This is a multi-part message in MIME format. ---------------9911301231132 Content-Type: text/plain; name="99-453.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-453.keywords" Fokker-Planck equation, non-Gaussian stochastic differential equations ---------------9911301231132 Content-Type: application/postscript; name="FP.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="FP.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: FP.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips FP -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.10.29:1339 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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 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Fi(\016)s(t)p Fm(\))r Fh(\000)r Fi(X)g Fm(\()p Fi(t)p Fm(\),)731 3571 y(corresp)r(onding)26 b(to)g(a)g(giv)n(en)g(time)f(lag)h Fi(\016)s(t)21 b(>)g Fm(0,)27 b(are)f(de\014ned)e(in)i(a)g(similar)g(w)n(a)n(y:)841 3823 y Fi(F)11 b Fm([)p Fi(p)p Fm(\()p Fi(x)16 b Fm(+)h Fi(\016)s(x;)12 b(t)17 b Fm(+)g Fi(\016)s(t)p Fh(j)p Fi(x;)12 b(t)p Fm(\)])83 b(=)g Fi(\016)s(Z)1921 3831 y Ff(X)1979 3823 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x;)g(t)p Fm(\))736 b(\(14\))1689 3939 y Fh(\021)83 b Fi(exp)p Fm(\()p Fi(\016)s(K)2083 3947 y Ff(X)2140 3939 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x;)f(t)p Fm(\)\))546 b(\(15\))1689 4056 y(=)83 b Fi(E)t Fm([)p Fi(exp)p Fm(\()p Fi(ik)r Fm(\()p Fi(X)6 b Fm(\()p Fi(t)16 b Fm(+)h Fi(\016)s(t)p Fm(\))g Fh(\000)f Fi(X)6 b Fm(\)\))p Fh(j)p Fi(X)g Fm(\()p Fi(t)p Fm(\))22 b(=)f Fi(X)6 b Fm(])-26 b(\(16\))731 4216 y(where)24 b Fi(k)j Fm(is)e(the)e(conjugate) j(v)l(ariable)f(of)g Fi(\016)s(x)p Fm(.)33 b(The)25 b(cum)n(ulan)n(ts)e (of)i(the)f(incremen)n(ts)731 4308 y Fi(C)786 4316 y Ff(n)853 4308 y Fm(are)j(the)e(co)r(e\016cien)n(ts)i(of)f(the)f(T)-6 b(a)n(ylor)27 b(expansion)e(of)i Fi(\016)s(K)2510 4316 y Ff(X)2568 4308 y Fm(:)1201 4512 y Fi(\016)s(K)1303 4520 y Ff(X)1361 4512 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x;)g(t)p Fm(\))21 b(=)g Fi(\016)s(t)1868 4436 y Fc(X)1869 4610 y Ff(n)p Fe(2)p Ff(J)2011 4463 y Fm(\()p Fi(ik)r Fm(\))2139 4432 y Ff(n)p 2011 4495 171 4 v 2062 4563 a Fi(n)p Fm(!)2191 4512 y Fi(C)2246 4520 y Ff(n)2289 4512 y Fm(\()p Fi(x;)12 b(t)p Fm(\))17 b(+)g Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))333 b(\(17\))846 4716 y(As)33 b(already)i(men)n (tioned,)g(the)e(classical)k(case)d(corresp)r(onds)h(to)f(an)g (analytic)731 4807 y(expansion)29 b(of)i Fi(\016)s(K)1285 4815 y Ff(X)1343 4807 y Fm(,)g(i.e.)47 b Fi(J)36 b Fh(\022)27 b Fn(N)p Fm(,)k(whereas)g(w)n(e)f(will)h(b)r(e)f(in)n(terested)g(b)n(y) e(a)i(non-)731 4898 y(analytic)c(expansion)g Fi(J)i Fm(=)21 b Fh(f)p Fm(1)p Fi(;)14 b(\013)p Fh(g)p Fm(.)1934 5147 y Fk(4)p eop %%Page: 5 5 5 4 bop 731 614 a Fj(3)134 b(Pro)t(cesses)31 b(with)f(stationary)i(and) d(indep)t(en-)731 764 y(den)l(t)45 b(incremen)l(ts)731 929 y Fm(Let)25 b(us)g(\014rst)g(consider)h(the)f(simple)g(sub-case)h (of)g(a)g(pro)r(cess)g(with)g(stationary)g(and)731 1020 y(indep)r(enden)n(t)32 b(incremen)n(ts.)57 b(It)33 b(corresp)r(onds)i (to)e Fi(C)2291 1028 y Ff(n)2334 1020 y Fm(\()p Fi(x;)12 b(t)p Fm(\))34 b Fh(\021)h Fi(C)2683 1028 y Ff(n)2759 1020 y Fm(=)g Fi(C)5 b(onst:)34 b Fm(in)731 1112 y(Eqs.7,)c(17)f(and)f (as)h(already)g(discussed)g(in)f(Sect.)43 b(1,)29 b(it)g(includes)f (the)g(linear)i(case)731 1203 y(\(Eq.2\))25 b(of)i(the)e(Langevin{lik)n (e)i(equation)e(\(Eq.1\).)846 1294 y(Ho)n(w)n(ev)n(er)c(w)n(e)h(b)r (eliev)n(e)g(that)f(the)g(follo)n(wing)j(deriv)l(ation)e(is)g(not)f (only)h(somewhat)731 1386 y(p)r(edagological)28 b(on)e(the)f(role)i(of) f(the)g(c)n(haracteristic)h(functions)f(for)h(the)e(nonlinear)731 1477 y(case,)i(but)d(terser)i(than)g(deriv)l(ations)g(previously)f 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Fg(0)2566 2311 y Fm(\))447 b(\(19\))1301 2428 y Fi(K)1366 2436 y Ff(X)1424 2428 y Fm(\()p Fi(k)r(;)14 b(t)i Fh(\000)h Fi(t)1680 2436 y Fg(0)1714 2428 y Fh(j)p Fi(x)1779 2436 y Fg(0)1814 2428 y Fi(;)c(t)1876 2436 y Fg(0)1910 2428 y Fm(\))83 b Fh(\021)g Fi(K)2231 2436 y Ff(X)2289 2428 y Fm(\()p Fi(k)r(;)13 b(t)k Fh(\000)g Fi(t)2545 2436 y Fg(0)2579 