THE FOKKER-PLANCK EQUATION - Imperial College …pavl/lec_fokker_planck.pdf · Existence and...

THE FOKKER-PLANCK EQUATION

Existence and Uniqueness of Solutions for the FP Equation

• Consider a diffusion process on Rd with time-independent driftand diffusion coefficients. The Fokker-Planck equation is

∂p

∂t= −

d∑

j=1

∂

∂xj(ai(x)p)+

12

d∑

i,j=1

∂2

∂xi∂xj(bij(x)p), t > 0, x ∈ Rd,

(1a)p(x, 0) = f(x), x ∈ Rd. (1b)

• Write it in non-divergence form:

∂p

∂t=

d∑

j=1

aj(x)∂p

∂xj+

12

d∑

i,j=1

bij(x)∂2p

∂xi∂xj+c(x)u, t > 0, x ∈ Rd,

(2a)p(x, 0) = f(x), x ∈ Rd, (2b)


• where

ai(x) = −ai(x) +d∑

j=1

∂bij

∂xj, ci(x) =

12

d∑

i,j=1

∂2bij

∂xi∂xj−

d∑

i=1

∂ai

∂xi.

• The diffusion matrix is always nonnegative. We will assumethat it is actually positive, i.e. we will impose the uniformellipticity condition:

d∑

i,j=1

bij(x)ξiξj > α‖ξ‖2, ∀ ξ ∈ Rd, (3)

• Furthermore, we will assume that the coefficients a, b, c aresmooth and that they satisfy the growth conditions

‖b(x)‖ 6 M, ‖a(x)‖ 6 M(1+‖x‖), ‖c(x)‖ 6 M(1+‖x‖2). (4)


• We will call a solution to the Cauchy problem for theFokker–Planck equation (2) a classical solution if:

i) u ∈ C2,1(Rd,R+).

ii) ∀T > 0 there exists a c > 0 such that

‖u(t, x)‖L∞(0,T ) 6 ceα‖x‖2

iii) limt→0 u(t, x) = f(x).


Theorem 1. Assume that conditions (3) and (4) are satisfied, andassume that |f | 6 ceα‖x‖2 . Then there exists a unique classicalsolution to the Cauchy problem for the Fokker–Planck equation.Furthermore, there exist positive constants K, δ so that

|p|, |pt|, ‖∇p‖, ‖D2p‖ 6 Kt(−n+2)/2 exp(− 1

2tδ‖x‖2

).

• This estimate enables us to multiply the Fokker-Planckequation by monomials xn and then to integrate over Rd andto integrate by parts.

The FP equation as a conservation law

• We can define the probability current to be the vector whoseith component is

Ji := ai(x)p− 12

d∑

j=1

∂

∂xj

(bij(x)p

).

• The Fokker–Planck equation can be written as a continuityequation:

∂p

∂t+∇ · J = 0.

• Integrating the FP equation over Rd and integrating by partson the right hand side of the equation we obtain

d

dt

∫

Rd

p(x, t) dx = 0.

• Consequently:

‖p(·, t)‖L1(Rd) = ‖p(·, 0)‖L1(Rd) = 1.

Examples of Diffusion Processes

• Set a(t, x) ≡ 0, b(t, x) ≡ 2D > 0. The Fokker-Planck equationbecomes:

∂p

∂t= D

∂2p

∂x2, p(x, s|y, s) = δ(x− y).

• This is the heat equation, which is the Fokker-Planck equationfor Brownian motion (Einstein, 1905). Its solution is

pW (x, t|y, s) =1√

2πD(t− s)exp

(− (x− s)2

2D(t− s)

).


• Assume that the initial distribution is

pW (x, s|y, s) = W (y, s).

• The solution of the Fokker-Planck equation for Brownianmotion with this initial distribution is

PW (x, t) =∫

p(x, t|y, s)W (y, s) dy.

• The Gaussian distribution is the fundamental solution(Green’s function) of the heat equation (i.e. the Fokker-Planckequation for Brownian motion).


