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

model of share prices

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

model of share prices Start with equation in differential form:

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:

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

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:

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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 + ½σ2

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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 + ½σ2

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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 + ½σ2

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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 + ½σ2 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 + ½σ2 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 - ½σ2, 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 ½σ2).

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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 ½σ2). 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 – gx3 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 – gx3 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 – gx3 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 G0: in time domain:

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**some definitions and notation**

Write this as where multiply by G0: in time domain: in frequency domain:

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**some definitions and notation**

Write this as where multiply by G0: 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: : x0 = + : x : G0 : -g

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iterate diagrams: = + 3 h

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iterate diagrams: = + 3 : ξ h

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iterate diagrams: = + 3 : ξ h +9 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:**

o o: = + 3

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**averaging the diagrams:**

o o: = + 3 o o +9

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**averaging the diagrams:**

o o: = + 3 o o +9 o +18 + … o

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correlation function o

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correlation function o o

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in algebra, o = + 3 o o +9 o +18 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 o o Σ = = 3 + 6 + … o

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**1st-order approximation**

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**1st-order approximation**

or, in time domain,

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**1st-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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**1st-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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**1st-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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**1st-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 C0 and G0 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 C0 and G0 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 C0 and G0 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 solution**

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**self-consistent solution**

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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