Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory

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1 Random Operators / Stochastic Eqs. 15 7, 5 c de Gruyter 7 DOI / ROSE.7.13 Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory Yuri A. Godin and Stanislav Molchanov Communicated by V. L. Girko Abstract. Approximation of one-dimensional stochastic differential equations and their additive functionals by dynamical systems with piecewise-constant random coefficients is obtained. We calculate asymptotic expansion of solution in terms of the step of discretization. Key words. Itô s stochastic equations; discretization; Lyapunov exponent; density of states. AMS classification. 58J5, 35J1, 35P99, 8B44, 35P5. 1. Introduction This paper was motivated by the analysis of one-dimensional Schrödinger operator d + ξt, dt with the white noise potential ξ. In this case one can introduce the usual phaseamplitude representation ψ = r λ sin ϑ λ, ψ = r λ cosϑ λ or use the Prüfer phase 1] for the solution of the Cauchy problem Hψ = λψ and get a stochastic differential equation SDE for ϑ λ t. The white noise potential here can be understood in several senses, including the Dirichlet form or Stratonovich SDE. Physical applications, however, are always dealing not directly with the white noise but rather with its regularization, i.e., a stationary Gaussian process having almost δ- correlator and, equivalently, a wide band spectrum. As an approximation, stochastic differential equations are obtained in the limit. To estimate the quality of this approximation, we have to calculate the remainder terms either in SDE for the phase or in stochastic integrals for the associated additive functional related to the Lyapunov exponent or the density of states, see ] 4]. Similar to ODEs, there exists an extensive literature on numerical methods for SDEs see for example 5] and references therein. However, unlike ODEs, we start from a dynamical system with discrete randomness that is more adequate physically, and approximate it or its functional by an SDE and stochastic integral. In doing so, we

2 6 Y. A. Godin and S. Molchanov consider only one-dimensional case. All calculations are performed for general equations in order to make them more transparent. The obtained result can also be applied in Monte Carlo simulation of SDEs or dynamical systems with discrete randomness. The second part of the paper will be devoted to the transition from localization pure point spectrum to the absolutely continuous spectrum in the spirit of the classical paper 6] and recent publications on the fast oscillating piecewise constant potentials 7] 8].. Description of the model Let w s = ws be the standard Brownian motion whose increment w n on the intervals of length 1 is a small parameter are independent N, random variables, w n = wn + 1 wn, n, w =. The simplest regularization ξ t of the white noise ξt = ẇt has the form ξ t = wn, t n,n + 1, n..1 If the origin t = is uniformly distributed on, ], the process ξ t will be a stationary Gaussian process with the correlator 1 t b t = E ξ ξ + t =, b t δ t if. in the sense of distributions. In what follows we put ξ t = w, t,..3 Assume now that Ax,Bx,x R, are smooth bounded functions. We consider the dynamical system ẋt = ξ tax + Bx..4 On each interval t n,n + 1] equation.4 has the form of usual ODE ẋt = wn Ax + Bx..5 The sequence xn, n, is the main goal of our study. The r.v. x n form usual Markov chain on R and its analytical properties are not simple. We prefer to work with SDE which has symbolically the form for dxt = Axtdw t + Bxtdt..6

3 Dynamical system with discrete randomness 7 Our goal is the analysis of the remainder in equation.6 which must be understood in the Stratonovich form 9]. We also want to study approximation by the stochastic integral of the following additive functional of x. Let i.e., for t = n y n = y n = ẏ t = ξ tfx t + Gx t,.7 n 1 k= w k k+1 k Fx s ds + k+1 In the first part of the paper we will construct the asymptotic expansion for or δxn = xn + 1 xn δyn = yn + 1 yn k Gx s ds..8 up to the order o using standard agreement that w = O 1/, w = O 3/, etc., since Ew =. Proposition.1. Let t = n,xs, ] be the solution of equation.6 on the interval n,n + 1]. Then for w = w n δx t = x t + x t = w A + B + w + 1 AA + 1 w A B + AB w3 A A + AA + 1 BB w AA B + AA B + A B + A B w4 A 3 A + 4A A + AA 3 + o..9 The remainder R in this formula has the form ER = E δx t w A w4 A 3 A + 4A A + AA 3] O 5, i.e., in fact R O 5/ in probability..1 The proof is based on direct calculations. Since solution xs, s, ], is a smooth function of the initial condition x and time t, it can be expanded into a Taylor series xt + s = x + ẋts xs ẍts 6 4 x4 s 4 + os 4..11