• Set a(t, x) = −αx, b(t, x) ≡ 2D > 0:

∂p

∂t= α

∂(xp)∂x

+ D∂2p

∂x2.

• This is the Fokker-Planck equation for the Ornstein-Uhlenbeckprocess (Ornstein-Uhlenbeck, 1930). Its solution is

pOU (x, t|y, s) =√

α

2πD(1− e−2α(t−s))exp

(−α(x− e−α(t−s)y)2

2D(1− e−2α(t−s))

).

• Proof: take the Fourier transform in x, use the method ofcharacteristics and take the inverse Fourier transform.


• Set y = 0, s = 0. Notice that

limα→0

pOU (x, t) = pW (x, t).

• Thus, in the limit where the friction coefficient goes to 0, werecover distribution function of BM from the DF of the OUprocesses.

• Notice also that

limt→+∞

pOU (x, t) =√

α

2πDexp

(−αx2

2D

).

• Thus, the Ornstein-Uhlenbeck process is an ergodic Markovprocess. Its invariant measure is Gaussian.

• We can calculate all moments of the OU process.


• Define the nth moment of the OU process:

Mn =∫

Rxnp(x, t) dx, n = 0, 1, 2, . . . .

• Let n = 0. We integrate the FP equation over R to obtain:∫

∂p

∂t= α

∫∂(xp)∂x

+ D

∫∂2p

∂x2= 0,

• after an integration by parts and using the fact that p(x, t)decays sufficiently fast at infinity. Consequently:

d

dtM0 = 0 ⇒ M0(t) = M0(0) = 1.

• In other words:d

dt‖p‖L1(R) = 0 ⇒ p(x, t) = p(x, t = 0).

• Consequently: probability is conserved.


• Let n = 1. We multiply the FP equation for the OU process byx, integrate over R and perform and integration by parts toobtain:

d

dtM1 = −αM1.

• Consequently, the first moment converges exponentially fastto 0:

M1(t) = e−αtM1(0).


• Let now n > 2. We multiply the FP equation for the OUprocess by xn and integrate by parts (once on the first term onthe RHS and twice on the second) to obtain:

d

dt

∫xnp = −αn

∫xnp + Dn(n− 1)

∫xn−2p.

• Or, equivalently:

d

dtMn = −αnMn + Dn(n− 1)Mn−2, n > 2.

• This is a first order linear inhomogeneous differential equation.We can solve it using the variation of constants formula:

Mn(t) = e−αntMn(0) + Dn(n− 1)∫ t

0

e−αn(t−s)Mn−2(s) ds.


• The stationary moments of the OU process are:

〈xn〉OU =√

α

2πD

∫

Rxne−

αx22D dx

= 1.3 . . . (n− 1)

(Dα

)n/2, n even,

0, n odd.

• We have thatlim

t→∞Mn(t) = 〈xn〉OU .

• If the initial conditions of the OU process are stationary, then:

Mn(t) = Mn(0) = 〈xn〉OU .


• set a(x) = µx, b(x) = 12σx2. This is the geometric

Brownian motion. The generator of this process is

L = µx∂

∂x+

σ2x2

2∂2

∂x2.

• Notice that this operator is not uniformly elliptic.

• The Fokker-Planck equation of the geometric Brownian motionis:

∂p

∂t= − ∂

∂x(µx) +

∂2

∂x2

(σ2x2

2p

).


• We can easily obtain an equation for the nth moment of thegeometric Brownian motion:

d

dtMn =

(µn +

σ2

2n(n− 1)

)Mn, n > 2.

• The solution of this equation is

Mn(t) = e(µ+(n−1) σ22 )ntMn(0), n > 2

• andM1(t) = eµtM1(0).

• Notice that the nth moment might diverge as t →∞,depending on the values of µ and σ:

limt→∞

Mn(t) =

0, σ2

2 < − µn−1 ,

Mn(0), σ2

2 = − µn−1 ,

+∞, σ2

2 > − µn−1 .

