Stochastic Processes. A. Bassi. RMP 85, April-June Appendix

Transcription

1 Sochasic Processes A. Bassi RMP 85, April-June Appendix

2 The bes known example of a sochasic process is Brownian moion : random moion of small paricles suspended in a liquid, under he influence of a viscous drag, and a flucuaing force resuling from collisions wih he molecules of his liquid. The quaniaive explanaions of Brownian moion by Einsein and Smoluchowski were simplified by Langevin, hrough his use of wha is now called he Langevin equaion. This dynamical equaion for he randomly evolving posiion of he suspended paricle is equivalen o he Fokker-Planck equaion for he ime evoluion of he probabiliy disribuion of he random variable. The Langevin equaion can be pu on a firm fooing as a sochasic differenial equaion using Iô s differenial calculus for sochasic variables. The brief review below inroduces sochasic processes, and should be of some assisance in undersanding he previous secions of his review. For a deailed survey he reader is referred o he books by (Arnold, 1971; Gardiner, 1983; Gikhman and Skorokhod, 1972; Grimme and Sirzaker, 21; Risken, 1996) and he review aricle by (Chandrasekhar, 1943). A. Some Basic Conceps from Probabiliy Theory Le Ω denoe a sample space, i.e. a se of all possible evens in an experimen. Then one can define he following srucures on Ω. Definiion 1. A collecion F of subses of Ω is called a σ-field if 1. φ F 2. if A 1, A 2 F hen i=1 A i F 3. if A F hen A C F The above properies also imply ha closed counable inersecions are also included in F. Examples: (a) F = {φ, Ω} is he smalles σ-field. (b) If A Ω hen F = {φ, A, A C, Ω} is a σ-field. Definiion 2. A probabiliy measure P on (Ω, F) is a funcion P F [, 1] saisfying: 1. P (φ) =, P (Ω) = 1

3 2. if A 1, A 2 is a collecion of disjoin members of F in ha A i A j = φ for all poins i, j saisfying i j hen P ( A i ) = i=1 i=1 P (A i ) The riple (Ω, F, P ) is defined o be he probabiliy space. B. Random Variables Definiion 3. A random variable is a funcion X Ω R wih he propery ha {ω Ω X(ω) x} F for each x R, such funcions are called F-measurable. 1. Disribuion of random variables X s called a discree random variable if i akes values in some counable subse {x 1, x 2,...} only of R. The X has probabiliy mass funcion f R [, 1] defined by f(x) = P (X = x). X is called a coninuous random variable if is disribuion funcion F R [, 1] given by F (x) = P (X x) can be expressed as x F (x) = f(u)du x R for some inegrable funcion f R [, ), where f is he probabiliy densiy funcion. Join disribuion Funcion : A random variable X = (X 1, X 2,..., X n ) on (Ω, F, P ) is a funcion F X R n [, 1] given by F X (x) = P (X x) for x R n. 2. Time Dependen Random Variables Le ξ ξ() be a ime dependen random variable. Assume an ensemble of sysems, such ha each sysem leads o a number ξ which depends on ime. The oucome for one sysem canno be precisely prediced, bu ensemble averages exis and can be calculaed. For fixed = 1 we define he probabiliy densiy by W 1 (x 1, 1 ) = δ(x 1 ξ( 1 )) (1) 2

4 where he angular brackes denoe he ensemble average. The probabiliy o find he random variable ξ( 1 ) in he inerval x 1 ξ( 1 ) x 1 +dx 1 is given by W 1 (x 1, 1 )dx 1,..., in inerval x n ξ( n ) x n + dx n is given by W n (x n, n ;...; x 1, 1 )dx n dx (n 1)...dx 1 where, W n (x n, n ;...; x 1, 1 ) = δ(x 1 ξ( 1 )) δ(x n ξ( n )) Given W n for all n, for every i in he inerval i + T he ime dependence of he process described by ξ() in he inerval [, + T ] can be know compleely. One can obain probabiliy densiies wih lower number of variables from hose of higher number of variables by inegraing as follows. For i < r W i (x 1, x 2,.., x i ) = W r (x 1, x 2,.., x i, x i+1,..., x r )dx i+1 dx r (2) Condiional Probabiliy densiy is defined as: P (x 1 x 2,..., x r ) = W r(x 1,..., x r ) W r 1 (x 2,..., x r ) where P (x 1 x 2,..., x r ) denoes he probabiliy densiy of x 1 given he x 2,..., x r. 3. Saionary Processes If he probabiliy densiies do no change by replacing i by i + T (T arbirary) he process is called a saionary process. In such a case W 1 does no depend on and W 2 can depend only on he ime difference 2 1. The Wiener-Khinchin Theorem : According o his heorem, he specral densiy is he Fourier Transform of he correlaion funcion for saionary processes.. Insead of he random variable ξ() one may consider is Fourier Transform ξ(ω) = exp ( iω)ξ()d 3

