A one- dimensional Brownian particle of mass m is placed in a fluid with temperature T. Its velocity v is determined by

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A one- dimensional Brownian particle of mass m is placed in afluid with temperature T. Its velocity v is determined by theLangevin equation
M(dv/dt) = -γv + √(2mꝩkT) ξ(t)
Where γ is the fractional coefficient. The Gaussian – whitenoise ξ(t) satisfies the following properties
<ξ(t)> = 0 , < ξ(t) ξ(t’)> = δ(t – t’ ) .
The initial velocity of the Brownian particle is set tov(0).
(a) Show that the mean velocity decays as < v(t) > =<v(0)> e-γt/m.
(b) Show that the correlation between final and initialvelocities also decays in the same way, < v(t)v(0) > =<v(0)2> e-γt/m.
(c ) Suppose that the Brownian particle is initially inthermal equilibrium. Hence , its initial velocity distribution isMaxwellian. Show that < v(t) > = 0 and < v(t)v(0) >α(=proportional to) T.
Could please answer these questions as soon as possible? Thankyou!