Gradient Flows

• Let V (x) = 12αx2. The generator of the OU process can be

written as:L = −∂xV ∂x + D∂2

x.

• Consider diffusion processes with a potential V (x), notnecessarily quadratic:

L = −∇V (x) · ∇+ D∆. (5)

• This is a gradient flow perturbed by noise whose strength isD = kBT where kB is Boltzmann’s constant and T theabsolute temperature.

• The corresponding stochastic differential equation is

dXt = −∇V (Xt) dt +√

2D dWt.

Gradient Flows

• The corresponding FP equation is:

∂p

∂t= ∇ · (∇V p) + D∆p. (6)

• It is not possible to calculate the time dependent solution ofthis equation for an arbitrary potential. We can, however,alwas calculate the stationary solution.

Gradient Flows

Proposition 1. Assume that V (x) is smooth and that

e−V (x)/D ∈ L1(Rd). (7)

Then the Markov process with generator (5) is ergodic. The uniqueinvariant distribution is the Gibbs distribution

p(x) =1Z

e−V (x)/D (8)

where the normalization factor Z is the partition function

Z =∫

Rd

e−V (x)/D dx.

Gradient Flows

• The fact that the Gibbs distribution is an invariant distributionfollows by direct substitution. Uniqueness follows from a PDEsargument (see discussion below).

• It is more convenient to ”normalize” the solution of theFokker-Planck equation wrt the invariant distribution.

Proposition 2. Let p(x, t) be the solution of the Fokker-Planckequation (6), assume that (7) holds and let ρ(x) be the Gibbsdistribution (8). Define h(x, t) through

p(x, t) = h(x, t)ρ(x).

Then the function h satisfies the backward Kolmogorovequation:

∂h

∂t= −∇V · ∇h + D∆h, h(x, 0) = p(x, 0)ρ−1(x). (9)

Gradient Flows

Proof. The initial condition follows from the definition of h. Wecalculate the gradient and Laplacian of p:

∇p = ρ∇h− ρhD−1∇V

and

∆p = ρ∆h− 2ρD−1∇V · ∇h + hD−1∆V ρ + h|∇V |2D−2ρ.

We substitute these formulas into the FP equation to obtain

ρ∂h

∂t= ρ

(−∇V · ∇h + D∆h

),

from which the claim follows.

Gradient Flows

• Consequently, in order to study properties of solutions to theFP equation, it is sufficient to study the backward equation (9).

• The generator L is self-adjoint, in the right function space.

• We define the weighted L2 space L2ρ:

L2ρ =

f |

∫

Rd

|f |2ρ(x) dx < ∞,

• where ρ(x) is the Gibbs distribution. This is a Hilbert spacewith inner product

(f, h)ρ =∫

Rd

fhρ(x) dx.

Gradient Flows

Proposition 3. Assume that V (x) is a smooth potential andassume that condition (7) holds. Then the operator

L = −∇V (x) · ∇+ D∆

is self-adjoint in L2ρ. Furthermore, it is non-positive, its kernel

consists of constants.

Proof. Let f, ∈ C20 (Rd). We calculate

(Lf, h)ρ =∫

Rd

(−∇V · ∇+ D∆)fhρ dx

=∫

Rd

(∇V · ∇f)hρ dx−D

∫

Rd

∇f∇hρ dx−D

∫

Rd

∇fh∇ρ dx

= −D

∫

Rd

∇f · ∇hρ dx,

from which self-adjointness follows.

Gradient Flows

If we set f = h in the above equation we get

(Lf, f)ρ = −D‖∇f‖2ρ,

which shows that L is non-positive.

Clearly, constants are in the null space of L. Assume thatf ∈ N (L). Then, from the above equation we get

0 = −D‖∇f‖2ρ,

and, consequently, f is a constant.

Remark 1. The expression (−Lf, f)ρ is called the Dirichlet formof the operator L. In the case of a gradient flow, it takes the form

(−Lf, f)ρ = D‖∇f‖2ρ. (10)

Gradient Flows

• Using the properties of the generator L we can show that thesolution of the Fokker-Planck equation converges to the Gibbsdistribution exponentially fast.

