分数阶时间导数方程和反常亚扩散过程——纪念茆诗松教授.pdf

应用概率统计第 40 卷第 2 期2024 年 4 月Chinese Journal of Applied Probability and StatisticsApr.,2024,Vol.40,No.2,pp.323-342doi:10.3969/j.issn.1001-4268.2024.02.007Time Fractional Equations and Anomalous Sub-Diffusions In Memory of Professor Shisong MaoCHEN Zhen-Qing(Department of Mathematics,University of Washington,Seattle,WA 98195,USA)Abstract:In this paper,we survey some recent progress in the study of time fractional equationsand its interplay with anomalous sub-diffusions,with some improvements and extensions.Keywords:time fractional derivative;time fractional equation;subordinator;inverse subordina-tor;strong and weak solution2020 Mathematics Subject Classification:primary 26A33,60H30;secondary 34K37Citation:CHEN Z-Q.Time fractional equations and anomalous sub-diffusionsJ.Chinese JAppl Probab Statist,2024,40(2):323342.1IntroductionSub-diffusions are random processes that can model the motions of particles that moveslower than Brownian motion(or the original underlying spatial motion),for example,dueto particle sticking and trapping.A prototype of anomalous sub-diffusions is modeled byYt=BLt,where B is a d-dimensional Brownian motion and Lt:=infr 0:Sr t,t 0,is the inverse of a-stable subordinator S with 0 0,which grows sub-linearly in t.Here():=0t1etdt is the Gamma function.On the other hand,fractional calculus has attracted lots of attentions in several fieldsincluding mathematics,physics,chemistry,engineering,hydrology and even finance andE-mail:zqchenuw.edu.Received December 1,2023.Revised January 23,2024.324Chinese Journal of Applied Probability and StatisticsVol.40social sciences;see,e.g.,3,57.The classical heat equation tu=u describes heatpropagation in homogeneous medium.The time-fractional diffusion equation tu=uwith 0 0 be a subordinator with S0=0.It is well known that there are aunique constant 0 and a unique L evy measure on(0,)satisfying0(1x)(dx)0,t 0,326Chinese Journal of Applied Probability and StatisticsVol.40Define for t 0,Lt=infs 0:Ss t,the inverse subordinator.Throughout thispaper,we assume the L evy measure of the subordinator S is infinite(which is equivalentto w(x):=(x,)being unbounded)on(0,)excluding compound Poisson processes.Under this assumption,almost surely,t 7 Stis strictly increasing and hence t 7 Ltiscontinuous.Let St:=St t.Clearly,S is a subordinator having L evy exponent0()=0(1 ex)(dx).For every a 0,by Fubini theorem,a0w(x)dx=a0(x,)(d)dx=0(a0dx)(d)=0(a)(d)0.(5)Remark 1For a left continuous decreasing functionwon(0,)withlimxw(x)=0,it uniquely determines a Radon measureon(0,)so that(x,)=w(x)forx 0.By(4),is the L evy measure of a subordinator if and only ifwis locally integrableon0,),andis an infinite measure if and only ifwis unbounded.For any locally bounded function f on 0,),we defined the Riemann-Liouville typeintegral byIwtf:=t0w(t s)f(s)ds=t0w(s)f(t s)ds,t 0.(6)In view of(5),the Laplace transform of Iwt(f)isLIwtf()=0()Lf()for 0.(7)Following 13,we define the generalized time fractional derivative with weight w bywtf(t):=ddtIwtf=ddtt0w(t s)f(s)f(0)ds,(8)whenever the right hand side is well defined.Its connection to the classical Caputo typefractional derivative,which is defined by the right hand side of(9),is given in the followinglemma.The advantage of our definition of the time fractional derivative wtf(t)is thatwe do not need to assume a priori the existence of f(s)for almost every s (0,t)nor theabsolute convergence of the integralt0w(t s)f(s)ds.No.2CHEN Z.-Q.:Time Fractional Equations and Anomalous Sub-Diffusions327Lemma 2Iffis a local Lipschitz function on0,),thenwtf(t)exists for almosteveryt 0andwtf(t)=t0w(t s)f(s)dsfor a.s.t 0.