2428 y Fm(\))434 b(\(20\))846 2590 y(On)29 b(the)h(other)g(hand,)g(the)g(indep)r(endence)f(of)h(the)g(incremen)n (ts)f(implies)h(that)731 2681 y(the)25 b(transition)i(probabilities)g (satisfy)g(a)f(con)n(v)n(olution)g(\(o)n(v)n(er)f(an)n(y)g(p)r(ossible) i(in)n(ter-)731 2772 y(mediate)e(p)r(osition)h Fi(y)s Fm(\))f(for)i(an)n(y)e(giv)n(en)h(time)e(lag)j Fi(\016)s(t)p Fm(:)769 3053 y Fh(8)p Fi(\016)s(t)21 b(>)g Fm(0)g(:)h Fi(p)p Fm(\()p Fi(x)16 b Fh(\000)h Fi(x)1331 3061 y Fg(0)1365 3053 y Fi(;)c(t)k Fm(+)g Fi(\016)s(t)f Fh(\000)h Fi(t)1707 3061 y Fg(0)1742 3053 y Fm(\))k(=)1874 2942 y Fc(Z)1970 3053 y Fi(dy)53 b(p)p Fm(\()p Fi(x)16 b Fh(\000)h 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y Fg(0)2968 4464 y Fm(\)+)p Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))3043 4602 y(\(24\))846 4694 y(This)30 b(demonstrates)g(\(in)f(the)g(limit)h Fi(\016)s(t)e Fh(!)f Fm(0\))j(Prop.2)h(and)e(Eq.6,)j(as)e(w)n(ell)h(as) 731 4785 y(Prop.3,)c(since)f(Eq.24)g(corresp)r(onds,)h(with)f(the)f (help)h(of)g(Eq.17,)h(to:)1934 5147 y Fk(5)p eop %%Page: 6 6 6 5 bop 731 772 a Fi(p)p Fm(\()p Fi(x;)12 b(t)p Fm(+)p Fi(\016)s(t)p Fh(j)p Fi(x)1095 780 y Fg(0)1128 772 y Fi(;)h(t)1190 780 y Fg(0)1225 772 y Fm(\))p Fh(\000)p Fi(p)p Fm(\()p Fi(x;)e(t)p Fh(j)p Fi(x)1553 780 y Fg(0)1587 772 y Fi(;)j(t)1650 780 y Fg(0)1684 772 y Fm(\))21 b(=)g Fi(\016)s(t)1894 695 y Fc(X)1934 869 y Ff(n)2014 772 y Fm([)p Fi(C)2090 780 y Ff(n)2142 723 y Fm(\()p Fh(\000)p Fm(1\))2300 691 y Ff(n)p 2142 755 201 4 v 2208 822 a Fi(n)p Fm(!)2365 661 y Fc(Z)2461 772 y Fi(dy)s(\016)2579 734 y Fg(\()p Ff(n)p Fg(\))2576 791 y Ff(x)p Fe(\000)p Ff(y)2696 772 y Fi(p)p Fm(\()p Fi(y)s(;)12 b(t)p Fh(j)p Fi(x)2932 780 y Fg(0)2966 772 y Fi(;)h(t)3028 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Fm(\(in)f(Eq.)61 b(17\).)h(Ho)n(w)n(ev)n(er,)37 b(b)n(y)c(the)i(v)n (ery)e(de\014nition)h(of)i(fractional)731 1145 y(di\013eren)n(tiation)f (\(e.g.[14)r(]\),)j(this)d(corresp)r(ondence)g(holds)h(also)g(for)g (non)e(in)n(teger)731 1236 y(orders.)h(Ho)n(w)n(ev)n(er,)25 b(there)h(is)g(not)f(a)h(unique)e(de\014nition)h(of)h(fractional)i (di\013eren)n(tia-)731 1328 y(tion)f(and)f(therefore,)i(as)g(discussed) f(in)g(some)f(details)i(in)f([6]\),)h(w)n(e)f(cannot)g(exp)r(ect)731 1419 y(to)e(ha)n(v)n(e)h(a)g(unique)e(expression)i(of)h(the)e(F)-6 b(ractional)27 b(F)-6 b(okk)n(er-Planc)n(k)25 b(equation.)846 1510 y(Since)33 b(it)g(will)h(b)r(e)f(su\016cien)n(t)f(for)i(the)f (follo)n(wing)i(to)e(consider)h(an)f(expansion)731 1602 y(of)24 b(the)f(c)n(haracteristic)i(function)f(in)n(v)n(olving)g (fractional)h(p)r(o)n(w)n(ers)g(of)f(only)f(the)h(w)n(a)n(v)n(e)731 1693 y(n)n(um)n(b)r(er)j Fh(j)p Fi(k)r Fh(j)p Fm(,)33 b(it)d(is)g(in)n(teresting)h(to)f(consider)g(Riesz's)h(de\014nition)f 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y(\(one-dimensional\))d (Laplacians)i(along)g(a)f(giv)n(en)f(direction)i Fi(u)p 2525 2561 45 4 v 25 w Fm(\()p Fh(j)22 b Fi(u)p 2667 2561 V 22 w Fh(j)p Fm(=)f(1\))26 b(:)1335 2731 y Fh(\000)p Fm(\()p Fh(\000)p Fm(\001)1549 2739 y Ff(u)p 1549 2752 41 4 v 1589 2731 a Fm(\))1619 2695 y Ff(\013=)p Fg(2)1725 2731 y Fi(f)8 b Fm(\()p Fi(x)p Fm(\))21 b(=)g Fi(F)2036 2695 y Fe(\000)p Fg(1)2119 2731 y Fm([)p Fh(j)p Fm(\()p Fi(k)p 2191 2744 43 4 v 2 w(;)13 b(u)p 2267 2744 45 4 v 1 w Fm(\))p Fh(j)2363 2695 y Ff(\013)2423 2711 y Fm(^)2407 2731 y Fi(f)8 b Fm(\()p Fi(k)r Fm(\)])468 b(\(33\))731 2855 y(where\(.,.\))33 b(denotes)22 b(the)f(scalar)h(pro)r(duct.)33 b(On)20 b(the)h(other)g(hand,)h(it)f(will)i(b)r(e)e(useful)731 2947 y(to)h(consider)g(the)f(fractional)j(p)r(o)n(w)n(er)e(of)h(the)e (con)n(traction)h(of)h(the)e(Laplacian)i(tensor)731 3038 y(\001)p 731 3051 64 4 v 731 3068 V(:)1697 3238 y(\001)1761 3246 y Ff(i;j)1856 3238 y Fm(=)1982 3189 y Fi(@)p 1947 3221 115 4 v 1947 3288 a(@)t(x)2036 3296 y Ff(i)2120 3189 y Fi(@)p 2082 3221 121 4 v 2082 3288 a(@)t(x)2171 3296 y Ff(j)3043 3238 y Fm(\(34\))731 3400 y(b)n(y)h(a)i(tensor)g Fi(\033)p 1132 3413 47 4 v 1132 3430 V 3 w Fm(,)g(with)g(the)f(follo)n (wing)j(de\014nition:)1036 3582 y Fh(\000)p Fm(\()p Fh(\000)p Fm(\001)p 1186 3595 64 4 v 1186 3612 V 21 w(:)22 b Fi(\033)p 1314 3595 47 4 v 1314 