• For this we need the following result.

Proposition 4. Assume that the potential V satisfies the convexitycondition

D2V > λI.

Then the corresponding Gibbs measure satisfies the Poincareinequality with constant λ:

∫

Rd

fρ = 0 ⇒ ‖∇f‖ρ >√

λ‖f‖ρ. (11)

Gradient Flows

Theorem 2. Assume that p(x, 0) ∈ L2(eV/D). Then the solutionp(x, t) of the Fokker-Planck equation (6) converges to the Gibbsdistribution exponentially fast:

‖p(·, t)− Z−1e−V ‖ρ−1 6 e−λDt‖p(·, 0)− Z−1e−V ‖ρ−1 . (12)

Proof. We Use (9), (10) and (11) to calculate

− d

dt‖(h− 1)‖2ρ = −2

(∂h

∂t, h− 1

)

ρ

= −2 (Lh, h− 1)ρ

= (−L(h− 1), h− 1)ρ = 2D‖∇(h− 1)‖ρ

> 2Dλ‖h− 1‖2ρ.Our assumption on p(·, 0) implies that h(·, 0) ∈ L2

ρ. Consequently,the above calculation shows that

‖h(·, t)− 1‖ρ 6 e−λDt‖H(·, 0)− 1‖ρ.

This, and the definition of h, p = ρh, lead to (12).

Gradient Flows

• The assumption∫

Rd

|p(x, 0)|2Z−1eV/D < ∞

• is very restrictive (think of the case where V = x2).

• The function space L(ρ−1) = L2(e−V/D) in which we proveconvergence is not the right space to use. Since p(·, t) ∈ L1,ideally we would like to prove exponentially fast convergence inL1.

• We can prove convergence in L1 using the theory oflogarithmic Sobolev inequalities. In fact, we can also proveconvergence in relative entropy:

H(p|ρV ) :=∫

Rd

p ln(

p

ρV

)dx.

Gradient Flows

• The relative entropy norm controls the L1 norm:

‖ρ1 − ρ2‖L1 6 CH(ρ1|ρ2)

• Using a logarithmic Sobolev inequality, we can proveexponentially fast convergence to equilibrium, assuming onlythat the relative entropy of the initial conditions is finite.

Theorem 3. Let p denote the solution of the Fokker–Planckequation (6) where the potential is smooth and uniformly convex.Assume that the the initial conditions satisfy

H(p(·, 0)|ρV ) < ∞.

Then p converges to the Gibbs distribution exponentially fast inrelative entropy:

H(p(·, t)|ρV ) 6 e−λDtH(p(·, 0)|ρV ).

Gradient Flows

• Convergence to equilibrium for kinetic equations, both linearand non-linear (e.g., the Boltzmann equation) has been studiedextensively in the last decade.

• It has been recognized that the relative entropy plays a veryimportant role.

• For more information see

• On the trend to equilibrium for the Fokker-Planck equation: aninterplay between physics and functional analysis by P.A.Markowich and C. Villani, 1999.

Eigenfunction Expansions

• Consider the generator of a gradient stochastic flow with auniformly convex potential

L = −∇V · ∇+ D∆. (13)

• We know that

i) L is a non-positive self-adjoint operator on L2ρ.

ii) It has a spectral gap:

(Lf, f)ρ 6 −Dλ‖f‖2ρwhere λ is the Poincare constant of the potential V .

• The above imply that we can study the spectral problem for−L:

−Lfn = λnfn, n = 0, 1, . . .


• The operator −L has real, discrete spectrum with

0 = λ0 < λ1 < λ2 < . . .

• Furthermore, the eigenfunctions fjinftyj=1 form an orthonormal

basis in L2ρ: we can express every element of L2

ρ in the form ofa generalized Fourier series:

φ =∞∑

n=0

φnfn, φn = (φ, fn)ρ (14)

• with (fn, fm)ρ = δnm.

• This enables us to solve the time dependent Fokker–Planckequation in terms of an eigenfunction expansion.

