[Q] Stochastic Nonlinear ODE

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

  1. 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.

    Thanks in advance!

    Cheong Siew Ann
    Siew-Ann Cheong, Oct 23, 2003
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  2. Siew-Ann Cheong

    simo.sarkka Guest

    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
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