5 For a saionary process ξ()ξ ( ) is a funcion only of he difference i.e., ξ()ξ ( ) = ξ( )ξ () Inroducing he new variables we have which gives where ξ(ω) ξ (ω ) = τ =, = + 2 exp ( i(ω ω ) )d exp ( i(ω + ω )τ/2) ξ(τ)ξ () dτ (3) langle ξ(ω) ξ (ω ) = πδ(ω ω )S(ω) S(ω) = 2 is he specral densiy. exp (iωτ) ξ(τ)ξ () dτ 1 Sochasic Processes and heir Classificaion Sochasic processes are sysems which evolve probabilisically, i.e., sysems in which a ime-dependen random variable exiss. The sochasic processes described by a random variable ξ can be classified as follows: Purely Random Process: If P n (n 2) does no depend on he values x i = ξ( i ), (i < n) a i < n, hen P (x n, n x n 1, n 1 ;..., ; x 1, 1 ) = P (x n, n ) hus W n (x n, n ;...; x 1, 1 ) = P (x n, n ) P (x 1, 1 ) 4

6 Hence, complee informaion of he process is conained in P (x 1, 1 ) = W 1 (x 1, 1 ). A purely random process canno describe physical sysems where he random variable is a coninuous funcion of ime. Markov Process: If he condiional probabiliy densiy depends only on he value ξ( n 1 ) = x n 1 a n 1, bu no on ξ( n 2 ) = x n 2 a n 2 and so on, hen such a process is known as a Markov Process. This is given by Then i follows ha W n (x n, n ;...; x 1, 1 ) = P (x n, n x n 1, n 1 ;...; x 1, 1 ) = P (x n, n x n 1, n 1 ) (4) P (x n, n x n 1, n 1 )P (x n 1, n 1 x n 2, n 2 ) P (x 2, 2 x 1, 1 )W 1 (x 1, 1 ) (5) For n = 2 P (x 2, 2 x 1, 1 ) = W 2(x 2, 2 ; x 1, 1 ) W 1 (x 1, 1 ) For a Markov process he complee informaion of he process is conained in W 2 (x 2, 2 ; x 1, 1 ). General Processes: There can be oher processes such ha he complee informaion is conained in W 3, W 4, ec. However his classificaion is no suiable o describe non-markovian processes. For describing non-markovian processes, more random variables (oher han ξ() = ξ 1 (), ξ 2 (),.., ξ r () can be considered. Then by proper choice of hese addiional variables, one can have a Markov process for r random variables. 2 Markov Processes The Markov assumpion is formulaed in erms of condiional probabiliies as follows : P (x 1, 1 ; x 2, 2 ;... y 1, τ 1 ; y 2, τ 2,...) = P (x 1, 1 ; x 2, 2 ;... y 1, τ 1 ) (6) From he definiion of condiional probabiliy P (x 1, 1 ; x 2, 2 ; y 1, τ 1 ) = P (x 1, 1 x 2, 2 ; y 1, τ 1 )P (x 2, 2 y 1, τ 1 ) (7) 5