• Consider the backward Kolmogorov equation (9).

• We assume that the initial conditions h0(x) = φ(x) ∈ L2ρ and

consequently we can expand it in the form (14).


• We look for a solution of (9) in the form

h(x, t) =∞∑

n=0

hn(t)fn(x).

• We substitute this expansion into the backward Kolmogorovequation:

∂h

∂t=

∞∑n=0

hnfn = L( ∞∑

n=0

hnfn

)(15)

=∞∑

n=0

−λnhnfn. (16)

• We multiply this equation by fm, integrate wrt the Gibbsmeasure and use the orthonormality of the eigenfunctions toobtain the sequence of equations

hn = −λnhn, n = 0, 1,


• The solution is

h0(t) = φ0, hn(t) = e−λntφn, n = 1, 2, . . .

• Notice that

1 =∫

Rd

p(x, 0) dx =∫

Rd

p(x, t) dx

=∫

Rd

h(x, t)Z−1eβV dx = (h, 1)ρ = (φ, 1)ρ

= φ0.


• Consequently, the solution of the backward Kolmogorovequation is

h(x, t) = 1 +∞∑

n=1

e−λntφnfn.

• This expansion, together with the fact that all eigenvalues arepositive (n > 1), shows that the solution of the backwardKolmogorov equation converges to 1 exponentially fast.

• The solution of the Fokker–Planck equation is

p(x, t) = Z−1e−βV (x)

(1 +

∞∑n=1

e−λntφnfn

).

Self-adjointness

• The Fokker–Planck operator of a general diffusion process isnot self-adjoint in general.

• In fact, it is self-adjoint if and only if the drift term is thegradient of a potential (Nelson, ≈ 1960). This is also true ininfinite dimensions (Stochastic PDEs).

• Markov Processes whose generator is a self-adjoint operator arecalled reversible: for all t ∈ [0, T ] Xt and XT−t have the sametransition probability (when Xt is statinary).

• Reversibility is equivalent to the invariant measure being aGibbs measure.

• See Thermodynamics of the general duffusion process:time-reversibility and entropy production, H. Qian, M. Qian, X.Tang, J. Stat. Phys., 107, (5/6), 2002, pp. 1129–1141.

Reduction to a Schrodinger Equation

• We can study the Fokker-Planck equation for a gradient flowby mapping it to a Schrodinger equation.

Proposition 5. The Fokker–Planck operator for a gradient flowcan be written in the self-adjoint form

∂p

∂t= D∇ ·

(e−V/D∇

(eV/Dp

)). (17)

Define now ψ(x, t) = eV/2Dp(x, t). Then ψ solves the PDE

∂ψ

∂t= D∆ψ − U(x)ψ, U(x) :=

|∇V |24D

− ∆V

2. (18)

Let H := −D∆ + U . Then L∗ and H have the same eigenvalues.The nth eigenfunction φn of L∗ and the nth eigenfunction ψn of Hare associated through the transformation

ψn(x) = φn(x) exp(

V (x)2D

).


Remarks

i) From equation (17) shows that the FP operator can be writtenin the form

L∗· = D∇ ·(e−V/D∇

(eV/D·

)).

ii) The operator that appears on the right hand side of eqn. (18)has the form of a Schrodinger operator:

−H = −D∆ + U(x).

iii) The spectral problem for the FP operator can be transformedinto the spectral problem for a Schrodinger operator. We canthus use all the available results from quantum mechanics tostudy the FP equation and the associated SDE.

iv) In particular, the weak noise asymptotics D ¿ 1 is equivalentto the semiclassical approximation from quantum mechanics.


Proof of Prop. 5. We calculate

D∇ ·(e−V/D∇

(eV/Df

))= D∇ ·

(e−V/D

(D−1∇V f +∇f

)eV/D

)

= ∇ · (∇V f + D∇f) = L∗f.

Consider now the eigenvalue problem for the FP operator:

−L∗φn = λnφn.

