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Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito calculus method small noise correlation time method (Stratonovich only) solution using Fokker-Planck equation Perturbation theory for nonlinear Langevin equations diagrammatic expansion self-consistent approximations

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention

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model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t => x(t) is log-normally distributed: Start with equation in differential form: Apply Ito’s lemma with model of share prices

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“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t => x(t) is log-normally distributed: Start with equation in differential form: Apply Ito’s lemma with model of share prices

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

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moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2 moments:

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geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2 moments:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model.

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) :

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

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GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables: as we got before

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GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written

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GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written

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GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables:

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GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables: y(t) is Gaussian with

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GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables: y(t) is Gaussian with as obtained from working with differentials and using Ito’s lemma

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GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention

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GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written

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GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written Here:

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GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written Here: But this is just the same equation as in the Stratonovich case, excpt for a reduced drift r - ½σ 2, in agreement with what we found using differentials and the Ito lemma.

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Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.

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Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other.

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Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other. Ito and Stratonovich problems are different (Stratonovich has a drift rate larger by ½σ 2 ).

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Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other. Ito and Stratonovich problems are different (Stratonovich has a drift rate larger by ½σ 2 ). I have shown this here for GBM, but it is true in general (except for constant G(x), in which case Stratonovich and Ito are equivalent).

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Perturbation theory for nonlinear Langevin equations (now back to additive noise)

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Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F :

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Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F :

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Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 :

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Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 :

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Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 : add an external driving force:

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some definitions and notation Write this as

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some definitions and notation Write this as where

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some definitions and notation Write this as where

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some definitions and notation Write this as where

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some definitions and notation Write this as where multiply by G 0 : in time domain:

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some definitions and notation Write this as where multiply by G 0 : in time domain: in frequency domain:

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some definitions and notation Write this as where multiply by G 0 : in time domain: in frequency domain: notation:

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iteration of equation of motion Define

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iteration of equation of motion Define equation of motion:

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iteration of equation of motion Define equation of motion:

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iteration of equation of motion Define equation of motion:

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iteration of equation of motion Define equation of motion: diagrammatic representation: key: =+ : x 0 : x : G 0 : -g

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iterate diagrams: =+ 3 h

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iterate diagrams: =+ 3 h : ξ

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iterate diagrams: = h : ξ h + …

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averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem)

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averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc.

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averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc. Define the Green’s function

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averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc. Define the Green’s function

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averaging the diagrams: =+ 3 o o:o:

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averaging the diagrams: = o o o o:o:

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averaging the diagrams: = o o o o o + … o:o:

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correlation function o

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o o

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in algebra, = o o o o o + …

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“self-energy” (“mass operator”) =+ + + …

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“self-energy” (“mass operator”) =+ + + … =+

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“self-energy” (“mass operator”) =+ + + … =+ Dyson equation

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“self-energy” (“mass operator”) =+ + + … =+ Dyson equation or

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“self-energy” (“mass operator”) =+ + + … =+ Dyson equation or = 3 o + 6 o o + … Σ =

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1 st -order approximation

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or, in time domain,

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1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G.

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1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ :

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1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ :

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1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ : increase in damping constant:

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Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams.

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Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams. lowest-order approximation (Hartree):

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Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams. lowest-order approximation (Hartree): Σ = … o o o o o o o o o o

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self-consistent solution

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self-consistent equation

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self-consistent solution self-consistent equation solution:

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self-consistent solution self-consistent equation solution: This solution is approximate. But it is exact if x is a vector with n components, with in the limit n -> ∞.

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