7 Using Markov propery one can wrie P (x 1, 1 ; x 2, 2 ; x 3, 3 ;...; x n, n ) = P (x 1, 1 x 2, 2 )P (x 2, 2 x 3, 3 )P (x 3, 3 x 4, 4 )...P (x n 1, n 1 x n, n )P (x n, n ) (8) provided n 1 n. 1. The Chapmann Kolmogorov Condiion The following condiion holds for all sochasic processes: P (x 1, 1 ) = dx 2 P (x 1, 1 ; x 2, 2 ) = dx 2 P (x 1, 1 x 2, 2 )P (x 2, 2 ) (9) Now P (x 1, 1 x 3, 3 ) = dx 2 P (x 1, 1 ; x 2, 2 x 3, 3 ) = dx 2 P (x 1, 1 x 2, 2 ; x 3, 3 )P (x 2, 2 x 3, 3 ) Inroducing he Markov assumpion, if 1 2 3, hen we can drop dependence on x 3, 3. Thus, P (x 1, 1 x 3, 3 ) = dx 2 P (x 1, 1 x 2, 2 )P (x 2, 2 x 3, 3 ) (1) This is he Chapmann-Kolmogorov Equaion. The differenial form of his equaion plays an imporan role in he descripion of sochasic processes ha follows. 2. The Fokker-Planck equaion From he definiion of ransiion probabiliy one can wrie W (x, + τ) = P (x, + τ x, )W (x, )dx (11) This connecs W (x, + τ) wih W (x, ). To obain a differenial equaion for he above, he following procedure can be carried ou. Assume ha all momens M n (x,, τ), n 1 are known M n (x,, τ) = [ξ( + τ) ξ()] n ξ()=x = (x x ) n P (x, + τ x, )dx 6

8 Inroducing = x x in equaion (11), he inegrand can be expanded in a Taylor series, which afer inegraion over δ can be pu in he following form: W (x, ) W (x, + τ) W (x, ) = + O(τ 2 ) = ( n x ) n=1 ( M n(x,, τ) ) W (x, ) (12) n! M n can be expanded ino Taylor series wih respec o τ (Risken, 1996) (n 1) M n (x,, τ) = D (n) (x, )τ + O(τ 2 ) n! Considering only linear erms in τ, Eqn. (12) can be wrien as where W (x, ) = n=1 ( x ) n D (n) (x, )W (x, ) = L KM W (x, ) (13) L KM = n=1 ( x ) n D (n) (x, ) The above is he Kramers-Moyal Expansion. Pawula Theorem saes ha for a posiive ransiion probabiliy P he Kramers-Moyal expansion sops afer he second erm; if no hen i mus conain an infinie number of erms. If he K-M expansion sops afer he second erm, hen i is called he Fokker -Planck equaion, given by W (x, ) = x D(1) (x, ) + 2 x 2 D(2) (x, ) (14) D (1) is called he drif coefficien, and D (2) is called he diffusion coefficien. The ransiion probabiliy P (x, x, ) is he disribuion W (x, ) for he iniial condiion W (x, ) = δ(x x ). Therefore he ransiion probabiliy mus also saisfy (13). Hence, P (x, x, ) = L KM (x, )P (x, x, ) (15) where he iniial condiion is given by P (x, x, ) = δ(x x ) 7

9 The Fokker-Planck equaion can also be wrien as W + S x = (16) where S(x, ) = [D (1) (x, ) x D(2) (x, )] W (x, ) Here S can be inerpreed as a probabiliy curren. We will discuss he one variable Fokker-Planck equaion wih ime-independen drif and diffusion coefficiens given by W (x, ) = x D(1) (x, ) + 2 x 2 D(2) (x, ) (17) For saionary processes he probabiliy curren S = consan. Mehods of soluion of such a F-P equaion for saionary processes are discussed in (Risken, 1996). Non-saionary soluions of he F-P equaion are in general difficul o obain. A general expression for nonsaionary soluion can be found only for special drif and diffusion coefficiens. The Fokker-Planck equaion can be aken as a saring poin for inroducing he concep of a sochasic differenial equaion. If he random variable ξ() saisfies he iniial condiion W (ξ(), y) = δ(ξ y) (18) ha is, i is sharply peaked a he value y, i can be shown by solving he Fokker-Planck equaion ha a shor ime laer, he soluion is sill sharply peaked, and is a Gaussian wih mean y + D (1) and variance D (2). The picure is ha of a sysem moving wih a sysemaic drif velociy D (1), and on his moion is superimposed a Gaussian flucuaion wih variance D (2). Thus, y( + ) = y() + D (1) + η() 1/2 (19) where η = and η 2 = D (2). This picure gives sample pahs which are always coninuous bu nowhere differeniable. As we will see shorly, his heurisic picure can be made much more precise and leads o he concep of he sochasic differenial equaion. 8