(9)ProofSince f is a local Lipschitz function on 0,),f(s)exists for a.e.s (0,)and for every t 0,|f(s)|is bounded over 0,t by a constant M(t)depending on t.Thust0w(t s)|f(s)|ds 0,which is a continuous function with G(0)=0;see 13;(2.2).Using the Fubini theorem and the integration by parts formula,we have for everyT 0,T0g(t)dt=T0Tsw(t s)dtf(s)ds=T0G(T s)f(s)ds=G(T s)f(s)?T0T0G(T s)f(s)ds=G(T)f(0)+T0w(T s)f(s)ds=T0w(T s)f(s)f(0)ds.This establishes the lemma.?Remark 3WhenSis a-stable subordinator with0 0 to every t 0.Note that in this paper,w(x)is defined to be(x,)rather thanits right continuous version(x,)as in 13.They differ only at possibly countablymany points on(0,).Proposition 4For everyt 0ands 0,P(Lt6 s)=P(Ss t)=s0Ew(t Sr)1tSrdr.(10)Consequently,0Ew(t Sr)1tSrdr=1for everyt 0.(11)ProofFor a 0,let f=1a,).Since S is a driftless subordinator with infiniteL evy measure and f is a right continuous non-decreasing function,we have by 20;328Chinese Journal of Applied Probability and StatisticsVol.40Corollary 2 thatEf(St)=f(0)+Et0(0,)f(Sr+z)f(Sr)(dz)dr.Hence we have by Fubinis theorem that for every a 0 and t 0,P(St a)=Et0(0,)1zaSr1Sr6a(dz)dr=Et0w(a Sr)1Sr6adr=t0Ew(a Sr)1Sr6adr.(12)As almost surely r 7 Sris strictly increasing,we have for each fixed t 0,Lt6 r=Sr t a.s.and soP(Lt6 r)=P(Sr t)=t0Ew(t Sr)1Srtdr.This in particular implies that Lthas a density function r 7 Ew(tSr)1Sr 0.The following lemma is takenfrom 13;(2.5)and Corollary 2.1(ii).It together with Proposition 4 played an importantrole in 13,18,19 for the probabilistic approach to time fractional equations.Proposition 5For everyt,s 0,t0w(t r)P(Ss r)dr=G(t)EG(t Ss)1tSs,0EG(t Sr)1Sr6tdr=tand0EG(t Sr)1tSrdr 6 t.3Time Fractional Parabolic EquationRecall that St;t 0 is a general subordinator with infinite L evy measure anddrift 0,whose Laplace exponent()is given by(3).Define w(x)=(x,)for x 0and 0():=0(1ex)(dx).Note that 0()is the Laplace exponent of the driftlesssubordinator St:=St t,t 0 having L evy measure.Clearly()=+0()and St=t+St.Since(0,)=,almost surely,t 7 Stis strictly increasing.Suppose that Tt;t 0 is a strongly continuous semigroup with infinitesimal gener-ator(L,D(L)in some Banach space(B,)with the property that supt0Tt 0Tt 0Ttf 0 of a strong Markov process X=Xt,t 0;Px,x X on a Lusin space X that has a weak dual with respect to some-finite referencemeasure m on X.For every p 1,Pt;t 0 is a strongly continuous semigroupin B:=Lp(E;m)with supt0Ptpp6 1.The infinitesimal generator(L,D(L)of Pt;t 0 in Lp(E;m)is called the Lp-generator of the Markov process X.(ii)Transition semigroup Pt;t 0 of a Feller process X=Xt,t 0;Px,x X ona locally compact separable metric space X.In this case,Pt;t 0 is a stronglycontinuous semigroup in the space(C(E),)of continuous functions on Xthat vanish at infinity equipped with uniform norm.The infinitesimal generator(L,D(L)of Pt;t 0 in B:=(C(X),)is called the Feller generator of X.(iii)Certain Feynman-Kac semigroups(can be non-local Feynman-Kac semigroups or evengeneralized Feynman-Kac semigroups)in Lp-space or in C(X)of a Hunt processX;cf.2123.For 0,let G:=0etTtdt be the resolvent of the semigroup Tt;t 0 onthe Banach space B.Then by the resolvent equation,D(L)=G(B)=G1(B),whichis dense in the Banach space(B,).Recall that Lt:=infs 0:Ss t,t 0,is theinverse of the subordinator S.Defineu(t)=ETLtf=0TsfdsP(Lt6 s)=0TsfdsP(Ss t).The following is essentially the main result of 13,Theorem 6 there,which gives theexistence and uniqueness of strong solutions to the time fractional parabolic equation(13).However,it contains an improvement(14)for the positive drift case of the subordinatorS.This improvement was given in the Appendix of the arXiv version of 13,added afterits publication.For the readers convenience,we reproduce its proof here.Theorem 6(Strong solution)Suppose that(L,D(L)is the infinitesimal generatorof a uniformly bounded strongly continuous semigroupTt;t 0in a Banach space(B,).For everyf D(L),u(t):=ETLtfis a solution in(B,)to(t+wt)u(t)=Lu(t)withu(0)=f(13)in the following sense:(i)supt0u(t)0withsupt0Lu(t)0,Iwt(u):=t0w(ts)u(s)f(x)dsis absolutely convergent in(B,)andlim01u(t+)u(t)+Iwt+(u)Iwt(u)=Lu(t)in(B,).330Chinese Journal of Applied Probability and StatisticsVol.40In addition,t 7 Lu(t)are continuous in(B,).When 0,t 7 u(t)is globallyLipschitz continuous in(B,),andbothtu(t)andddtIwt(u)exists as a continuous function taking values in(B,).