Variational Principles in Conformation Dynamics - FU Berlin, FB MI

A. AppendixA.1. Diffusion in a quadratic potentialLet us prove the properties of the diffusion in a quadratic potential as introduced in sectionIII.4. First,letusderiveananalyticexpressionfortheproblem’ssolution.Onstochasticintegrals and Ito’s formula, see [Evans, ch.4].Lemma A.1: If the process is initially distributed according to X 0 ,thenthedistributionattime t is given by the stochastic integralX t = e −θt X 0 + σ t0dB s e −θ(t−s) .(A.1)Proof. The proof can also be found on [Wikipedia]. Let X t be the solution of the correspondingstochastic differential equation, i.e.dX t = −θX t dt + σdB t .(A.2)Consider the function f(X t ,t):=e θt X t . According to Ito’s formula, f(X t ,t) satisfies theSDE:d(f(X t ,t)) = ∂f ∂fdt +∂t ∂x dX t + 1 ∂ 2 f2 ∂x 2 σ2 dt(A.3)= θe θt X t dt − θe θt X t dt + e θt σdB t = e θt σdB t . (A.4)The process f(X t ,t) is therefore given by the stochastic integrale θt X t = f(X t ,t)=X 0 + σ t0dB s e θs .(A.5)