Path Integral methods for Stochastic DifferentialEquations

Carson C Chow and Michael A Buice

TNJC 11th December 2012

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 1 / 36

I Introduction

MotivationStudy SDEs of the form

dxdt

= f (x) + g(x)(t)

Want to know moments (e.g. x(t), x(t)x(t )) and probability densityfunction (pdf, p(x , t))Can use Langevin and Fokker-Planck equations to study, but perturbationmethods can be difficult to apply

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 2 / 36

I Introduction

Outline2 Introduce moment generating functionals (Z []), distribution of functions

(P[x(t)])3 Path integrals to compute moment generating functional of SDE, using

Ornstein-Uhlenbeck process as example4 Perturbation methods using Feynman diagrams5 Derive equations for the density p(x , t)

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 3 / 36

II Moment generating functionals

Moment generating functionFor a single random variable X , the moments (X =

xnP(x) dx) are

obtained from the MGF

Z () = ex =

exP(x) dx

by taking derivatives

X n = 1Z (0)

dn

dnZ ()

=0

MGF contains all information about RV X , alternative to studying the pdfdirectly.

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 4 / 36

II Moment generating functionals

Cumulant generating functionDefine W () = log Z (), then

X nC =dn

dnW ()|=0

are the cumulants of RV XAs with MGF, contains all information about X , and is sometimes moreconvenient. For example

X C = X X 2C = X 2 X 2 = var(x) = 2nd central momentX 3C = X 3 3X 2X + 2X 3 = 3rd central moment

...

Higher order cumulants are neither moments or central moments

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 5 / 36

II Moment generating functionals

Moment generating function(al)For an n-dimensional random variable x = (x1, . . . , xn), the generatingfunctional is

Z () = ex = n

i=1

dxiexP(x)

for = (1, . . . , n).k th order moments are obtained via

ki=1

x(i)

=

1Z (0)

ki=1

n

(i)Z ()

=0

As before, the cumulant generating function is W () = log Z ()

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 6 / 36

II Moment generating functionals

Moment generating functionalIdentify with each xi in x a time, t = ih, such that xi = x(ih) and let totaltime T = nh, splitting interval [0,T ] into n segments of length hTake limit n (with h = T/n) such that xi x(ih) = x(t), i (t)and P(x) P[x(t)] = exp(S[x(t)]) for some functional S[x ], called theaction

Instead of summing over all points in Rn( n

i=1

dxi

), sum over all

curves(Dx(t)

), giving the MGF:

Z [] =Dx(t) eS[x ]+

(t)x(t) dt

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 7 / 36

II Moment generating functionals

Moment generating functionalNote: inner product becomes x y

x(t)y(t) dt

Moments can be obtained viak

i=1

x(t(i))

=

1Z [0]

ki=1

(t(i))Z []

(t)=0

Cumulant generating functional again

W [] = log(Z [])

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 8 / 36

II Moment generating functionals

The functional derivative F []

Extension of directional derivative for functions f : Rn R:

v f (x) = f v = lim0

f (x + v)

gives rate of change in direction vector v at point x.Can compute using a test function f (x)

F, f (x)

=

dd

F [+ f ]=0

Example:

W [] =

122(t) dt ;

W

= (t)

F [] = e(x)g(x) dx ;

F

= g(x)e(x)g(x) dx

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 9 / 36

II Moment generating functionals

Gaussian RVs in one dimensionRV X N(a, 2) has MGF

Z () =

exp[(x a)2

22+ x

]dx =

2 exp(a + 22/2),

obtained by completing the square.And has cumulant GF

W () = a +1222 + log(Z (0))

so cumulants are xC = a, x2C = var X = 2 and xk C = 0 for allk > 2.

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 10 / 36

II Moment generating functionals

Gaussian RVs in n dimensionsThe n dimensional RV X N(0,K ), with covariance matrix K , has MGF

Z () =

e12

jk xj K1jk xk +

j j xj dx

Since K is symmetric positive definite (then so is K1) we candiagonalise in orthonormal coordinates, giving

Z () = [2 det(K )]n/2e12

jk j Kjkk

Infinite dimensionalMGF is

Z [] =Dx(t)e 12

x(s)K1(s,t)x(t)dsdt+

(t)x(t)dt = Z [0]e

12

(s)K (s,t)(t)dsdt

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 11 / 36

II Moment generating functionals

Gaussian RVs and Wicks theoremRelate higher order central moments of multivariate normally distributed xto products of second order central moments

ki=1

x(i)

=

{0, k oddA K(1)(2)K(3)(4) K(k1)(k), k even

for A = {all pairings of x(i)}. For k = 2l sum will contain(2l 1)!/[2l1(l 1)!] terms.Example: for k = 4

x1x2x3x4 = K12K34 + K13K24 + K14K23

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 12 / 36

II Moment generating functionals

Quantum mechanicsSum over paths x(t): K (a,b) =

Dx(t)e2iS[x ]/h where |K (a,b)|2 gives the

probability particle with action S = tb

taL(x(t), x(t), t) dt travels from point a to

b.