10 3. Wiener Process A process which is described by Eqn. (17) wih D (1) = and D (2) (x) = D = consan, is called a Wiener process. Then he equaion for ransiion probabiliy P = P (x, x, ) is he diffusion equaion, given by P = D 2 P x 2 (2) wih he iniial condiion P (x, x, ) = δ(x x ). Then he soluion for > is given by he gaussian disribuion P (x, x, ) = 1 4πD( ) exp ( (x x ) 2 4D( ) ) (21) Thus he general soluion for probabiliy densiy wih iniial disribuion W (x, ) is given by W (x, ) = P (x, x, )W (x, )dx An iniially sharp disribuion spreads in ime. The one-variable Wiener process is ofen simply called Brownian moion, since i obeys he same differenial equaion of moion as Brownian moion. 4. Ornsein-Uhlenbeck Process This process is described by Eqn. (17) when he drif coefficien is linear and diffusion coefficien is consan, i.e., D (1) (x) = γx, D (2) (x) = D = consan The Fokker-Planck equaion hen can be wrien as P = γ 2 (xp ) + D x x P (22) 2 wih iniial condiion P (x, x, ) = δ(x x ). The above equaion can be solved by aking a Fourier ransform w.r.. x i.e., P (x, x, ) = 1 2π e ikx P (k, x, )dk 9

11 Thus i resuls in he following equaion P = γk P k Dk2 P wih iniial condiion P (k, x, ) = e ikx for >. Then one ges P (k, x, ) = exp [ ikx e γ( ) Dk2 2γ (1 e 2γ( ) )] (23) By applying he inverse Fourier ransform one finally obains he soluion of he Fokker Planck equaion describing he Ornsein Uhlenbeck process P (x, x, γ ) = 2πD(1 e 2γ( ) ) exp [ γ(x e γ( ) x ) 2 2D(1 e 2γ( ) ) ] (24) In he limi γ we ge he Gaussian disribuion for a Wiener Process. Eqn. (24) is valid for posiive and negaive values of γ. For posiive γ and large ime difference γ( ) >> 1 he equaion passes over o he saionary disribuion given by W s = γ 2πD For γ no saionary soluion exiss. 3 Langevin Equaion exp [ γx2 2D ] (25) We inroduce sochasic inegraion via he Langevin equaion. In he presence of a viscous drag linearly proporional o velociy, he equaion of moion for a paricle of mass m is given by m v + αv = (26) or v + γv = 1

12 where γ = α/m = 1/τ, τ being he relaxaion ime. The soluion of he above equaion is given by v() = v()e /τ = v()e γ If he mass of he paricle is small, so ha he velociy due o hermal flucuaions is no negligible, hen v h = kt v 2 = m is observable and hence he velociy of he small paricle canno be described by Eqn. (26). Thus his equaion has o be modified as follows: v + γv = Γ() (27) where Γ() = F f ()/m is he sochasic erm, F f () is he flucuaing force acing on he paricle. This is he Langevin equaion. 1. Brownian Moion The Langevin equaion for Brownian moion is given by Eqn. (27) where Γ describes he Langevin force wih Γ() =, Γ()Γ( ) = qδ( ) such ha all he higher momens are given in erms of he wo poin correlaion funcion. In oher words, Γ is gaussian disribued wih zero mean. The specral densiy of noise, described below, gives color of he noise. The dela correlaed noise is referred o as whie noise. This model of noise in he Langevin equaion fully describes ordinary Brownian Moion of a paricle. The specral densiy for dela correlaed noise is given by S(ω) = 2q e iωτ δ(τ)dτ Since i is independen of ω i is called whie noise. In general he specral densiy depends on ω; in such a case, he noise is called colored noise. 11