3612 V 2 w(:\033)p 1381 3595 V 1381 3612 V 1428 3547 a Fe(\003)1464 3582 y Fm(\))1504 3527 y Fb(\013)p 1504 3536 38 4 v 1509 3568 a Fd(2)1577 3582 y Fh(\021)g Fi(F)1719 3547 y Fe(\000)p Fg(1)1801 3582 y Fm([)p Fh(j)g Fm(\()p Fi(k)p 1895 3595 43 4 v 2 w(;)13 b(\033)p 1971 3595 47 4 v 1971 3612 V 3 w(:\033)p 2039 3595 V 2039 3612 V 2085 3547 a Fe(\003)2121 3582 y Fi(:k)p 2142 3595 43 4 v 24 w Fh(j)2237 3527 y Fb(\013)p 2238 3536 38 4 v 2243 3568 a Fd(2)2290 3582 y Fm(])21 b(=)g Fi(F)2473 3547 y Fe(\000)p Fg(1)2556 3582 y Fm([)p Fh(j)h Fi(\033)p 2620 3595 47 4 v 2620 3612 V 2666 3547 a Fe(\003)2702 3582 y Fi(:k)p 2723 3595 43 4 v 24 w Fh(j)2808 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Fi(\013)22 b Fm(=)f(1)h(:)73 b Fi(!)s Fm(\()p Fi(k)r(;)12 b(\013)p Fm(\))22 b(=)2729 4673 y Fi(\031)p 2729 4704 47 4 v 2733 4772 a Fm(2)2785 4721 y Fi(l)q(og)s Fh(j)p Fi(k)r Fh(j)73 b Fm(\(37\))1934 5147 y Fk(7)p eop %%Page: 8 8 8 7 bop 846 614 a Fm(Considering)24 b(an)f(Ito-lik)n(e)f(forw)n(ard)i (in)n(tegration)g(of)f(Eq.1,)h(the)f(incremen)n(ts)f Fi(\016)s(L)731 706 y Fm(generates)33 b(the)f(follo)n(wing)j (\(\014rst\))c(c)n(haracteristic)j(function)f(for)g(the)f(incremen)n (ts)731 797 y Fi(\016)s(X)f Fm(of)c(the)e(motion)g Fi(X)6 b Fm(\()p Fi(t)p Fm(\):)1026 980 y Fi(\016)s(Z)1115 988 y Ff(X)1173 980 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x)k Fh(\000)g Fi(\016)s(x;)12 b(t)p Fm(\))21 b(=)g Fi(e)1813 944 y Ff(ik)q(m)p Fg(\()p Ff(X)q(;t)p Fg(\))2069 980 y Fi(\016)s(Z)2158 988 y Ff(\033)r(L)2242 980 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x;)f(t)p Fm(\))17 b(+)g(+)p Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))158 b(\(38\))731 1099 y(whic)n(h)22 b(yields)g(the)f(follo)n(wing)k(elemen)n(tary)c(cum) n(ulan)n(t)f(generating)j(function)f Fi(\016)s(K)3101 1107 y Ff(X)3158 1099 y Fm(:)1175 1343 y Fi(\016)s(K)1277 1351 y Ff(X)1335 1343 y Fm(\()p Fi(k)r(;)13 b(\016)s(t)p Fh(j)p Fi(x;)f(t)p Fm(\))21 b(=)g Fi(\016)s(t)p Fm([)p Fi(ik)r(m)p Fm(\()p Fi(x;)13 b(t)p Fm(\))k(+)f Fi(ik)r(\015)t(\033)s Fm(\()p Fi(x;)d(t)p Fm(\))473 b(\(39\))1177 1495 y Fh(\000)p Fi(D)r Fh(j)p Fi(k)r Fh(j)1386 1459 y Ff(\013)1431 1495 y Fm(\(1)17 b Fh(\000)g Fi(i\014)1698 1446 y(k)p 1676 1478 85 4 v 1676 1545 a Fh(j)p Fi(k)r Fh(j)1771 1495 y Fi(!)s Fm(\()p Fi(k)r(;)c(\013)p Fm(\)\))p Fi(\033)s Fm(\()p Fi(x;)f(t)p Fm(\))2249 1459 y Ff(\013)2293 1495 y Fm(])17 b(+)g Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))473 b(\(40\))731 1689 y(and)29 b(whic)n(h)g(is)h(of)g(the)f(same)h(t)n(yp)r (e)e(as)i(Eq.17,)i(with)d Fi(J)35 b Fm(=)28 b Fh(f)p Fm(1)p Fi(;)14 b(\013)p Fh(g)p Fm(.)46 b(Therefore,)32 b(as)731 1781 y(discussed)18 b(in)f(Sect.5,)j(w)n(e)e(ha)n(v)n(e)f 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Fm(\()p Fi(t)p Fm(\))p Fi(;)13 b(t)p Fm(\).)731 2680 y Fj(7)134 b(Extension)46 b(to)g(v)l(ector)f(pro)t(cesses)731 2845 y Fm(With)27 b(but)h(one)g(imp)r(ortan)n(t)f(exception,)i(the)f (extension)g(of)h(the)f(previous)g(results)731 2937 y(to)d(higher)g (dimensions)f(is)i(rather)e(straigh)n(tforw)n(ard.)37 b(The)25 b(starting)g(p)r(oin)n(t)g(of)h(this)731 3028 y(extension)f(is)h(the)g(follo)n(wing)i(nonlinear)e(sto)r(c)n(hastic)h (equation)f(\()p Fi(X)p 2640 3041 70 4 v 6 w Fm(\()p Fi(t)p Fm(\))21 b Fh(2)g Fi(R)2949 2996 y Ff(d)2985 3028 y Fm(\):)1333 3211 y Fi(dX)p 1373 3224 V 6 w Fm(\()p Fi(t)p Fm(\))g(=)g Fi(m)p 1632 3224 68 4 v -1 w Fm(\()p Fi(X)p 1729 3224 70 4 v 7 w Fm(\()p Fi(t)p Fm(\))p Fi(;)12 b(t)p Fm(\))p Fi(dt)k Fm(+)h Fi(\033)p 2139 3224 47 4 v 2139 3240 V 3 w Fm(\()p Fi(X)p 2216 3224 70 4 v 6 w Fm(\()p Fi(t)p Fm(\))p Fi(;)12 b(t)p Fm(\))p Fi(:dL)p 2525 3224 53 4 v 466 w Fm(\(41\))731 3330 y(where)37 b Fi(m)p 964 3343 68 4 v 36 w Fm(and)f Fi(\033)p 1228 3343 47 4 v 1228 3360 V 39 w Fm(are)i(the)e(natural)h(v)n(ector,)j (resp)r(ectiv)n(ely)c(tensor,)k(extensions)731 3422 y(of)e(the)f (deterministic-lik)n(e)g(trend,)k(resp)r(ectiv)n(ely)d(mo)r(dulation)f (of)h(the)g(random)731 3513 y(driving)26 b(force.)40 b Fi(L)p 1214 3526 53 4 v 27 w Fm(is)27 b(a)g(d-dimensional)g(Levy)f (stable)h(motion)g(and,)g(as)g(discussed)731 3604 y(b)r(elo)n(w,)43 b(the)38 b(expression)g(of)i(its)f(c)n(haracteristic)h(function)e (corresp)r(onds)h(to)g(the)731 3695 y(source)24 b(of)h(the)f (di\016cult)n(y)g(in)g(extending)g(the)f(scalar)j(results)f(to)f(high)h (dimensions.)731 3787 y(On)h(the)g(con)n(trary)-6 b(,)27 b(it)g(is)g(straigh)n(tforw)n(ard)i(to)e(c)n(hec)n(k)f(that)h(Props.)38 b(2,)28 b(3)f(are)h(v)l(alid)731 3878 y(in)d(the)h(d-dimensional)f (case,)i(with)f(the)f(follo)n(wing)k(extensions)d(\()p Fi(x)p 2660 3891 44 4 v 21 w Fh(2)c Fi(R)2857 3846 y Ff(d)2892 3878 y Fm(\))k(for)h(Eq.)731 3969 y(6:)1147 4144 y Fi(@)t(p)p 1147 4176 84 4 v 1152 4243 a(@)t(t)1241 4193 y Fm(\()p Fi(x)p 1271 4206 44 4 v -1 w(;)13 b(t)p Fh(j)p Fi(x)1441 4201 y Fg(0)1475 4193 y Fi(;)g(t)1537 4201 y Fg(0)1572 4193 y Fm(\))21 b(=)1704 4082 y Fc(Z)1800 4193 y Fi(dy)1890 4144 y(@)1951 4129 y Fc(e)1935 4144 y Fi(K)p 1890 4176 116 4 v 1911 4243 a(@)t(t)2015 4193 y Fm(\()p Fi(x)p 2045 4206 44 4 v 17 w Fh(\000)c Fi(y)p 2183 4220 41 4 v 2 w Fh(j)p Fi(y)p 2244 4220 V 3 w(;)c(t)p Fm(\))p Fi(p)p Fm(\()p Fi(y)p 2446 4220 V 2 w(;)g(t)p Fh(j)p Fi(x)2613 4201 y Fg(0)2647 4193 y Fi(;)g(t)2709 4201 y Fg(0)2743 4193 y Fm(\))270 b(\(42\))731 4377 y(and)25 b(for)h(Eq.)35 b(7)26 b(\()p Fi(n)p 1240 4390 47 4 v 21 w Fh(2)c Fi(J)28 b Fh(\022)22 b Fn(N)1601 4345 y Ff(d)1637 4377 y Fi(;)13 b Fh(j)22 b Fi(n)p 1714 4390 V 21 w Fh(j)p Fm(=)1884 4317 y Fc(P)1971 4337 y Ff(d)1971 4404 y(i)p Fg(=1)2087 4377 y Fi(n)2133 4385 y Ff(i)2160 4377 y Fm(\):)916 4607 y Fi(@)t(p)p 916 4638 84 4 v 921 4706 a(@)t(t)1009 4655 y Fm(\()p Fi(x)p 1039 4668 44 4 v(;)13 b(t)p Fh(j)p Fi(x)p 1166 4668 V 1209 4676 a Fg(0)1244 4655 y Fi(;)g(t)1306 4663 y Fg(0)1340 4655 y Fm(\))22 b(=)1473 4578 y Fc(X)1473 4752 y Ff(n)p 1473 4765 39 4 v 1 w Fe(2)p Ff(J)1820 4607 y Fi(@)1865 4575 y Fe(j)p Ff(n)p 1884 4588 V Fe(j)p 1615 4638 536 4 v 1615 4707 a Fi(@)t(x)1704 4671 y Ff(n)1742 4681 y Fd(1)1704 4726 y Fg(1)1779 4707 y Fi(@)t(x)1868 4671 y Ff(n)1906 4681 y Fd(2)1868 4726 y Fg(2)1942 4707 y Fi(::@)t(x)2073 4669 y Ff(n)2111 4681 y Fb(d)2073 4729 y Ff(d)2160 4655 y Fm([)p Fi(A)2239 4663 y Ff(n)p 2239 4676 43 4 v 2282 4655 a Fm(\()p Fi(x)p 2312 4668 44 4 v -1 w(;)13 b(t)p Fm(\))p Fi(p)p Fm(\()p Fi(x)p 2516 4668 V -1 w(;)g(t)p Fh(j)p Fi(x)p 2642 4668 V 21 x Fg(0)2720 4655 y Fi(;)g(t)2782 4663 y Fg(0)2817 4655 y Fm(\)])175 b(\(43\))731 4898 y(the)25 b(relation)i(to)f(the)f(cum)n(ulan)n(ts)f Fi(C)1793 4906 y Ff(n)p 1793 4919 43 4 v 1861 4898 a Fm(of)j(the)e(incremen)n(ts)g(is)h(no)n(w:)1934 5147 y Fk(8)p eop %%Page: 9 9 9 8 bop 1365 741 a Fi(A)1423 749 y Ff(n)p 1423 762 43 4 v 1465 741 a Fm(\()p Fi(x)p 1495 754 44 4 v(;)13 b(t)p Fm(\))21 b(=)1889 692 y(\()p Fh(\000)p Fm(1\))2047 660 y Fe(j)p Ff(n)p 2066 673 39 4 v Fe(j)p 1743 724 531 4 v 1743 791 a Fm(\()p Fi(n)1819 799 y Fg(1)1853 791 y Fm(\)!\()p Fi(n)1980 799 y Fg(2)2015 791 y Fm(\)!)p Fi(::)p Fm(\()p Fi(n)2184 800 y Ff(d)2222 791 y Fm(\)!)2283 741 y Fi(C)2338 749 y Ff(n)p 2338 762 43 4 v 2380 741 a Fm(\()p Fi(x)p 2410 754 44 4 v -1 w(;)14 b(t)p Fm(\))497 b(\(44\))846 904 y(On)25 b(the)g(other)h(hand,)f(Eq.)34 b(41)27 b(yields)f(the)f (follo)n(wing)j(extension)e(to)g(Eq.38:)1247 1086 y Fi(\016)s(Z)1336 1094 y Ff(X)p 1336 1107 58 4 v 1394 1086 a Fm(\()p Fi(k)p 1424 1099 43 4 v 2 w(;)13 b(\016)s(t)p Fh(j)p Fi(x)p 1586 1099 44 4 v(;)g(t)p Fm(\))21 b(=)g Fi(e)1860 1050 y Ff(i)k(k)p 1907 1063 34 4 v 1 w(:m)p 1960 1063 55 4 v 1 w Fg(\()p Ff(x)p 2039 1063 35 4 v -1 w(;t)p Fg(\))2144 1086 y Fi(\016)s(Z)2233 1094 y Ff(\033)p 2233 1107 38 4 v 2233 1124 V 2 w(:L)p 2290 1107 43 4 v 2336 1086 a Fm(\()p Fi(k)p 2366 1099 V 2 w(;)13 b(\016)s(t)p Fh(j)p Fi(x)p 2528 1099 44 4 v(;)g(t)p Fm(\))379 b(\(45\))731 1221 y(and)25 b(therefore)h(w)n(e)h(ha)n(v)n(e:)1022 1404 y Fi(\016)s(K)1124 1412 y Ff(X)p 1124 1425 58 4 v 1182 1404 a Fm(\()p Fi(k)p 1212 1417 43 4 v 2 w(;)13 b(\016)s(t)p Fh(j)p Fi(x)p 1374 1417 44 4 v -1 w(;)h(t)p Fm(\))20 b(=)i Fi(i)j(k)p 1663 1417 43 4 v 3 w(:m)p 1727 1417 68 4 v Fm(\()p Fi(x)p 1825 1417 44 4 v -1 w(;)13 b(t)p Fm(\))k(+)g Fi(\016)s(K)2156 1412 y Ff(L)p 2156 1425 47 4 v 2202 1404 a Fm(\()p Fi(\033)p 2232 1417 V 2232 1434 V 2278 1368 a Fe(\003)2314 1404 y Fi(:k)p 2335 1417 43 4 v 3 w(;)c(\016)s(t)p Fh(j)p Fi(x)p 