Set φn = ψn exp(− 1

2DV). We calculate −L∗φn:

−L∗φn = −D∇ ·(e−V/D∇

(eV/Dψne−V/2D

))

= −D∇ ·(

e−V/D

(∇ψn +

∇V

2Dψn

)eV/2D

)

=(−D∆ψn +

(−|∇V |2

4D+

∆V

2D

)ψn

)e−V/2D = e−V/2DHψn.

From this we conclude that e−V/2DHψn = λnψne−V/2D from whichthe equivalence between the two eigenvalue problems follows.


Remarks

i) We can rewrite the Schrodinger operator in the form

H = DA∗A, A = ∇+∇U

2D, A∗ = −∇+

∇U

2D.

ii) These are creation and annihilation operators. They canalso be written in the form

A· = e−U/2D∇(eU/2D·

), A∗· = eU/2D∇

(e−U/2D·

)

iii) The forward the backward Kolmogorov operators have thesame eigenvalues. Their eigenfunctions are related through

φBn = φF

n exp (−V/D) ,

where φBn and φF

n denote the eigenfunctions of the backwardand forward operators, respectively.

The Ornstein–Uhlenbeck Process and Hermite Polynomials

• The generator of the OU process is

L = −yd

dy+ D

d2

dy2(19)

• The OU process is an ergodic Markov process whose uniqueinvariant measure is absolutely continuous with respect to theLebesgue measure on R with density ρ(y) ∈ C∞(R)

ρ(y) =1√2πD

e−y2

2D .

• Let L2ρ denote the closure of C∞ functions with respect to the

norm‖f‖2ρ =

∫

Rf(y)2ρ(y) dy.

• The space L2ρ is a Hilbert space with inner product

(f, g)ρ =∫

Rf(y)g(y)ρ(y) dy.


• Consider the eigenvalue problem for L:

−Lfn = λnfn.

• The operator L has discrete spectrum in L2ρ:

λn = n, n ∈ N .

• The corresponding eigenfunctions are the normalized Hermitepolynomials:

fn(y) =1√n!

Hn

(y√D

), (20)

where

Hn(y) = (−1)ney2

2dn

dyn

(e−

y2

2

).


• The first few Hermite polynomials are:

H0(y) = 1,

H1(y) = y,

H2(y) = y2 − 1,

H3(y) = y3 − 3y,

H4(y) = y4 − 3y2 + 3,

H5(y) = y5 − 10y3 + 15y.

• Hn is a polynomial of degree n.

• Only odd (even) powers appear in Hn when n is odd (even).


Lemma 1. The eigenfunctions fn(y)∞n=1 of the generator of theOU process L satisfy the following properties.

i) They form an orthonormal set in L2ρ:

(fn, fm)ρ = δnm.

ii) fn(y) : n ∈ N is an orthonormal basis in L2ρ.

iii) Define the creation and annihilation operators on C1(R) by

A+φ(y) =1√

D(n + 1)

(−D

dφ

dy(y) + yφ(y)

), y ∈ R

and

A−φ(y) =

√D

n

dφ

dy(y).

ThenA+fn = fn+1 and A−fn = fn−1. (21)


iv) The eigenfunctions fn satisfy the following recurrence relation

yfn(y) =√

Dnfn−1(y) +√

D(n + 1)fn+1(y), n > 1. (22)

v) The function

H(y; λ) = eλy−λ22

is a generating function for the Hermite polynomials. Inparticular

H

(y√D

; λ)

=∞∑

n=0

λn

√n!

fn(y), λ ∈ C, y ∈ R.

The Klein-Kramers-Chandrasekhar Equation

• Consider a diffusion process in two dimensions for the variablesq (position) and momentum p. The generator of this Markovprocess is

L = p · ∇q −∇qV∇p + γ(−p∇p + D∆p). (23)

• The L2(dpdq)-adjoint is

L∗ρ = −p · ∇qρ−∇qV · ∇pρ + γ (∇p(pρ) + D∆pρ) .

• The corresponding FP equation is:

∂p

∂t= L∗p.

