b(unctions.)28 b Fc(CR)n(C)i(Pr)l(ess)e Ft(\(1992\).)0 3894 y([30])41 b(M.E.)19 b(Gurtin:)33 b(Thermomec)n(hanics)19 b(of)g(ev)n(olving)g(phase)g(b)r(oundaries)g (in)h(the)g(plane.)f Fc(Oxfor)l(d)k(Scienc)l(e)g(Public)l(ations)150 3994 y Ft(\(1993\).)0 4164 y([31])41 b(M.)e(Kac,)i(G.)e(Uhlen)n(b)r(ec) n(k,)j(P)-7 b(.C.)39 b(Hemmer:)60 b(On)38 b(the)i(v)-5 b(an)39 b(der)f(W)-7 b(aals)39 b(theory)f(of)h(v)-5 b(ap)r(or-liquid)38 b(equilibrium.)150 4264 y(I.)h(Discussion)g(of)g(a)g(one)g(dimensional) g(mo)r(del)g Fc(J.)i(Math.)h(Phys.)g Fb(4)d Ft(\(1963\),)i(216{228.)36 b(I)r(I.)k(Discussion)e(of)i(the)150 4363 y(distribution)25 b(functions.)h Fc(J.)h(Math.)i(Phys.)g Fb(4)24 b Ft(\(1963\))g (229{247.)e(I)r(I)r(I.)k(Discussion)e(of)h(the)h(critical)e(region.)g Fc(J.)k(Math.)150 4463 y(Phys.)j Fb(5)c Ft(\(1964\),)g(60{74.)0 4634 y([32])41 b(M.A.)29 b(Katsoulakis,)d(P)-7 b(.E.)28 b(Souganidis:)37 b(Generalized)28 b(motion)g(b)n(y)g(mean)g(curv)-5 b(ature)27 b(as)h(a)g(macroscopic)e(limit)j(of)150 4733 y(sto)r(c)n(hastic)34 b(Ising)g(mo)r(dels)g(with)h(long)f(range)f(in)n (teractions)h(and)g(Glaub)r(er)g(dynamics.)h Fc(Commun.)h(Math.)i (Phys.)150 4833 y Fb(169)27 b Ft(\(1995\))f(61{97.)0 5003 y([33])41 b(M.A.)24 b(Katsoulakis,)f(P)-7 b(.E.)24 b(Souganidis:)34 b(Sto)r(c)n(hastic)24 b(Ising)g(mo)r(dels)g(and)g (anisotropic)e(fron)n(t)i(propagation.)e Fc(J.Stat.)150 5103 y(Phys.)31 b Fb(87)c Ft(\(1997\))g(63{89.)1882 5507 y Fi(24)p eop %%Page: 25 25 25 24 bop 0 241 a Ft([34])41 b(J.L.)22 b(Leb)r(o)n(witz,)i(A.)f(Mazel,) g(E.)g(Presutti:)34 b(Rigorous)21 b(pro)r(of)h(of)h(a)f(liquid-v)-5 b(ap)r(or)22 b(phase)h(transition)f(in)h(a)f(con)n(tin)n(uum)150 340 y(particle)32 b(system.)h Fc(Phys.)j(R)l(ev.)f(L)l(etters)f Fb(80)e Ft(\(1998\))g(4701.)f(Liquid-v)-5 b(ap)r(or)32 b(phase)g(transitions)g(for)g(systems)h(with)150 440 y(\014nite-range)26 b(in)n(teractions.)h Fc(J.)j(Stat.)f(Phys.)i Fb(94)c Ft(\(1999\))g(955{1025.)0 611 y([35])41 b(J.)22 b(Leb)r(o)n(witz,)h(O.)e(P)n(enrose:)33 b(Rigorous)20 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