2498 1417 44 4 v -1 w(;)g(t)p Fm(\))k(+)g Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))154 b(\(46\))846 1528 y(Let)24 b(us)g(recall)h(that)f(a)g(stable) h(L)n(\023)-36 b(evy)23 b(v)n(ector)h(in)g(the)g(classical)j(sense)d ([9)q(,)g(15)q(,)h(16])731 1620 y(\(see)f([17)q(])h(for)g(a)f (discussion)h(and)f(a)h(generalization\))h(corresp)r(onds)f(to)f(the)g (limit)g(of)731 1711 y(a)29 b(sum)e(of)j(jumps,)f(with)f(a)i(p)r(o)n(w) n(er-la)n(w)f(distribution,)h(along)g(random)e(directions)731 1802 y Fi(u)p 731 1815 45 4 v 36 w Fh(2)36 b Fi(@)t(B)1001 1810 y Fg(1)1035 1802 y Fm(,)h Fi(B)1151 1810 y Fg(1)1220 1802 y Fm(b)r(eing)e(the)f(unit)g(ball,)k(distributed)c(according)h(to) g(a)f(\(p)r(ositiv)n(e\))731 1894 y(measure)18 b Fi(d)p Fm(\006\()p Fi(u)p 1148 1907 V 1 w Fm(\).)32 b(The)19 b(latter,)i(whic)n(h)f(generalizes)h(the)e(scale)h(parameter)f Fi(D)i Fm(of)f(the)731 1985 y(scalar)28 b(case,)h(is)f(the)f(source)h (of)g(the)f(di\016cult)n(y)g(since)h(in)f(general)i(the)e(probabilit)n (y)731 2076 y(distribution)18 b(of)g(a)h(stable)f(L)n(\023)-36 b(evy)17 b(v)n(ector)h(dep)r(ends)g(on)g(this)g(measure,)h(and)f (therefore)731 2168 y(is)24 b(a)g(non)f(parametric)h(distribution.)34 b(Ho)n(w)n(ev)n(er,)25 b(as)f(discussed)g(b)r(elo)n(w,)i(there)d(is)h (at)731 2259 y(least)i(a)g(trivial)h(exception:)34 b(the)25 b(case)i(of)f(isotropic)h(stable)g(L)n(\023)-36 b(evy)25 b(v)n(ectors.)846 2350 y(Corresp)r(onding)19 b(to)f(our)h(previous)f (remarks,)h(a)f(\(classical\))j(stable)e(L)n(\023)-36 b(evy)17 b(v)n(ector)731 2441 y(has)26 b(the)f(follo)n(wing)j(\(F)-6 b(ourier\))26 b(cum)n(ulan)n(t)e(generating)i(function:)1119 2652 y Fi(K)1184 2660 y Ff(L)p 1184 2673 47 4 v 1231 2652 a Fm(\()p Fi(k)p 1261 2665 43 4 v 2 w Fm(\))21 b(=)g Fi(\016)s(t)p Fm([)p Fi(i)p Fm(\()p Fi(k)p 1577 2665 V 2 w(;)13 b(\015)p 1653 2680 44 4 v 4 w Fm(\))k Fh(\000)1821 2541 y Fc(Z)1867 2730 y Ff(u)p 1867 2743 37 4 v Fe(2)p Ff(@)s(B)2028 2740 y Fd(1)2064 2652 y Fm(\()p Fi(ik)p 2120 2665 43 4 v 3 w(;)c(u)p 2197 2665 45 4 v Fm(\))2271 2616 y Ff(\013)2315 2652 y Fi(d)p Fm(\006\()p Fi(u)p 2440 2665 V Fm(\)])18 b(+)e Fi(o)p Fm(\()p Fi(\016)s(t)p Fm(\))252 b(\(47\))731 2840 y(whic)n(h)25 b(yields)h(with)g(the)f(help) h(of)g(the)g(Eq.46:)831 3073 y Fi(@)p 817 3104 73 4 v 817 3172 a(@)t(t)916 3106 y Fc(e)900 3121 y Fi(K)965 3129 y Ff(X)p 965 3142 58 4 v 1023 3121 a Fm(\()p Fi(k)p 1053 3134 43 4 v 2 w Fm(\))21 b(=)g Fh(\000)p Fi(div)s Fm(\()p Fi(m)p 1423 3134 68 4 v 16 w Fm(+)c Fi(\033)p 1584 3134 47 4 v 1584 3151 V 3 w(:\015)p 1652 3149 44 4 v 4 w Fm(\))g Fh(\000)g Fi(F)1880 3085 y Fe(\000)p Fg(1)1962 3121 y Fm([)1983 3010 y Fc(Z)2029 3199 y Ff(u)p 2029 3212 37 4 v Fe(2)p Ff(@)s(B)2190 3209 y Fd(1)2226 3121 y Fm(\()p Fi(i\033)p 2282 3134 47 4 v 2282 3151 V 2329 3085 a Fe(\003)2365 3121 y Fm(\()p Fi(x)p 2395 3134 44 4 v -1 w(;)d(t)p Fm(\))p Fi(:k)p 2552 3134 43 4 v 2 w(;)f(u)p 2628 3134 45 4 v Fm(\))2702 3085 y Ff(\013)2746 3121 y Fi(d)p Fm(\006\()p Fi(u)p 2871 3134 V Fm(\)])77 b(\(48\))846 3350 y(The)26 b(scalar)h(case)f(\(Eq.36\))h(corresp)r (onds)f(to:)769 3603 y(0)c Fh(\024)f Fi(p)g Fh(\024)g Fm(1)g(:)h Fi(\014)k Fm(=)21 b(2)p Fi(p)13 b Fh(\000)h Fm(1)p Fi(;)37 b(d)p Fm(\006\()p Fi(u)p Fm(\))21 b(=)h Fi(D)r(cos)p Fm(\()2076 3555 y Fi(\031)s(\013)p 2076 3586 96 4 v 2105 3654 a Fm(2)2182 3603 y(\)[)p Fi(p\016)2306 3615 y Fg(\()p Ff(u)p Fe(\000)p Fg(1\))2487 3603 y Fm(+)13 b(\(1)h Fh(\000)f Fi(p)p Fm(\))p Fi(\016)2818 3615 y Fg(\()p Ff(u)p Fg(+1\))2983 3603 y Fm(])39 b(\(49\))846 3785 y(F)-6 b(or)24 b(an)n(y)g(dimension)g(d,)h(the)f(second)h(term)e (on)i(the)f(righ)n(t)h(hand)f(side)h(of)g(Eq.48)731 3876 y(corresp)r(onds)30 b(to)g(a)f(fractional)j(di\013eren)n(tiation)e(op)r (erator)g(of)h(order)f Fi(\013)p Fm(.)46 b(This)30 b(op-)731 3967 y(erator)c(can)g(b)r(e)f(sligh)n(tly)h(re-arranged.)35 b(With)26 b(the)f(help)g(of)i(the)e(o)r(dd)g Fi(d)p Fm(\006)2874 3936 y Fe(\000)2927 3967 y Fm(\()p Fi(u)p 2957 3980 45 4 v Fm(\))g(and)731 4059 y(ev)n(en)g Fi(d)p Fm(\006)1001 4027 y Fg(+)1052 4059 y Fm(\()p Fi(u)p 1082 4072 V Fm(\))g(parts)h(of)g (the)g(measure)f Fi(d)p Fm(\006\()p Fi(u)p 2022 4072 V Fm(\),)863 4312 y(2)h Fi(d)p Fm(\006)1022 