13 2. Soluion of he Langevin equaion The formal soluion of he Langevin equaion is given by v() = v e γ + e γ( ) Γ( )d (28) By using he whie noise model one can obain he correlaion funcion of velociy v( 1 )v( 2 ) = v 2 e γ( 1+ 2 ) + which afer solving for he double inegral gives 1 2 e γ( ) qδ( 1 2)d 1d 2 (29) v( 1 )v( 2 ) = v 2 e γ( 1+ 2 ) + q 2γ (e γ 1 2 e γ( 1+ 2 ) ) (3) For large 1 and 2, i.e., γ 1, γ 2 >> 1 he correlaion is independen of v and is a funcion of he ime difference. v( 1 )v( 2 ) = q 2γ e γ 1 2 Now in he saionary sae for a Brownian paricle. E = 1 2 m v()2 = 1 2 m q 2γ According o he law of equipariion E = 1 2 kt and comparing wih he earlier expression we ge 3. Overdamped Langevin equaion q = 2γkT m The overdamped Langevin Equaion looks like v() = 1 γ Γ() 12

14 where he acceleraion erm is dropped because of he prominence of damping. In he large ime limi, he wo poin correlaion for velociy is given by v( 1 )v( 2 ) 1 γ 2 Γ( 1)Γ( 2 ) = q γ 2 δ( 1 2 ) 4. Non-linear Langevin Equaion A nonlinear Langevin equaion has he following form ξ = h(ξ, ) + g(ξ, )Γ() (31) Here Γ() is assumed o be Gaussian random variable wih zero mean and δ correlaion funcion. Inegraing Eqn. (31) Γ() =, Γ()Γ( ) = 2δ( ) ξ = h(ξ, )d + g(ξ, )Γ()d (32) Here g could be a consan or can depend on ξ. Consan g gives addiive noise, while if i depends on ξ i is referred o as muliplicaive noise. For ξ dependen g in he above equaion, since Γ() has no correlaion ime, i is no clear which value of ξ one has o use in g while evaluaing he inegral. Physiciss use an approximaion of he δ funcion o ensure ha one ges appropriae resuls. Bu from a purely mahemaical poin of view one canno answer his quesion, unless some addiional specificaion is given. Thus, his gives rise o he requiremen of using he sochasic inegrals, namely he Iô and Sraonovich inegrals for solving such equaions. In he following secion, we give a brief inroducion o hese Sochasic Inegrals. Prior o his we sae an example of a paricular form of noise and solve he nonlinear Langevin equaion for his paricular case. Example: For g = aξ where a is a consan ξ = aξγ() 13

15 The formal soluion of he above equaion is given by Assuming ξ() = x ξ() 7 x exp [a ξ() = x exp [a Γ( )d ] Γ( )d ] = x [1 + a 1 2! a2 Γ( 1 ) d 1 + Γ( 1 ) ] (33) Since Γ is dela correlaed Gaussian whie noise, all he higher correlaions can be expressed in erms of wo poin correlaions, herefore he above equaion can be wrien in he following form = (2n)! 2 n n! [ = exp [ 1 2 a2 n φ( 1 2 )d 1 d 2 ] (34) Γ( 1 )Γ( 2 ) d 1 d 2 ] (35) For dela correlaed Langevin force he double inegral gives and ξ() = x exp (a 2 ) ξ() = a 2 ξ() wih ξ() = x d d ξ() = = a2 x is called spurious or noise induced drif. 4 Sochasic inegraion, Iô calculus and sochasic differenial equaions Consider a Langevin equaion of he form dx d = a(x, ) + b(x, )ζ() (36) 14

16 for a ime-dependen variable x, where a(x, ) and b(x, ) are known funcions, and ζ() is he rapidly flucuaing random erm which induces sochasiciy in he evoluion. We wan o examine he mahemaical saus of his equaion as a differenial equaion. Since we expec i o be inegrable, he inegral u() = d ζ( ) (37) should exis. If we demand ha u is a coninuous funcion of, hen i is a Markov process whose evoluion can be described by a Fokker-Planck equaion, which can be shown o have zero drif, and diffusion uniy. Hence, u() is a Wiener process, denoed say by W (); bu we know ha W () is no differeniable. This would imply ha in a mahemaical sense he Langevin equaion does no exis! However, he corresponding inegral equaion x() x() = a[x(s), s]ds + can be inerpreed consisenly. We make he replacemen b[x(s), s]ζ(s)ds (38) dw () W ( + d) w() = ζ()d (39) so ha he second inegral can be wrien as b[x(s), s]dw (s) (4) which is a sochasic Riemann-Sieljes inegral, which we now define. Given ha G() is an arbirary funcion of ime and W () is a sochasic process, he sochasic inegral G( )dw ( ) is defined by dividing he inerval (, ) ino n sub-inervals ( i 1, i ) such ha 1 2 (41) and defining inermediae poins τ i such ha i 1 τ i i. The sochasic inegral is defined as a limi of parial sums S n = n i=1 G(τ i )[W ( i ) W ( i 1 ] (42) The challenge is ha, G() being funcion of a random variable, he limi of S n depends on he paricular choice of inermediae poin τ i! Differen 15

