ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS

Available online at www.sciencedirect.com.MathemiqlaStientia”ScienceDirect数学物理学报Acta Mathematica Scientia 2007 ,27B(1):145-152http:// actams.wipm.ac.cnENTROPY PRODUCTION RATE OF THE MINIMALDIFFUSION PROCESS ,Zhang Puxi(章复熹) Qian Min(钱敏)LMAM, School of Mathernatical Sciences, Peking University, Bejing 100871, ChinaE- mail: zhangfrxi@math. pku. edu.cnAbstract The entropy production rate of stationary minimal difusion processes withsmooth coficients is calculated. As a byproduct, the continuity of paths of the minimaldifusion processes is discussed, and that the point at infinity is absorbing is proved.Key words Minimal difusion process, entropy production rate, continuous path2000 MR Subject Classification 60J60, 60G101 IntroductionIn statistical physics, the entropy production rate (or EPR for short) measures how fara specifc state of a system is away from its equilibrium state [1-6]. When physics systemsare modelled by stochastic processes, reversible processes and stationary processes may beinterpreted as systems being in equilbrium states and in steady states. To gauge the differenceof a stationary process from its time reversal, a unified measure- theoretic definition of the EPRof a stochastic process is proposed. As suggested by following works, the EPR is the time-averaged relative entropy of the probability distribution P of the process on the path spacewith respect to that of its time-reversal P- , that is,dP|F.epr:= H(P|r,P-|x)= EplogdP-lxIn particular, an irreversible stationary process has a positive EPR.For a Markov chain {Xn :n≥0} in a finite state space with irreducible transition proba-bility matrix (pi) and initial invariant measure {π}, the time-reversed process is still a Markovchain with transition probability pij = πjPji/πi Assume Pij>0台Pji> 0 for any i,j. Thenthe relative entropy over a finite time interval is中国煤化工Elog I. PXk-1Xk=n> πpPij logπiPijCH一2(TiPij一"IjPji)lUg ;CNMHGm;pijPXx-1Xki力jπjPji2的TjPji* Received December 12, 2004. This work is supported by NSFC (10271008 and 10531070)146ACTA MATHEMATICA SCIENTIAVol.27 Ser.Bwith the convention log0/0= 0. The EPR of {Xn:n≥0} is shown in [7] to beepr=52 (T:P:j - πj)}logπjPjii≠jIt follows immediately that a Markov chain is reversible if and only if its EPR vanishes. Equiv-alently, it is in detailed balance, that is, TiPij = πjPji for any i,j.For a Markov chain with continuous time parameter {Xt :t≥0} in a finite state space withirreducible transition rate (qij) and initial invariant measure { πi }, the time reversed process isa Markov chain with transition rate qij = πj9ji/πi. Assume qij > 0台9ji >0 for any i,j. TheEPR of {Xp:t≥0} is shown in [7] to beepr= 2 (πiGij - πjGj;)logπq9ijTj9jii#j{Xt:t≥0} is reversible if and only if its EPR vanishes. Equivalently, it is in detailed balance,that is, Ti9ij = πjGji for any i,j.For a diffusion process {Xt : t≥0} with generator L = VAV + bV and invariantinitial measure 0(x)dx, the time-reversed process is a difusion process with generator C* =VAV + 6V, whereb= AV logθ - b. Assume A, b are bounded, then the logarithm Radon-Nikodym derivative of P with respect to P- is the sum of a martingale and号Jo(cT Ac)(X,)ds,wherec= 2A-1b- V logθ. Hence the relative entropy over [0,t] is吃f(exe)(Xx)ds=专()()()x0.by stationarity. The EPR is shown in [8, 9] to beepr=引/(e" A()'()dx.(1.1)The diffusion process is reversible if and only if its EPR vanishes. Equivalently, it is in detailedbalance, that is, V logθ = 2A-1b.The purpose of this article is to extend (1.1) to a more general setting when the cefficientsare not bounded, In physical systems or practical models, boundedness condition might be toorestrictive, while the coefficients are generally assumed to be smooth. Without boundednesscondition, the uniqueness of the distribution on the path space is not available， Hence wediscuss the minimal diffusion process with smooth coefficients as a substitution. Since the EPRis defined for a stationary process, we suppose there is an invariant distribution 0(dx). In thiscase, the process is unexploded. By the Weyl Lemma, 0(dx) has a smooth density 0(x), whichis positive by the Strong Extremum Priciple of the particial differential equation. Moreover,the time reversed process is still a difusion process with generator L* = VAV + 