4276 y Fg(+)1072 4312 y Fm(\()p Fi(u)p 1102 4325 V 1 w Fm(\))21 b(=)g Fi(d)p Fm(\006\()p Fi(u)p 1404 4325 V Fm(\))c(+)g Fi(d)p Fm(\006\()p Fh(\000)p Fi(u)p 1757 4325 V Fm(\);)90 b(2)26 b Fi(d)p Fm(\006)2101 4276 y Fe(\000)2153 4312 y Fm(\()p Fi(u)p 2183 4325 V Fm(\))21 b(=)g Fi(d)p Fm(\006\()p Fi(u)p 2484 4325 V 1 w Fm(\))c Fh(\000)f Fi(d)p Fm(\006\()p Fh(\000)p Fi(u)p 2837 4325 V Fm(\))132 b(\(50\))731 4474 y(and)25 b(the)g(iden)n(tit)n(y)g(\()p Fi(\022)i Fm(b)r(eing)f(the)g (Hea)n(viside)g(function\):)1335 4657 y(\()p Fi(ik)r Fm(\))1463 4621 y Ff(\013)1529 4657 y Fm(=)21 b Fh(j)p Fi(k)r Fh(j)1694 4621 y Ff(\013)1739 4657 y Fm([)p Fi(\022)r Fm(\()p Fi(k)r Fm(\))p Fi(e)1936 4621 y Ff(i)1968 4601 y Fb(\013\031)p 1968 4610 74 4 v 1991 4642 a Fd(2)2073 4657 y Fm(+)c Fi(\022)r Fm(\()p Fh(\000)p Fi(k)r Fm(\))p Fi(e)2386 4621 y Fe(\000)p Ff(i)2466 4601 y Fb(\013\031)p 2466 4610 V 2489 4642 a Fd(2)2554 4657 y Fm(])468 b(\(51\))731 4781 y(one)25 b(can)h(write)h(the)e(extension)h(of)g(Eq.4)g(under)f (the)g(follo)n(wing)k(form:)1934 5147 y Fk(9)p eop %%Page: 10 10 10 9 bop 1259 739 a Fi(@)p 1245 770 73 4 v 1245 838 a(@)t(t)1327 787 y(p)p Fm(\()p Fi(x)p 1396 800 44 4 v -1 w(;)13 b(t)p Fh(j)p Fi(x)p 1522 800 V 21 x Fg(0)1601 787 y Fi(;)g(t)1663 795 y Fg(0)1697 787 y Fm(\))21 b(=)g Fh(\000)p Fi(div)s Fm([)p Fi(m)p 2016 800 68 4 v Fm(\()p Fi(x)p 2114 800 44 4 v(;)13 b(t)p Fm(\))j(+)h Fi(\033)p 2343 800 47 4 v 2343 817 V 3 w Fm(\()p Fi(x)p 2420 800 44 4 v -1 w(;)c(t)p Fm(\))p Fi(:\015)p 2576 815 V 4 w Fm(\)])p Fi(p)p Fm(\()p Fi(x)p 2740 800 V -1 w(;)g(t)p Fh(j)p Fi(x)p 2866 800 V 21 x Fg(0)2945 787 y Fi(;)g(t)3007 795 y Fg(0)3041 787 y Fm(\)])731 959 y Fh(\000)p Fm([)p Fi(<)21 b Fm(\()p Fh(\000)p Fm(\001)p 983 972 64 4 v 983 989 V 21 w(:)g Fi(\033)p 1110 972 47 4 v 1110 989 V 3 w(:\033)p 1178 972 V 1178 989 V 1225 923 a Fe(\003)1261 959 y Fm(\))1301 904 y Fb(\013)p 1301 913 38 4 v 1306 945 a Fd(2)1374 959 y Fi(>)1434 975 y Fg(\006)1477 961 y Fd(+)1549 959 y Fh(\000)g Fi(<)g Fm(\()p Fh(r)p 1741 972 64 4 v Fi(:\033)p 1826 972 47 4 v 1826 989 V 1873 923 a Fe(\003)1909 959 y Fm(\))p Fi(:)p Fm(\()p Fh(\000)p Fm(\001)p 2050 972 64 4 v 2050 989 V 21 w(:)h Fi(\033)p 2178 972 47 4 v 2178 989 V 3 w(:\033)p 2246 972 V 2246 989 V 2293 923 a Fe(\003)2329 959 y Fm(\))2369 899 y Fb(\013)p Fa(\000)p Fd(1)p 2368 913 112 4 v 2410 945 a(2)2515 959 y Fi(>)2575 975 y Fg(\006)2618 961 y Fa(\000)2671 959 y Fm(])p Fi(p)p Fm(\()p Fi(x)p 2761 972 44 4 v(;)13 b(t)p Fh(j)p Fi(x)p 2888 972 V 2931 980 a Fg(0)2966 959 y Fi(;)g(t)3028 967 y Fg(0)3062 959 y Fm(\))30 b(\(52\))731 1115 y(where)c(the)f(symmetric) f(and)h(an)n(tisymmetric)f(op)r(erators)j(are)f(de\014ned)e(,)i (similarly)731 1206 y(to)f(Eq.35,)i(in)f(the)f(follo)n(wing)j(manner:) 1055 1481 y Fh(\000)21 b Fi(<)g Fm(\()p Fh(\000)p Fm(\001)p 1307 1494 64 4 v 1307 1511 V 21 w(:)h Fi(\033)p 1435 1494 47 4 v 1435 1511 V 2 w(:\033)p 1502 1494 V 1502 1511 V 1549 1445 a Fe(\003)1585 1481 y Fm(\))1625 1426 y Fb(\013)p 1625 1435 38 4 v 1630 1467 a Fd(2)1698 1481 y Fi(>)1758 1497 y Fg(\006)1801 1483 y Fd(+)1936 1481 y Fm(=)2078 1370 y Fc(Z)2124 1559 y Ff(u)p 2124 1572 37 4 v Fe(2)p Ff(@)s(B)2285 1569 y Fd(1)2334 1481 y Fi(d)p Fm(\006)2429 1445 y Fg(+)2480 1481 y Fm(\()p Fi(u)p 2510 1494 45 4 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Fm(of)h(the)e(measure)h Fi(d)p Fm(\006)g(is)g(isotropic.)36 b(Indeed,)25 b(the)731 2304 y(in)n(tegration)h(o)n(v)n(er)g(directions)g(yields)g(only)g(a)g (prefactor)g Fi(D)r Fm(:)1198 2551 y Fi(<)21 b Fh(\000)p Fm(\(\001)p 1369 2564 64 4 v 1369 2581 V 21 w(:)h Fi(\033)p 1497 2564 47 4 v 1497 2581 V 2 w(:\033)p 1564 2564 V 1564 2581 V 1611 2515 a Fe(\003)1647 2551 y Fm(\))1687 2495 y Fb(\013)p 1687 2504 38 4 v 1692 2536 a Fd(2)1760 2551 y Fi(>)1820 2567 y Fg(\006)1863 2553 y Fd(+)1914 2551 y Fm(=)f Fi(D)28 b Fm(\()p Fh(\000)p Fm(\001)p 2176 2564 64 4 v 2176 2581 V 21 w(:)22 b Fi(\033)p 2304 2564 47 4 v 2304 2581 V 3 w(:\033)p 2372 2564 V 2372 2581 V 2419 2515 a Fe(\003)2455 2551 y Fm(\))2495 2495 y Fb(\013)p 2494 2504 38 4 v 2499 2536 a Fd(2)1505 2728 y Fi(D)i Fm(=)1673 2617 y Fc(Z)1719 2806 y Ff(u)p 1719 2819 37 4 v Fe(2)p Ff(@)s(B)1880 2816 y