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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 3

For
$$
P(\mathbf x)=Z^{-1}\exp\left(-\frac12D_{ij}x_ix_j\right)
$$
with symmetric $D$,
$$
\frac{\partial P}{\partial x_j}=-D_{jk}x_kP.
$$
Substitution into the current from part (d) yields
$$
J_i=\left(a_{ik}+\frac12b_{ij}D_{jk}\right)x_kP.
$$
It vanishes for all $\mathbf x$ exactly when
$$
\boxed{a+\frac12bD=0,\qquad D=-2b^{-1}a.}
$$
Thus the stated <multivariate normal distribution> is a stationary solution whenever the zero-current assumption makes $D$ symmetric positive definite. This is the <Stationary Gaussian density for a linear Fokker-Planck equation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 13C / e / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 13C / e / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 13C / e / Solution

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