(14)Conversely,ifu(t)is a solution to(13)in the sense of(i)and(ii)above withf D(L),thenu(t)=ETLtf(x)inBfor everyt 0.ProofAll except(14)are established in 13;Theorem 2.3.We point out that in theoriginal statement of 13;Theorem 2.1,the continuity of t 7 Lu(t)in(B,)is statedas a part of the definition of the strong solution.But one can see from the uniquenesspart of the proof of 13;Theorem 2.1 that the continuity of t 7 Lu(t)in(B,)isnot used.The improved version(14)is given in the Appendix of the arXiv version of 13after its publication.For the readers convenience,we spell out its proof here.When 0,it is shown in 13;Theorem 2.3 that t 7 u(t)is globally Lipschitzcontinuous in(B,).Thus tu(t)exists in(B,)for a.e.t 0,and there is M 0so that tu(t)6 M for a.e.t 0.Hence g(t):=t0w(t s)su(s)ds is well-defined asan element in(B,).We claim t 7 g(t)is uniformly continuous in(B,).This isbecause for any t 0 and 0,we have by(12)and Fubini theorem,g(t+)g(t)6 Mt0|w(t+s)w(t s)|ds+Mt+tw(t+s)ds=Mt0(r,r+dr+M0w(r)dr=M0106r6t01r6r+(d)dr+M0()(d)=Mt+0(t()+dr)(d)+M0()(d)6 2M0()(d),which goes to zero as 0 uniformly in t as0(1)(d)0,we have by Fubini theorem and integration by parts thatt0g(s)ds=t0s0w(s r)ru(r)drds=t0trw(s r)dsru(r)dr=t0G(t r)ru(r)dr=G(0)u(t)G(t)u(0)+t0w(t r)u(r)dr=t0w(t r)u(r)u(0)dr=Iwt(u).No.2CHEN Z.-Q.:Time Fractional Equations and Anomalous Sub-Diffusions331Since g(s)is continuous in(B,),wtu(t):=(d/dt)Iwt(u)exists and t 7 wtu is contin-uous in(B,).Now it follows from(ii)of Theorem 6 thattu(t):=lim01u(t+)u(t)=1Lu(t)wtu(t)exists and t 7 u(t)is continuous in(B,).Hence u(t):=ETLtf(x)satisfies(13).?Remark 7We take this opportunity to point out that there is a critical error inthe proof of the main result,Theorem 5.1,in 24.In Definition 2.4 in 24,the fractionalderivativefDtu(t)is defined to befDtu(t)=ddtu(t)+t0ddtu(t s)(ds).In the proof of Theorem 5.1 in 24,the existence of the fractional derivativefDtTtuon topof p.134 of 24 is not justified,nor is the interchange of limit and integration in the secondequality on p.134.The main problem is that,in the setting and the approach of 24,onedoes not know ift 7 Ttuis locally Lipschitz int 0,).We consider the following assumption.Assumption 8Suppose thatBis a Banach space over a metric spaceX.Forinstance,B=Lp(X;m)for somep 1and a-finite measuremwith full support on aLusin spaceX,orB=C(X),the space of continuous functions on a locally compactseparable metric spaceXthat vanish at infinity equipped with uniform norm.Assume thatthe uniformly bounded and strongly continuous semigroupTt;t 0in(B,)has adensity functionp0(t,x,y)with respect to some-finite measuremonXwith full support,such that(i)Ttf(x)=Xp0(t,x,y)f(y)m(dy)forf B;(ii)for eachx,y X,t 7 p0(t,x,y)is Borel measurable on(0,).Remark 9Suppose that the subordinatorSis driftless(that is,=0)and Assump-tion 8 holds.Then for any bounded functionf D(L),by Fubinis theorem,u(t,x):=ETLtf(x)=EXp0(Lt,x,y)f(y)m(dy)=Xf(y)Ep0(Lt,x,y)m(dy),x X.(15)This says thatp(t,x,y):=Ep0(Lt,x,y),x,y Xandt 0,(16)332Chinese Journal of Applied Probability and StatisticsVol.40is the“fundamental solution”to the homogeneous time fractional equationwtu(t,x)=Lu(t,x)withu(0,x)=(x).