Quantum field theorySum over fields (x, t):

S[] =(t)K1(t, t)(t)dd tdd t + g

4(t)dd t

Statistical mechanicsThe sum over all states Z =

q eS[q] is called the partition function. Z [J] is

the partition function of QFT

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 13 / 36

II Moment generating functionals

Quantum mechanicsWhen S is large compared with h/2 the integral

Dx(t)e2iS[x ]/h is a rapidly

oscillating exponential method of stationary phase says only curves for

whichSx

= 0 contribute principle of least action in classical mechanics

Quantum field theoryFor the case where we can describe state of system in terms ofmechanical variables (e.g. atoms in a periodic latte, crystal), each point isdescribed by a quantum harmonic oscillator. If sensible to take acontinuum approximation quantum field theory. Higher modes, excitedstates, in the coupled oscillators have particle like behaviour.In other cases, each point in space is described other variables, e.g.electromagnetism, but may still be quantised as oscillators. Excitedstates of these fields are called bosons, such as the photon.

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 14 / 36

III Application to SDEs

Repeat for an SDEConstruct generating functional for SDEs of the form

dxdt

= f (x , t) + g(x)(t) + y(t t0),

for t [0,T ].Discretize in time steps h (Ito interpretation)

xi+1 xi = fi (xi )h + gi (xi )wi

h + yi,0

Each wi is Guassian with wi = 0 and wiwj = ij

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 15 / 36

III Application to SDEs

Probability generating functionalPDF given the random walk wi :

P[x |w ; y ] =n

i=0

[xi+1 xi + fi (xi )h gi (xi )wi

h yi,0]

Take Fourier transform

P[x |w ; y ] = N

j=0

dkj2

ei

j kj (xj+1xjfj (xj )hgj (xj )wj

hyj,0)

Using the law of total probability and completing the square:

P[x |y ] = N

j=0

dkj2

e

j (ikj )( xj+1xj

h fj (xj )yj,0/h)

h+

j12 g

2j (xj )(ikj )

2h

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 16 / 36

III Application to SDEs

Continuum limit

Again let h 0 with N = T/h, replace iki with x(t) andxj+1 xj

hwith

x(t):

P[x(t)|y , t0] =Dx(t)e

[x(t)(x(t)f (x(t),t)y(tt0)) 12 x

2g2(x(t),t)]dt

x(t) represents wave number k , proportional to momentum p, thus canwrite down a moment generating functional for position andmomumentum space:

Z [J, J] =Dx(t)Dx(t)eS[x,x ]+

Jx dt+

Jx dt

Typo in set of equations below (6)?

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 17 / 36

III Application to SDEs

More generally...Instead of g(x)(t), with (t) white noise, an SDE having noise processwith cumulant W [(t)] will have PDF:

P[x(t)|y , t0] =D(t)[x(t) f (x , t) (t) y(t t0)]eS[(t)]

=

D(t)Dx(t)e0

x(t)(x(t)f (x,t)y(tto)) dt+W [x(t)]

If (t) is delta correlated ((t)(t ) = (t t )) then W [x(t)] can beTaylor expanded in both x(t) and x(t):

W [x(t)] =

n=1,m=0

vnmn!

xn(t)xm(t) dt

summation over n starts at one because W [0] = log(Z [0]) = 0? delta correlated means no mixed derivative terms in finite dimensionalequivalent?

Ben Lansdell () Path Integrals and SDEs TNJC 11th December 2012 18 / 36

A. Ornstein-Uhlenbeck Process

For example...The OU process has the action

S[x , x ] = [

x(t)(x(t) + ax(t) y(t t0))D2

x2(t)]

dt

G and OUs moments could found immedately, since action is quadratic,demonstrate perturbation method to motivate the method more generallyBreak action into free and interacting terms. Free terms wouldrepresent a particle without any interaction with a field or potential, andwould have a quadratic action

Ben Lansdell () Path Integrals and SDEs