Fd(1)1929 2728 y Fi(d)p Fm(\006)2024 2692 y Fg(+)2075 2728 y Fm(\()p Fi(u)p 2105 2741 45 4 v Fm(\))d Fh(j)h Fm(\()p Fi(u)p 2273 2741 V 21 x Fg(1)2351 2728 y Fi(;)13 b(u)p 2385 2741 V 1 w Fm(\))21 b Fh(j)2502 2692 y Ff(\013)3043 2728 y Fm(\(55\))731 2951 y(and)32 b(for)i Fi(\013)h Fm(=)e(2)h(this)f(corresp)r(onds)h(to)f (the)g(classical)j(term)c(\(\001)p 2612 2964 64 4 v 2612 2981 V 33 w(:)j Fi(\033)p 2765 2964 47 4 v 2765 2981 V 2 w(:\033)p 2832 2964 V 2832 2981 V 2879 2920 a Fe(\003)2915 2951 y Fm(\))e(of)h(the)731 3043 y(standard)d(d-dimensional)g(F)-6 b(okk)n(er-Planc)n(k)31 b(equation.)53 b(If)32 b Fi(d)p Fm(\006)f(itself)j(is)e(rotation)731 3134 y(in)n(v)l(arian)n(t,)41 b(then)c(the)g(asymmetric)g(op)r(erator)i(v)l(anishes,)i(since)e Fi(d)p Fm(\006)2802 3102 y Fe(\000)2896 3134 y Fm(=)j(0.)72 b(If)731 3225 y(furthermore,)34 b Fi(\033)p 1186 3238 V 1186 3255 V 35 w Fm(is)f(rotation)g(in)n(v)l(arian)n(t,)h(i.e.)56 b Fi(\033)p 2155 3238 V 2155 3255 V 36 w Fm(=)32 b Fi(\033)s Fm(1)p 2374 3238 39 4 v 2374 3255 V(,)j(then)c(one)i(obtains)g(the)731 3317 y(follo)n(wing)28 b(F)-6 b(ractional)27 b(F)-6 b(okk)n(er-Planc)n (k)24 b(equation:)1014 3533 y Fi(@)p 1000 3564 73 4 v 1000 3632 a(@)t(t)1083 3581 y(p)p Fm(\()p Fi(x)p 1152 3594 44 4 v -1 w(;)13 b(t)p Fh(j)p Fi(x)p 1278 3594 V 21 x Fg(0)1356 3581 y Fi(;)g(t)1418 3589 y Fg(0)1453 3581 y Fm(\))83 b(=)f Fh(\000)p Fi(div)s Fm([)p Fi(\033)p 1895 3594 47 4 v 1895 3611 V 3 w(:\015)p 1963 3609 44 4 v 4 w Fm(\()p Fi(x)p 2037 3594 V(;)13 b(t)p Fm(\))j(+)h Fi(m)p 2266 3594 68 4 v Fm(\()p Fi(x)p 2364 3594 44 4 v -1 w(;)c(t)p Fm(\)])p Fi(p)p Fm(\()p Fi(x)p 2589 3594 V -1 w(;)g(t)p Fh(j)p Fi(x)p 2715 3594 V 21 x Fg(0)2794 3581 y Fi(;)g(t)2856 3589 y Fg(0)2890 3581 y Fm(\))123 b(\(56\))1566 3738 y Fh(\000)82 b Fi(D)28 b Fm([\()p Fh(\000)p Fm(\001\))2004 3702 y Ff(\013=)p Fg(2)2111 3738 y Fm(])p Fi(\033)s Fm(\()p Fi(x;)12 b(t)p Fm(\))2344 3702 y Ff(\013)2388 3738 y Fi(p)p Fm(\()p Fi(x)p 2457 3751 V -1 w(;)h(t)p Fh(j)p Fi(x)p 2583 3751 V 21 x Fg(0)2661 3738 y Fi(;)g(t)2723 3746 y Fg(0)2758 3738 y Fm(\))255 b(\(57\))846 3894 y(Therefore,)24 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y(the)27 b(the)h(measure)g Fi(d)p Fm(\006)g(do)r(es)h(if)g(for)g (classical)h(stable)f(L)n(\023)-36 b(evy)27 b(v)n(ectors.)42 b(This)29 b(there-)731 4716 y(fore)j(diminishes)e(the)h(imp)r(ortance)g (of)g(the)g(asymmetry)e(of)j(the)e(latter.)51 b(Indeed,)731 4807 y(the)32 b(comp)r(onen)n(ts)h(of)h(a)f(generalized)i(stable)f(L)n (\023)-36 b(evy)32 b(v)n(ector)h(do)g(not)g(ha)n(v)n(e)g(neces-)731 4898 y(sarily)f(the)e(same)h(L)n(\023)-36 b(evy)30 b(stabilit)n(y)i (index,)g(the)f(latter)g(b)r(eing)h(generalized)g(in)n(to)f(a)1914 5147 y Fk(10)p eop %%Page: 11 11 11 10 bop 731 614 a Fm(second)32 b(rank)f(tensor.)53 b(Similarly)-6 b(,)33 b(the)e(di\013eren)n(tial)i(op)r(erators)g(in)n (v)n(olv)n(ed)d(in)i(the)731 706 y(corresp)r(onding)23 b(F)-6 b(ractional)23 b(F)-6 b(okk)n(er-Planc)n(k)21 b(equation)h(ha)n(v)n(e)g(no)g(longer)h(a)g(unique)731 797 y(order)k(of)h(di\013eren)n(tiation.)39 b(This)28 b(is)g(rather)f(easy)h(to)f(c)n(hec)n(k)f(in)i(case)g(of)g(a)f (discrete)731 888 y(measure)e Fi(d)p Fm(\006\()p Fi(u)p 1155 901 45 4 v Fm(\))h(and)f(w)n(e)h(will)h(explore)f(elsewhere)h(the) e(general)i(case.)731 1149 y Fj(8)134 b(Conclusion)731 1314 y Fm(W)-6 b(e)22 b(ha)n(v)n(e)h(deriv)n(ed)f(a)i(F)-6 b(ractional)24 b(F)-6 b(okk)n(er-Planc)n(k)22 b(equation,)i(i.e.)35 b(a)23 b(kinetic)g(equa-)731 1406 y(tion)k(whic)n(h)g(in)n(v)n(olv)n (es)g(fractional)i(deriv)l(ativ)n(es,)f(for)g(the)f(ev)n(olution)g(of)h (the)f(proba-)731 1497 y(bilit)n(y)f(distribution)g(of)i(nonlinear)f (sto)r(c)n(hastic)g(di\013eren)n(tial)g(equations)g(driv)n(en)f(b)n(y) 731 1588 y(non-Gaussian)34 b(Levy)g(stable)h(noises.)61 b(W)-6 b(e)34 b(\014rst)g(established)h(this)g(equation)f(in)731 1680 y(the)23 b(scalar)i(case,)g(where)f(it)f(has)h(a)g(rather)f (compact)g(expression)h(with)g(the)f(help)g(of)731 1771 y(fractional)g(p)r(o)n(w)n(ers)f(of)g(the)f(Laplacian,)j(and)d(then)g (discussed)g(its)h(extension)g(to)f(the)731 1862 y(v)n(ector)29 b(case.)45 b(This)30 b(F)-6 b(ractional)31 b(F)-6 b(okk)n(er-Planc)n(k) 