4 8 Y. A. Godin and S. Molchanov But ẋt + s = w Axt + s + B xt + s,.1 and for s = we have ẋt = w Ax + B x..13 Similarly, ẍ = w A + B ẋ = w A + B w A + B.14 and so on. Then.9 is obtained by substitution of these formulas in.11 and truncating the terms of the order O α, α 5/. In the case of additive functional given by.7 calculations are similar: δy t = y n+1 y n = y n + 1 yn ] n+1 w n = Fx s + Gx s ds n = ẏn + 1 ÿn yn y4 n + os For t = n, x = x n, w = w n the increments of the derivatives can be calculated as follows The final result represents Proposition.. ẏ = w Fx + Gx ÿ = w F x + G x ẋ = w F x + G x w Ax + Bx = w AF + w F B + AG + BG. δy t = w Fx + Gx + 1 w AF + 1 w BF + AG + 1 BG w3 A F + AA F w ABF + A BF + AB F + AA G + A G w4 A 3 F + 3A A F + A A F + AA F + o..16

5 Dynamical system with discrete randomness 9 Observe that both δx n and δy n are polynomials of only one Gaussian random variable w n, that makes numerical simulation Monte Carlo of the dynamical system x and the additive functional y very simple in contrast to simulation of SDEs see below. Consider a general SDE on the probabilistic space of the same Brownian motion W s,s, as our dynamical system x t dx t = αx t dw t + βx t dt, y = x,.17 with smooth coefficients α and β. We will select them in the form from the condition that the expression E δx n δxn ] αx = α x + α 1 x + o,.18 βx = β x + β 1 x + o,.19 x = E n + 1 xn + 1 xn=xn=x]. attains minimum value. On the interval t,t + ] = n,n + 1], with condition xn = x equation.17 for xs, s t, t + ], has integral Itô s form xs = x + αx s dw s + which can be solved by successive approximations as follows: x s = x, x 1 s = x + αxw s + βxs, x s = x + αx + αxw u + βxudw u + = x + αxw s + βxs + α x + β x αx + αxβx + 1 β xα x αx βx s ds,.1 βx + αxw u + βxudu, w u dw u + βx w u du + 1 βxs ] + 1 α x uw u dw u ] w u du α xα 3 x α x udw u ] w u dw u w 3 u du + o..

6 1 Y. A. Godin and S. Molchanov The third approximation gives the estimate to the order O One can easily see x 3 s = x + x 3 s = x s + α x + + α xβx + αxα xβ x + α xα xα x αx udw u + αxα x u ] v dw v dw u u w v dw v du βx udu..3 u w v dw v dw u u w u w v dw v dw u + o..4 Arranging in.,.4 the terms in their order in, we obtain Proposition.3. Expansion of δxn for x = xn, s, ] has the form δxn = xn + 1 xn = αxw + βx + αxα x + α xβx sdw s + αxβ x w s ds + 1 α xα x + αxα x w u dw u dw s + 1 βxβ x + αxα xβx α3 xα x + αxα xβ x + α xα xα x sw s dw s + 1 α xβ x w 3 s dw s + α x βx w u dw u ds ws w u dw u w s sw s + o. w s dw s w s dw s ds.5 udw u dw s Stochastic integrals in.5 can be written in the form containing random variable w and uncorrelated terms or even independent on w. Proposition.4. Integrals in.5 can be transformed using the formulas below. Proofs are based on the Itˆo formula and integration by parts.