• The corresponding stochastic differential equations is theLangevin equation

Xt = −∇V (Xt)− γXt +√

2γDWt. (24)

• This is Newton’s equation perturbed by dissipation and noise.


• The Klein-Kramers-Chandrasekhar equation was first derivedby Kramers in 1923 and was studied by Kramers in his famouspaper ”Brownian motion in a field of force and the diffusionmodel of chemical reactions”, Physica 7(1940), pp. 284-304.

• Notice that L∗ is not a uniformly elliptic operator: there aresecond order derivatives only with respect to p and not q. Thisis an example of a degenerate elliptic operator. It is, however,hypoelliptic: we can still prove existence and uniqueness ofsolutions for the FP equation, and obtain estimates on thesolution.

• It is not possible to obtain the solution of the FP equation foran arbitrary potential.

• We can calculate the (unique normalized) solution of thestationary Fokker-Planck equation.


Proposition 6. Assume that V (x) is smooth and that

e−V (x)/D ∈ L1(Rd).

Then the Markov process with generator (23) is ergodic. The uniqueinvariant distribution is the Maxwell-Boltzmann distribution

ρ(p, q) =1Z

e−βH(p,q) (25)

where

H(p, q) =12‖p‖2 + V (q)

is the Hamiltonian, β = (kBT )−1 is the inverse temperatureand the normalization factor Z is the partition function

Z =∫

R2d

e−βH(p,q) dpdq.


• It is possible to obtain rates of convergence in either a weightedL2-norm or the relative entropy norm.

H(p(·, t)|ρ) 6 Ce−αt.

• The proof of this result is very complicated, since the generatorL is degenerate and non-selfadjoint.

• See F. Herau and F. Nier, Isotropic hypoellipticity and trend toequilibrium for the Fokker-Planck equation with a high-degreepotential, Arch. Ration. Mech. Anal., 171(2),(2004), 151–218.


• The phase-space Fokker-Planck equation can be written in theform

∂ρ

∂t+ p · ∇qρ−∇qV · ∇pρ = Q(ρ, fB)

• where the collision operator has the form

Q(ρ, fB) = D∇ · (fB∇(f−1

B ρ))

.

• The Fokker-Planck equation has a similar structure to theBoltzmann equation (the basic equation in the kinetic theory ofgases), with the difference that the collision operator for theFP equation is linear.

• Convergence of solutions of the Boltzmann equation to theMaxwell-Boltzmann distribution has also been proved. See

• L. Desvillettes and C. Villani: On the trend to globalequilibrium for spatially inhomogeneous kinetic systems: theBoltzmann equation. Invent. Math. 159, 2 (2005), 245-316.


• We can study the backward and forward Kolmogorov equationsfor (24) by expanding the solution with respect to the Hermitebasis.

• We consider the problem in 1d. We set D = 1. The generatorof the process is:

L = p∂q − V ′(q)∂p + γ(−p∂p + ∂2

p

).

=: L1 + γL0,

• where

L0 := −p∂p + ∂2p and L1 := p∂q − V ′(q)∂p.

• The backward Kolmogorov equation is

∂h

∂t= Lh. (26)


• The solution should be an element of the weighted L2-space

L2ρ =

f |

∫

R2|f |2Z−1e−βH(p,q) dpdq < ∞

.

• We notice that the invariant measure of our Markov process isa product measure:

e−βH(p,q) = e−β 12 |p|2e−βV (q).

• The space L2(e−β 12 |p|2 dp) is spanned by the Hermite

polynomials. Consequently, we can expand the solution of (26)into the basis of Hermite basis:

h(p, q, t) =∞∑

n=0

hn(q, t)fn(p), (27)

• where fn(p) = 1/√

n!Hn(p).


• Our plan is to substitute (27) into (26) and obtain a sequenceof equations for the coefficients hn(q, t).

• We have:

L0h = L0

∞∑n=0

hnfn = −∞∑

n=0

nhnfn

• FurthermoreL1h = −∂qV ∂ph + p∂qh.

