Stochastic Processes

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Wednesday - September 8, 2010

Stochastic Variables
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Random Events
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Stochastic Processes
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Markov Processes
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The Master Equation
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One-Step Processes
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Chemical Reactions
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The Fokker-Planck Equation
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The Langevin Approach
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Expansion of the Master Equation
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The Diffusion Type
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First-Passage Problems
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Unstable Systems
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Fluctuations in Continuous Systems
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The Statistics of Jump Events
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Stochastic Differential Equations
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Stochastic Behavior of Quantum Systems
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Exercise. Let $ψ (t)$ be a given function of $t$ and $X$ a given random variable. Then

Y (t) = ψ (t) X

is a stochastic process. Compute its $n$ -time averages, its variance and autocorrelation function (i.e., express them in the stochastic properties of $X$ ).

Solution. The $n$ -time average is:

<Y (t_{1}) \dots Y (t_{n})> = \int_{- \infty}^{\infty} ψ (t_{1}) \dots ψ (t_{n}) X^{n} P_{X} (x) d x = <X^{n}> \prod_{i = 1}^{n} ψ (t_{i})

The variance is:

σ^{2} (t) = <{(Y (t) - <Y (t)>)}^{2}> = <Y {(t)}^{2}> - {<Y (t)>}^{2}

σ^{2} (t) = \int_{- \infty}^{\infty} Y {(t)}^{2} P_{X} (x) d x - {(\int_{- \infty}^{\infty} Y (t) P_{X} (x) d x)}^{2}

σ^{2} (t) = ψ {(t)}^{2} <X^{2}> - ψ {(t)}^{2} {<X>}^{2} = ψ {(t)}^{2} σ_{X}^{2}

The autocorrelation function is:

κ (t_{1}, t_{2}) = <Y (t_{1}) Y (t_{2})> - <Y (t_{1})> <Y (t_{2})>

κ (t_{1}, t_{2}) = ψ (t_{1}) ψ (t_{2}) <X^{2}> - ψ (t_{1}) ψ (t_{2}) {<X>}^{2} = ψ (t_{1}) ψ (t_{2}) σ_{X}^{2}