7 Dynamical system with discrete randomness 11 a b c d e f g h i w s dw s = 1 w sw s dw s = 1 w sdw s = w udw u w s ds dw s = w 4 3 ws dw s = 1 3 w3 w s ds w u dw u dw s = 1 6 w3 1 w w s ds w s ds ws 3 dw s = 1 4 w4 3 ws ds w u dw u ds = 1 ws ds 1 4 w u dw u dw s = 1 4 w4 3 4 w w s ds.6 Observe that along with and w, δxn contains additional random variables w s ds and w s ds. These random variables have a joint Gaussian distribution with covariance matrix cov = Hence, standard calculation gives w s dw s = 1 w + where w s is an independent copy of w s, and it easy to see that E w s ds 1 w w = w,.8 and var w s ds = var 1 w + var 3 6 w = 3 3.

8 1 Y. A. Godin and S. Molchanov Note also that w is uncorrelated with all terms of order O in formulas.5.6. Indeed, after the transformation w s w s the terms of order O preserve their sign. For example, in.6 Ew s Therefore, w 3 s dw s = E w s Ew s w s 3 d w s = Ew s w 3 s dw s =. w 3 s dw s..9 Now we are ready to find optimal expressions for the diffusion coefficients αx and βx in in terms of the coefficients Ax and Bx of the dynamical system.4. To this end, we have to compare expansions in Proposition.1 with those in Proposition.3, taking into account formulas.6. Since for t = n and x t + x t = δx = Axw + Bx + AxA x + w AxA x + o.3 xt + xt = δx = αxw + βx + w we have Thus, we put αx = Ax + O, βx = Bx + 1 AxA x + o. αxα x + o,.31 α x = Ax,.3 β x = Bx + 1 AxA x..33 Note that the drift coefficient βx contains the Stratonovich term 1 AxA x. That means that in the transition from the difference relation δx t = w Axt + Bxt, t = n,.34 when we obtain in the leading term not the Itô quation dxt = Axtdw t + Bxtdt,.35

9 Dynamical system with discrete randomness 13 but rather the Stratonovich one dxt = Axt dw t + Bxtdt = Axtdw t + Bxt + 1 AA xt dt..36 In order to find next corrections α 1 and β 1, we put αx = α x + α 1 x = Ax + α 1 x + o, βx = β x + β 1 x = Bx + 1 AxA x + β 1 x + o. Compare now the terms of order O 3/ in.9 and.5. In.9 they are while in.5 we have 1 w A B + AB w3 A A + AA.37 w α 1 x + α β + 1 α α sdw s + α β w s ds ws dw s + α α w u dw u dw s..38 Using formulas.6 we transform the last expression to the following 1 α α + α α + w α β 1 α α 6 w3 + α β α β 1 α α w s ds..39 The terms containing w 3 are the same in both expressions.37 and.39. Comparison of the coefficients of w with.8 gives or α 1 x+α β + 1 α α + 1 α β α β 1 α α = 1 A B + AB.4 α α β + α β 1 4 α α + α α = α A B + 1 AA + 1 B A + 1 AA + A 1 AA + A A 4 = α A B + AB from.37 = 1 A B + AB..41

10 14 Y. A. Godin and S. Molchanov Thus, α 1 x =..4 However, we have a discrepancy between.37 and.39. The expression 3 6 w α β α β 1 α α = 3 6 AB A B. It gives contribution to the diffusion coefficient of the order O. Indeed, Dx = 1 varaxw + w Dx = A x + D 1 x = A x + O. Now we can find correction β 1 x to the drift coefficient.43 βx = Bx + 1 AxA x + β 1 x = β x + β 1 x..44 Let us take expectations in.9 and. and compare the terms of order O. In.9 such terms are In. they are 1 BB AA B + AA B + A B + A B A 3 A + 4A A A + AA β 1 x + 1 β β α β = β 1 x + 1 B + 1 AA B + 1 AA + A A B + 1 ] A A + AA,.46 where we used the identities E Ew 4 =, w ds = 1. From one can find the following expression for β 1 : β 1 x = 1 AA B A B + AA B A B Thus, we have proved the following