17 choices of he inermediae poin give differen resuls for he inegral. Denoing = max( i i 1 ), he Iô sochasic inegral is defined by aking τ i = i 1 and aking he limi of he sum: τ n G( )dw ( ) = lim G( i 1 )[W ( i ) W ( i 1 ] (43) i=1 As an example, i can be shown ha W ( )dw ( ) = 1 2 [W ()2 W ( ) 2 ( )] (44) Noe ha he resul for he inegraion is no longer he same as he ordinary Riemann- Sieljes inegral, where he erm ( ) would be absen; he reason being ha he difference W ( + ) W () is almos always of he order, which implies ha, unlike in ordinary inegraion, erms of second order in W () do no vanish on aking he limi. An alernaive definiion of he sochasic inegral is he Sraanovich inegral, denoed by S, and is such ha he anomalous erm above, ( ), does no occur. This happens if he inermediae poin τ i is aken as he mid-poin τ i = ( i + i+1 )/2, and i can hen be shown ha S = W ( )dw ( ) = 1 2 [W ()2 W ( ) 2 ] (45) For arbirary funcions G() here is no connecion beween he Iô inegral and he Sraanovich inegral. However, in cases where we can specify ha G() is relaed o some sochasic differenial equaion, a formula can be given relaing he wo differenial equaions. 1. Rules of Iô calculus Mean square limi or he limi in he mean is defined as follows: Le X n (ω) be a sequence of random variables X n on he probabiliy space Ω, where ω are he elemens of he space which have probabiliy densiy p(ω). Thus one can say ha X n o X in he mean square if lim dω p(ω)[x n (ω) X(ω)] 2 lim (X n X) 2 = (46) n 16

18 This is wrien as Rules ms lim n X n = X (a) dw () 2 = d (b) dw 2+N () = The above formulae mean he following [dw ( )] 2+N G( ) ms lim n i G i 1 W 2+N i = for an arbirary non-anicipaing funcion G. Proof: For N = consider he following sum I = lim n [ i = lim n i d G( ) for N = for N > (47) 2 G i 1 ( Wi 2 i )] (48) (G i 1 ) 2 ( Wi 2 i ) 2 + 2G i 1 G j 1 ( Wj 2 j )( Wi 2 i ) (49) i<j Using he following resuls menioned earlier one ges This can be wrien as W 2 i = i ( W 2 i i ) i I = 2 lim n [ i 2 i (G i 1 ) 2 ] ms lim (G i 1 W 2 n i G i 1 i ) i 17

19 since we ge ms lim G i 1 i = n d G( ) [dw ( )] 2 G( ) = d G( ) from his we see ha dw ( ) 2 = d. Similarly one can show ha dw () 2+N (N > ). Anoher imporan resul ha can be proved by he above mehod is G( )dw ( ) ms lim n G i 1 W i i = (5) Rule for inegraion of polynomials: W ( ) n dw ( 1 ) = n + 1 [W ()(n+1) W ( ) (n+1) ] n 2 W () (n 1) d (51) General rule for differeniaion; df[w (), ] = ( f f f ) d + dw () (52) 2 W 2 W 2. Sochasic differenial equaions The Iô inegral is mahemaically he mos saisfacory, bu no always he mos naural physical choice. The Sraanovich inegral is he naural choice for an inerpreaion where ζ() is a colored (no whie) noise. Also, unlike in he Iô inerpreaion, he Sraanovich inerpreaion enables he use of ordinary calculus. From he mahemaical poin of view, i is more convenien o define he Iô SDE, develop is equivalence wih he Sraanovich SDE, and use eiher form depending on circumsances. A sochasic quaniy x() obeys an Iô differenial equaion if for all and x() = x( ) + dx() = a[x(), ]d + b[x(), ]dw () (53) d a[x( ), ] + dw ( )b[x( ), ] (54) 18

