# [Q] Stochastic Nonlinear ODE

Discussion in 'Math Research' started by Siew-Ann Cheong, Oct 23, 2003.

1. ### Siew-Ann CheongGuest

Hi,

In the Langevin equation,

dv/dt = -gamma v + eta(t)

where v is the velocity of a Brownian particle, gamma is the damping
coefficient, and eta(t) a stochastic noise term, satisfying

<eta(t)> = 0 and <eta(t)eta(t')> = delta(t-t'),

the average velocity is

<v>(t) = v_0 exp(-gamma t),

which is the same as the noiseless case.

What if the equation is nonlinear, say, something like

dv/dt = -gamma v^2 + eta(t)?

Is the average velocity still the same as the noiseless case? I am
wondering whether the average velocity can get 'shifted' compared to
the noiseless case when the equation of motion is nonlinear.

Cheong Siew Ann

Siew-Ann Cheong, Oct 23, 2003

2. ### simo.sarkkaGuest

You can infer this by taking averages of both sides in
Langevin equation:

E[dv/dt] = E[-gamma v + eta(t)]

and due to linearities and E[eta(t)] = 0 you get

dE[v]/dt = -gamma E[v]

which has the same form as noiseless equation.
If we try the same for this:

E[dv/dt] = E[-gamma v^2 + eta(t)]
dE[v]/dt = -gamma E[v^2]

It can be seen that derivative of velocity mean depends on
average squared velocity of the particle. This can be
also written in form

dE[v]/dt = -gamma E[v]^2 - gamma E[(v-E[v])^2]

which shows that in this case average velocity 'shift' is
proportional to variance of the particle.

simo.sarkka, Nov 2, 2003