11 Dynamical system with discrete randomness 15 Theorem.5. The best approximation to the order of the dynamical system ẋ = ξ tax t + Bx t, by the SDE is given by Here ξ t = wn, t n,n + 1],.48 dxt = αx t dw w + βx t dt.49 αx = Ax + o,.5 βx = Bx + 1 AA + 1 E δx n δxn ] AA B A B + AA B A B + o. = E x n + 1 x n xn + 1 xn ] = O 4,.51 that is, x t xt = O in probability. The differential operator associated with the diffusion process xt has the form A = 1 α x d dx + βx d ] 1 d dx = A x + o dx.5 + Bx + 1 AA x + AA B A B + AA B A B ] d + o 1 dx. 3. Approximation of the additive functional Let us apply the techniques developed in the previous section to approximation of the additive functional y t = y + t ξ sfx sds + t Gx sds, 3.1 δy t = y n + 1 y n, t = n. 3. We have to expand δyt = yn + 1 yn

12 16 Y. A. Godin and S. Molchanov with respect to the diffusion process xt δyt = fδxsdw s + gδxs ds, 3.3 and then find fx = f x + f 1 x + o, 3.4 gx = g x + g 1 x + o, 3.5 from the condition that E δy δy = min. 3.6 Calculation of δyt is straightforward δyn = fx + δxsdw s + gx + δssds, 3.7 where δxs = αxw s + βxs + αxα x + αxβ x w u du + 1 α xα x w u dw u + α xβx w u du udw u u + αxα x dw u w v dw v + o 3.8 and αx = Ax + o, βx = Bx + 1 AxA x + β 1 x. 3.9

13 Dynamical system with discrete randomness 17 Evaluation of the first integral in 3.7 gives = fx + δxsdw s fx + f xδxs + 1 f xδ xs + 1 ] 6 f xδ 3 xs + o dw s = fxw + f x αx + αxα x + αxβ x w s dw s + βx dw s w u dw u + α xβx dw s sdw s w u du + 1 α xα x dw s udw u dw s wu dwu + αxα x s u dw s dw u w v dw v +o ]+ 1 f α x + αxβx f xα 3 x sw s dw s +α xα x ] w s dw s w u dw u + o w s dw s w 3 s dw s + o. 3.1 The second integral in 3.7 is expanded as follows gx + δxsds = gx + g x αx + αxα x + 1 g x α x ds w s ds + 1 βx ] w u dw u +o ] ws ds + o Similar to the case of xt, we have δyt = fxw + gx 1 αxf x + 1 f xαxw + o. 3.1 Due to Proposition., δy t = Fxw + Gx + 1 AxF xw + o. 3.13

14 18 Y. A. Godin and S. Molchanov This means that fx = Fx + o = f x + o, gx = Gx + 1 AxF x + o = g x + o, 3.14 f x = Fx, g x = Gx + 1 AxF x. Let us find the next corrections fx = f x + f 1 x + o, 3.15 gx = g x + g 1 x + o The terms of order O 3/ in Proposition. are the following 1 w AxG x+bxf x w3 A xf x+axa xf x Using Proposition.3 we can find terms of the same order in w f 1 x + f xβ x sdw s + αxα xf x + 1 f xα x ws dw s + α xg x w s ds = w f 1 x + f β w + 1 w f xα 3 x 3 dw s w u dw u w w s ds + α xα xf x 3 6 w w s ds + α xg x w s ds

