Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 3 / d

Table of contents
- Solution d

Solution

The supplied derivative identity and the Cauchy-Schwarz inequality give

U_{x} (t, x)^{2} \leq 4 U (t, x) E [f^{'} (x + B_{1 - t})^{2}] .

Therefore

\frac{1}{2} E \int_{0}^{1} \frac{U _{x} ( t , W _{t} ) ^{2}}{U ( t , W _{t} )} d t \leq 2 \int_{0}^{1} E [f^{'} (W_{t} + B_{1 - t})^{2}] d t .

The sum

W_{t} + B_{1 - t}

is standard normal for every

t

, so the right side is

2 E f^{'} (W_{1})^{2}

. Part c proves the Gaussian logarithmic Sobolev inequality

E [f (W_{1})^{2} lo g f (W_{1})^{2}] \leq E [f (W_{1})^{2}] lo g E [f (W_{1})^{2}] + 2 E [f^{'} (W_{1})^{2}] .

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