(17)Note that in the PDE literatures,the most standard approach to analyze the fundamentalsolution of(17)is to use the Mittag-Leffler function,and then take the inverse Fouriertransform whenLis a second order of elliptic operator having constant coefficients.Onecan then extend it to a general second order of elliptic operatorLhaving H older continuouscoefficients by a parametrix method;see 25 whenStis a-stable subordinator.Weemphasize that the expression(23)is more intuitive,simple,and general.Moreover,itis amenable for estimation.We refer the reader to 18 for two-sided estimates of thefundamental solutionp(t,x,y)under various cases of the semigroupTt;t 0and thesubordinatorS.Using the ideas from the proof of Theorem 6,we can similarly establish the existenceand uniqueness of weak solution to the time fractional equation(13)when(L,D(L)isthe infinitesimal generator of a uniformly bounded strongly continuous symmetric semi-group Tt;t 0 in a Hilbert space(H,).The following result extends 18;Theorem2.4 in which(L,D(L)is the infinitesimal generator of an m-symmetric strong Markovprocess on a locally compact separable metric space X with H=L2(X;m).Theorem 10(Weak solution)Suppose that(L,D(L)is the infinitesimal generatorof a uniformly bounded strongly continuous symmetric semigroupTt;t 0in a HilbertspaceHequipped with inner product,andSis a driftless subordinator with infiniteL evy measure.For everyf H,u(t):=ETLtfis a weak solution inHto(13)in thefollowing sense:(i)t 7 u(t)is continuous inH.Consequently,for everyt 0,Iwt(u):=t0w(t s)u(s)f(x)dsis absolutely convergent inH.(ii)For everyg D(L)andt 0,ddtIwt(u),g=u(t),Lg(x).(18)Conversely,ifu(t)is a weak solution inHto(13)in the sense of(i)and(ii)above withf H,thenu(t)=ETLtfonHfor everyt 0.No.2CHEN Z.-Q.:Time Fractional Equations and Anomalous Sub-Diffusions333Proof1)(Existence)Since Tt:t 0 is a uniformly bounded strongly continuoussymmetric semigroup in H and t 7 Ltis continuous a.s.,we have by the boundedconvergence theorem that t 7 u(t)=ETLtf is continuous in H and u(t)6 Mf,where M=supt0Tt.Since by(4),t0w(s)ds=0(z t)(dz)0,(19)Iwt(u)is absolutely convergent in H for every t 0 with Iwt(u)6 2Mft0w(s)ds t).(20)Here and in what follows,drdenotes the Lebesgue-Stieltjes differential in r-variable.By(20),Proposition 5 and the self-adjointness of L in H,we have for every t 0 andg D(L),Iwt(u),g=t0w(t r)u(r,x)u(0,x)dr,g=t0w(t r)0(Tsf,g f,g)dsP(Ss r)dr=0Tsf f,gdst0w(t r)P(Ss r)dr=0Tsf f,gdsEG(t Ss)1Ss6t=0EG(t Ss)1Ss6tLTsf,gds=0EG(t Ss)1Ss6tTsf,Lgds.On the other hand,according to(20)and Proposition 4,we find that for every t 0 andg D(L),t0u(s),Lg(x)ds=t00TufduP(Su s),Lgds=0Tuf,Lgdut0P(Su s)ds=0Tuf,Lgt0Ew(s Su)1Su6sdsdu=0Tuf,LgEG(t Su)1Su6tdu.334Chinese Journal of Applied Probability and StatisticsVol.40Thus we conclude that for every t 0,Iwt(u),g=t0u(s),Lg(x)ds.This establishes(18)as s 7 u(s)is continuous in H.2)(Uniqueness)Suppose that u(t)is a weak solution to(13)in the sense of(i)and(ii)with f H.Then v(t):=u(t)ETLtf is a weak solution to(13)with v(0)=0.Note that by(19),limt0Iwt(v)6 2 maxs0,1v(s)limt0t0w(s)ds=0.Hence we have for every t 0 and g D(L),t0w(t r)v(r,x)dr,g=t0v(s,x)ds,Lg(x).(21)Let V():=0etv(t)dt,0,be the Laplace transform of t 7 v(t).Taking theLaplace transform in t on both sides of(21)yields that for every 0,V()0esw(s)ds,g=1V(),Lg.Note that by(5),the Laplace transform of w(t)is()/.We get from the above displaythat V(),()Lg=0 for every 0.Let G;0 be the resolvents of L.Foreach fixed 0 and h H,take g:=G()h,which is in D(L).Since()Lg=h,we deduce that V(),h=0.Since this holds for every h H,we have V()=0 forevery 0.By the uniqueness of the Laplace transform and the fact that t 7 v(t)is continuous in H,it follows that v(t)=0 in H for every t 0.In other words,u(t)=ETLtf in H for every t 0.This esta