6V, whereb=AVlogθ一b(see[10]).In this article, we first discuss the continuity of中国煤化Ifusion process.Section 2 is devoted to find a version of the minimalYHC N M H Gntinuous pathson the compact Polish space obtained from Rd by one point compactification. We prove thatthe point at infinity is an absorbing state, and there is no explosion if the transition probabilityin Rd is conservative. In Section 3, by the uniqueness of the path distribution of a difusionNo.1Zhang & Qian: ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS 147process in a finite domain with absorbing boundary, we get a local Radon-Nikodym derivative.It converges as the domain enlarges to the whole space, and the limit is the Radon-Nikodymderivative of P with respect to P-.Theorem 1.1 Assume A is a smooth positively definite matrix, and b is a smooth vector.Let {Xi : t≥0} be the minimal diffusion process with coficients A, b and initial measureθ(x)dx. Suppose that 0(x)dx is an invariant measure of {Xt : t≥0} and 0(x) is a smoothpositive density. Letc= 2A- -1b- V logθ. Assume f(cT Ac)(x)(x)dx <∞. Then for anyt> 0,三Ep log;cr Ac)(x)0(x)dx.dP-lF.Hence the EPR of a stationary minimal difusion process with smooth coefficients existsand the expression coincides with (1.1). The minimal process is reversible if and only if theEPR vanishes. Equivalently, it is in detailed balance, that is, V logθ= 2A-1b.2”Continuous Paths of the Minimal ProcessFor the sake of the reader, we recall sorne facts first. Detailed proofs are referred to [10]. .LetL=VAV+bVbe the generator of a difusion process on Rd, where A = A(x) and b = b(x) are smooth positivedefinite matrix and smooth vector. DenoteC%(R4) = the set of bounded continuous functions in Rd,Cδ(R)= the set of smooth functions with cormpact supports in Rd,Co(Bn)={f∈C6(R):f|B; =0},where Bn={x∈Rd:|x| 0,t≥0,s> 0,P(Xt = 8,Xt+s t Bm) = 0({})i(s,0,{}) = 0({}),(2.2)where 0 is the distribution of Xt. On the other hand, for any E > 0, there exist finitely manyxi∈Bm such that {B(x;, e/4)} covers Bm. Choose smooth function f; such that0≤fi≤1,fi|B(xs,c/4)=1，fl|(x,e/2)e= 0.Then fr(u)≤1{|v- -x| 0} is the resolvent of {S(t):t≥0}. by a sught moaincation of the proof in[11], we can show that {Frn,n : m,n∈N} separates points in E and that {e mtpm,n(Xt):t≥0}is a supermartingale for any m, n. Following the steps in Section 2.2 of [11], we can obtain thefollowing. ，万万数据No.1Zhang & Qian: ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS 149Proposition 2.2 Pa.s., Xt+ exists for any t≥0 and Xt- exists for any t> 0.The above two propositions yield the followingCorollary 2.3 Xt+ ∈Ft and P(Xt+ = Xt)=1 for anyt≥0.By Proposition 2.2, P almost surely limXs+ = Xt- for anyt> 0. Hence {Xt+:t≥0} isa version of the minimal process with right continuous paths having left limits. Now we discussthe absorbing property of the state 8. Denote by Bn the closure of Bn. Definen:= inf{t≥0:Xt+≠Bn}，T:= lim Tn.For any t≥0,{mn≥t}={Xs+∈Br,Vs∈{0,t)}= n {Xr+∈ Bn}.r∈Q(0,t)Hence Tn and r are stopping times. To prove that 8 is absorbing, we defnex=Xt+,t 0, defne the time-reversal operator asREVi : C([0,∞),R)→C([0,t],R)，(REV(w))。= X;_ .s(u), Vs∈[0,t].Denote by P- the unique probability on C([0, ∞), Rd) whose marginal on C([0,t],R") is P。REV:- 1. Then P- is the distribution of the time-reversed process of {Xt :t≥0} with initialmeasure 0(x)dxr. .Choose bounded smooth A(m), b(m) and 0(m) such that they are equal to A, b and θ in Bnrespectively, and b(m)≌A(n)V log 0(n) - b(n) is also bounded. Then the diffusion process withcoefficients (A(m), b(n) ( resp. (A(n), 6b(m)) has a unique distribution on the path space withinitial measure 0(x)dx, which is denoted by Pn (resp. Pn). By Theorem 2.5, without loss ofgenerality, we choose probabilities Pn and Pn on C([0,∞),E) with日as an absorbing state. .Since the coefficients are equal within Bn,中国煤化工PnlFrnrn =PIzom，PnlFinrn(3.1)YHCNMHGFrom the Cameron -Martin- Girsanov formula [12],r(,)= BdPnIF,=exp{{- J°inx.X.- (2(000urum (3.2)No.1 Zhang & Qian: ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS 151where则=X- ( +((X.)dsis a martingale under Pn andcm)=2(A()厂6m)- Vlog(1m).By (2.4), Xu∈Bn ifu≤Tn. Hence x{n+1)= x) for anyu≤Tn. Notice that Pn+1lF. =Pnlrrn. It follows thatYn+1(tNTn,w)= Yn(t^Tn,w)，P-a.s..This ensures thatxY(t,w):= lim Yn(,w)1{t n,Ep|Mm(t,w) - Mn(,w)|2= EocT Ac) (Xxu)1{rnSu