15 Dynamical system with discrete randomness 19 = w f x + f xβ x 1 α xα xf x w3 + α xα xf x + α xf x w s ds α xg xβ xf x 1 f xα x = w f 1 x + f xβ x 1 α xα xf x + 1 α xg x 1 β xf x 1 4 f xα x w3 + α xα xf x + α xf x 3 6 w α xg x β xf x f xα x The terms containing w 3 in are the same, f x = Fx, α x = Ax Thus, for determining f 1 x we obtain the equation f 1 x F x Bx + 1 AxA x + Ax Gx + 1 AxF x AxA xf x A xf x 4 = AxG x + BxF x Hence, as in the previous case, f 1 x =. 3. In order to find the correction to the drift g 1 x, we have to compute the mean values of δyt and δy t at t = n. Eδy t = Gx + 1 axf x + 1 BxG x + AxBxF x 6 + AxB xf xa xbxf xaxa xg x + A xg x + A 3 xf x + 3A xa xf x + AxA x F x 8 + A xa xf x + o 3.1

16 Y. A. Godin and S. Molchanov and Eδyt = g x + g 1 x + g xβ x + α xg x 4 = + + g 1 x Gx + AxA x G x + A xf x + AxF x Bx + AxA x + 4 From the relation one can find expression for g 1 x G + A xf x + A xf x + AxF x Eδy t = Eδyt. 3. g 1 x = 1 AxBxF + AxB xf x A xbxf x AxA xg x A xg x. 3.3 Theorem 3.1. The best approximation of the additive functional δy t by the stochastic integral of xt has the form for δyt = fx = Fx + o, fx s dw s + gx s ds 3.4 gx = Gx + 1 AxF x + g 1 x + o, 3.5 g 1 x = 1 AxBxF + AxB xf x A xbxf x AxA xg x A xg x. Similar to Theorem.5, E δy n δyn = O 4, 3.6 that is, δy n δyn = O in probability.

17 Dynamical system with discrete randomness 1 Let dx t = αx t dwt + βx t dt 3.7 be a stochastic equation and yt = y + t fx s dw s + t gx s ds 3.8 be a linear functional. Below we formulate a proposition regarding the expectation λ y + ut,x = E x e t fx s dw s + which is a combination of Kac Feynman and Girsanov s formulas. t ] gx s ds, 3.9 Proposition 3.. The function ut, x is a solution of the following parabolic problem ut, x t = α u,x = e λy. To prove 3.3 observe that λ y + ut +,x = E x e That means u x + β + λfα u x + λg + λ fα u, 3.3 t fx s dw s + t ] gx s ds ut,x = E x e λαxfxw + λgx + f α w ] + o ut,x = E x 1 + λαfw + λgx + f α w + 1 λ α f w + o ] ut,x + u x αw + β + 1 ] u xx α w + o ut +,x ut,x Passing here to the limit we obtain 3.3. = λgxu + u xβ + λα fu x + 1 u xxα + 1 λ α f + o. 3.3

18 Y. A. Godin and S. Molchanov References 1. P. Hartman, Ordinary Differential Equations. John Wiley, New York, R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger operators. Birkhäuser, Boston, L. Pastur, A. Figotin, Spectra of Random and Almost Periodic Operators, Springer-Verlag, New York, S. A. Molchanov, Ideas in the theory of random media. Acta Appl. Math. 1991, P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York, S. Kotani, N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials. Communications in Mathematical Physics , A. Laptev, S. Naboko, O. Safronov, Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials. Communications in Mathematical Physics 53 5, J. Holt, S. Molchanov, O. Safronov, Cwikel and quasi-szegö type estimates for random operators, to appear in Communications in Partial Differential Equations. 9. I. I. Gihman, A. V. Skorohod, Stochastic Differential Equations. Springer-Verlag, New York, 197. Received 1 May, 6; revised 5 March, 7 Author information Yuri A. Godin, Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 83, USA. ygodin@uncc.edu Stanislav Molchanov, Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 83, USA. smolchan@uncc.edu