28 b(equation)h(generalizes)i(broadly)731 1954 y(previous)25 b(results)i(obtained)e(for)i(a)f(linear)g(Langevin-lik)n(e)g(equation)g (with)g(a)g(L)n(\023)-36 b(evy)731 2045 y(forcing,)35 b(as)e(w)n(ell)g(as)g(the)e(standard)i(F)-6 b(okk)n(er-Planc)n(k)31 b(equation)h(for)h(a)f(nonlinear)731 2136 y(Langevin)25 b(equation)h(with)g(a)g(Gaussian)h(forcing.)731 2397 y Fj(9)134 b(Ac)l(kno)l(wledgmen)l(ts)731 2562 y Fm(W)-6 b(e)25 b(w)n(ould)i(lik)n(e)f(to)h(thank)e(Dr.)36 b(James)26 b(Brannan)g(for)h(helpful)g(discussions.)37 b(P)n(art)731 2653 y(of)29 b(this)f(w)n(ork)g(w)n(as)i(p)r(erformed)e(while)h(Daniel) f(Sc)n(hertzer)g(w)n(as)i(visiting)e(Clemson)731 2745 y(Univ)n(ersit)n(y)-6 b(.)731 3005 y Fj(References)769 3171 y Fm([1])43 b(F)-6 b(ogedb)n(y)g(,)24 b(H.C.,)j Fl(Phys.R)l(ev.E,)h(50)p Fm(,)e(2,)g(1657-1660,)j(\(1994\))769 3292 y([2])43 b(Chec)n(hkin,)g(A.V.,)i(D.)40 b(Sc)n(hertzer,)k(A.V.)c (T)-6 b(ur,)45 b(V.V.)40 b(Y)-6 b(ano)n(vsky)g(,)42 b Fl(Ukr.)f(J)892 3383 y(Phys.,)27 b(40)p Fm(,)f(5,)h(434-439)g (\(1995\).)769 3504 y([3])43 b(Compte,)25 b(A)g Fl(Phys.R)l(ev.E,)j(53) p Fm(,)e(4,)g(4191-4193,)j(\(1996\))769 3625 y([4])43 b(Marsan,)25 b(D.,)g(D.)f(Sc)n(hertzer,)g(S.)g(Lo)n(v)n(ejo)n(y)-6 b(,)24 b(J.)h(Geoph)n(ys.)f(Res.,)h Fn(31D)p Fm(,)f(26333-)892 3716 y(26346)j(\(1996\).)769 3837 y([5])43 b(Cha)n(v)n(es,)26 b(A.S.,)g(Ph)n(ys.)f(Letters)h(A,)g Fn(239)p Fm(,)g(13-16)h(\(1998\)) 769 3958 y([6])43 b(Y)-6 b(ano)n(vsky)g(,V.)27 b(V.,)k(A.)e(V.)h(Chec)n (hkin,)g(D.)g(Sc)n(hertzer)f(and)g(A.)h(V.)f(T)-6 b(ur,)31 b(sub-)892 4049 y(mitted)24 b(to)i Fl(Physic)l(a)i(A)p Fm(,)e(1999.)769 4170 y([7])43 b(F)-6 b(ristedt,)36 b(B.,)h(L.)e(Gra)n (y)-6 b(,)36 b(A)e(Mo)r(dern)g(Approac)n(h)f(to)i(Probabilit)n(y)g (Theory)-6 b(,)892 4262 y(Birkhauser,)26 b(Boston)h(\(1997\).)769 4382 y([8])43 b(Gardiner,)20 b(C.W.,)g(Handb)r(ok)d(of)i(Stao)r(c)n (hastic)g(Metho)r(ds)f(for)h(Ph)n(ysics,)h(Chem-)892 4474 y(istry)25 b(and)h(the)f(Natural)h(Sciences,.)g(Springer-V)-6 b(erlag,)26 b(Berlin)h(\(1985\).)769 4595 y([9])43 b(L)n(\023)-36 b(evy)28 b(P)-6 b(.,)31 b(Theorie)g(de)f(l"Addition)f(des)h(V)-6 b(ariables)31 b(Al)n(\023)-36 b(eatoires.)31 b(Gauthier{)892 4686 y(Villars,)c(P)n(aris)g(\(1937\).)731 4807 y([10])43 b(Khin)n(tc)n(hine)d(A.Y.)h(and)g(L)n(\023)-36 b(evy)40 b(P)-6 b(.,)46 b(C.)c(R.)f(Acad.)h(Sci.\(P)n(aris\))g Fn(202)p Fm(,)47 b(374)892 4898 y(\(1936\).)1914 5147 y Fk(11)p eop %%Page: 12 12 12 11 bop 731 614 a Fm([11])43 b(Gnedenk)n(o)23 b(B.V.)j(and)e (Kolmogoro)n(v)h(A.N.,)g(Limit)f(Distributions)h(for)g(Sums)892 706 y(of)33 b(Indep)r(enden)n(t)d(Random)h(V)-6 b(ariables,)35 b(Addison)c(W)-6 b(esley)g(,)34 b(Reading,)g(MA,)892 797 y(\(1954\).)731 922 y([12])43 b(F)-6 b(eller)32 b(W.,)i(An)d(In)n (tro)r(duction)g(to)g(Probabilit)n(y)i(Theory)f(and)f(its)h(Applica-) 892 1013 y(tions.)26 b(John.Wiley)g(&)g(Sons,)g(New.Y)-6 b(ork)25 b(\(1966\).)731 1137 y([13])43 b(Zolotarev)23 b(V.M.,)h(One-dimensional)d(Stable)h(Distributions,)i(MM,)f(V)-6 b(ol.)23 b(65,)892 1229 y(AMS)i(\(1986\).)731 1353 y([14])43 b(Miller,)34 b(K.S.,)g(An)c(In)n(tro)r(duction)h(to)h(the)f(F)-6 b(ractional)33 b(Calculus)g(and)e(F)-6 b(rac-)892 1445 y(tional)26 b(Di\013eren)n(tial)g(Equations,)h(J.)f(Wiley)g(&)f(sons,)i (New)f(Y)-6 b(ork)24 b(\(1993\).)731 1569 y([15])43 b(P)n(aulausk)l(as) 23 b(V.J.)f(\(1976\).)i(Some)d(remarks)h(on)g(Multiv)l(ariate)h(Stable) g(Distri-)892 1660 y(butions.)i Fl(J.)j(Multivariate)g(A)n(nal.)p Fm(,)p Fn(6)e Fm(356-368)731 1785 y([16])43 b(Nikias)26 b(C.L.)h(and)f(M.)h(Shao)f(\(1995\).)i Fl(Signal)g(Pr)l(o)l(c)l(essing) i(with)e(alpha-stable)892 1876 y(distributions)h(and)f(applic)l(ations) p Fm(,)f(John)e(Wiley)h(and)g(Sons,)f(New-Y)-6 b(ork.)731 2001 y([17])43 b(Sc)n(hertzer,)24 b(D.,)h(M.)h(Larc)n(hev)n(^)-36 b(eque,)24 b(J.)h(Duan,)g(S.)f(Lo)n(v)n(ejo)n(y)-6 b(,)25 b(Submitted)d(to)j(J.)892 2092 y(Multiv)l(ariate)h(Anal.,)g(July)g(27)g (1999.)731 2217 y([18])43 b(Oksendal,)25 b(B.)i(Sto)r(c)n(hastic)f (Di\013eren)n(tial)g(Equations,)h(Springer-V)-6 b(erlag,)26 b(5th)892 2308 y(ed.,)g(324)g(pp.)g(\(1998\).)1914 5147 y Fk(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9911301231132--