• We calculate each term on the right hand side of the aboveequation separately. For this we will need the formulas

∂pfn =√

nfn−1 and pfn =√

nfn−1 +√

n + 1fn+1.


p∂qh = p∂q

∞∑n=0

hnfn = p∂ph0 +∞∑

n=1

∂qhnpfn

= ∂qh0f1 +∞∑

n=1

∂qhn

(√nfn−1 +

√n + 1fn+1

)

=∞∑

n=0

(√

n + 1∂qhn+1 +√

n∂qhn−1)fn

• with h−1 ≡ 0.

• Furthermore

∂qV ∂ph =∞∑

n=0

∂qV hn∂pfn =∞∑

n=0

∂qV hn

√nfn−1

=∞∑

n=0

∂qV hn+1

√n + 1fn.


• Consequently:

Lh = L1 + γL1h

=∞∑

n=0

(− γnhn +√

n + 1∂qhn+1

+√

n∂qhn−1 +√

n + 1∂qV hn+1

)fn

• Using the orthonormality of the eigenfunctions of L0 we obtainthe following set of equations which determine hn(q, t)∞n=0.

hn = −γnhn +√

n + 1∂qhn+1

+√

n∂qhn−1 +√

n + 1∂qV hn+1, n = 0, 1, . . .

• This is set of equations is usually called the Brinkmanhierarchy (1956).


• We can use this approach to develop a numerical method forsolving the Klein-Kramers equation.

• For this we need to expand each coefficient hn in anappropriate basis with respect to q.

• Obvious choices are other the Hermite basis (polynomialpotentials) or the standard Fourier basis (periodic potentials).

• We will do this for the case of periodic potentials.

• The resulting method is usually called the continued fractionexpansion. See Risken (1989).


• The Hermite expansion of the distribution function wrt to thevelocity is used in the study of various kinetic equations(including the Boltzmann equation). It was initiated by Gradin the late 40’s.

• It quite often used in the approximate calculation of transportcoefficients (e.g. diffusion coefficient).

• This expansion can be justified rigorously for theFokker-Planck equation. See

• J. Meyer and J. Schroter, Comments on the Grad Procedure forthe Fokker-Planck Equation, J. Stat. Phys. 32(1) pp.53-69(1983).

Boundary conditions for the Fokker–Planck equation

• So far we have been studying the FP equation on Rd.

• The boundary condition was that the solution decayssufficiently fast at infinity.

• For ergodic diffusion processes this is equivalent to requiringthat the solution of the backward Kolmogorov equation is anelement of L2(µ) where µ is the invariant measure of theprocess.

• We can also study the FP equation in a bounded domain withappropriate boundary conditions.

• We can have absorbing, reflecting or periodic boundaryconditions.


• Consider the FP equation posed in Ω ⊂ Rd where Ω is abounded domain with smooth boundary.

• Let J denote the probability current and let n be the unitnormal vector to the surface.

i) We specify reflecting boundary conditions by setting

n · J(x, t) = 0, on ∂Ω.

ii) We specify absorbing boundary conditions by setting

p(x, t) = 0, on ∂Ω.


iii) When the coefficient of the FP equation are periodic functions,we might also want to consider periodic boundary conditions,i.e. the solution of the FP equation is periodic in x with periodequal to that of the coefficients.

• Reflecting BC correspond to the case where a particle whichevolves according to the SDE corresponding to the FP equationgets reflected at the boundary.

• Absorbing BC correspond to the case where a particle whichevolves according to the SDE corresponding to the FP equationgets absorbed at the boundary.

• There is a complete classification of boundary conditions in onedimension, the Feller classification: the BC can be regular,exit, entrance and natural.

REFERENCES

• A lot of information about methods for solving theFokker-Planck equation can be found in Risken TheFokker-Planck equation, Springer, Berlin 1989.

• See also Gardiner Handbook of Stochastic Methods,Horsthemke and Lefever Noise Induced Transitions.

• For a mathematically rigorous treatment see the books ofFriedman, Arnold on SDES.

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