20 If f[x()] is an arbirary funcion of x() hen Iô s formula gives he differenial equaion saisfied by f: df[x()] = (a[x(), ]f [x()] b[x(), ]2 f [x()]) d + b[x(), ]f [x()]dw () (55) Thus change of variables is no given by ordinary calculus unless f[x()] is linear in x(). Given he ime developmen of an arbirary f[x()], he condiional probabiliy densiy p(x, x, ) for x() can be shown o saisfy a Fokker-Planck equaion wih drif coefficien a(x, ) and diffusion coefficien b(x, ) 2. The sochasic differenial equaion sudied in deail in he QMUPL model dψ = [ i hhd + λ(q q )dw λ 2 (q q ) 2 d] ψ (56) is an Iô differenial equaion of he ype (53). If we formally rea his as an equaion for ln ψ hen upon comparison we see ha ln ψ is x, and a[x(), ] = i hh λ 2 (q q ) 2, b = λ(q q ) (57) The real par of he drif coefficien a is relaed o he diffusion coefficien b 2 by 2a = b 2 (58) This non-rivial relaion beween diffusion and drif is wha gives he equaion is norm-preserving maringale propery which evenually leads o he Born rule. Why he drif and diffusion mus be relaed his way is a presen no undersood in Trace Dynamics and here probably is some deep underlying reason for his relaion. Sraonovichs sochasic differenial equaion : Given he Iô differenial equaion (53) is soluion x() can also be expressed in erms of a Sraonovich inegral x() = x + d α[x( ), ] + S dw ( )β[x( ), ] (59) 19

21 where α(x, ) = a(x, ) 1 2 b(x, ) xb(x, ), β(x, ) = b(x, ) (6) In oher words, he Iô SDE (53) is he same as he Sraonovich SDE dx = [a(x, ) 1 2 b(x, ) xb(x, )] d + bdw () (61) or conversely, he Sraonovich SDE is he same as he Iô SDE dx = αd + βdw () (62) dx = [α(x, ) β(x, ) xβ(x, )] d + βdw () (63) I can be shown ha in a Sraonovich SDE he rule for change of variable is he same as in ordinary calculus. 5 Maringales Maringales play a role in sochasic processes roughly similar o ha played by conserved quaniies in dynamical sysems. Unlike a conserved quaniy in dynamics, which remains consan in ime, a maringale s value can change; however, is expecaion remains consan in ime. A maringale is defined as follows: A discree ime maringale is a discree ime sochasic process, X 1, X 2,..., ha saisfies for any ime n, E( X n ) <, E(X n+1 X 1,..., X n ) = X n where E(X) denoes he expecaion of X. Maringale Sequence wih Respec o Anoher Sequence: A sequence Y 1, Y 2,... is said o be a maringale w.r.. anoher sequence X 1, X 2, X 3,... if for all n E( Y n ) <, E(Y n+1 X 1,..., X n ) = 2

22 In oher words, a maringale is a model of a fair game, where no knowledge of pas evens can help o predic fuure winnings. I is a sequence of random variables for which a a paricular ime in a realized sequence, he expecaion of he nex value in he sequence is equal o he presen observed value even given knowledge of all prior observed value a curren ime. References Arnold, L. (1971), Sochasic Differenial Equaions: Theory and Applicaions (Wiley, New York). Chandrasekhar, S. (1943), Rev. Mod. Phys. 15, 1. Gardiner, C. W. (1983), Handbook of Sochasic Mehods for Physics, Chemisry and he Naural Sciences (Springer-Verlag). Gikhman, I. I., and A. V. Skorokhod (1972), Sochasic differenial equaions (Springer Verlag, Berlin). Grimme, G. R., and D. Sirzaker (21), Probabiliy and random processes, 3rd ed. (Oxford Universiy Press). Risken, H. (1996), The Fokker-Planck Equaion: Mehods of Soluions and Applicaions (Springer Series in Synergeics). 21