The Csiszár-Kullback inequality

Today we give an inequality, which is very useful in proving the convergence of a solution to PDEs tends to the stationary solution as time tends to infinity.

Lemma. Let \Omega\in\mathbb R^d be a domain and let f, g\in L^1(\Omega) satisfy f\geq 0, g>0 and \int_\Omega fdx = \int_\Omega gdx = 1. Furthermore, let \Phi\in C^1(\mathbb R) satisfy

\Phi(s) \geq \Phi(1) + \Phi'(1)(s-1) + \gamma^2(s-1)^21_{\{s<1\}}

for all s\in\mathbb R and some \gamma>0, where 1_{\{s<1\}} is the characteristic function on A\subset \mathbb R. Finally, let

H_\Phi[f] = \int_\Omega \Phi\left(\dfrac{f}{g}\right)gdx.


\|f - g\|^2_{L^1(\Omega)} \leq \dfrac{4}{\gamma^2}(H_\Phi[f] - H_\Phi[g])

Proof. First, we try to estimate H_\Phi[f] - H_\Phi[g].

H_\Phi[f] - H_\Phi[g] = \int_{\Omega}(\Phi(f/g) - \Phi(1))gdx

\geq \int_{\Omega}(\Phi'(1)(f/g-1) + \gamma^2(f/g-1)^21_{\{f/g<1\}})gdx

= \Phi'(1)\int_{\Omega}(f - g)dx + \gamma^2\int_{f<g}(f/g-1)^2gdx

= \gamma^2\int_{f<g}(f/g-1)^2gdx (since \int_\Omega fdx = \int_\Omega gdx =1)

Now, it is sufficient to prove that

\gamma^2\int_{f<g}(f/g-1)^2gdx \geq \dfrac{\gamma^2}{4}\|f-g\|^2_{L^1(\Omega)}

or equivalently

4\int_{f<g}(f/g-1)^2gdx \geq \|f-g\|^2_{L^1(\Omega)}

We have

\|f - g\|_{L^1(\Omega)} = \int_{\Omega}|f-g|dx

= \int_{f<g}(g-f)dx + \int_{f\geq g}(f-g)dx

= \int_{f<g}(g-f)dx + \int_{f\geq g}fdx - \int_{f\geq g}gdx

= \int_{f<g}(g-f)dx +(\int_{\Omega}fdx - \int_{f<g}fdx) - (\int_{\Omega}gdx - \int_{f<g}gdx)

= 2\int_{f<g}(g-f)dx (since \int_\Omega fdx = \int_\Omega gdx = 1).

We now have to show that

\int_{f<g}(f/g-1)^2gdx \geq (\int_{f<g}(g-f)dx)^2.

Using Hölder’s inequality and \int_\Omega gdx = 1, we have

\int_{f<g}(g-f)dx = \int_{f<g}(1-f/g)gdx

\leq \int_{f<g}|(1-f/g)\sqrt{g}||\sqrt{g}|dx

\leq \left(\int_{f<g}|(1-f/g)^2g|dx\right)^{1/2}\left(\int_\Omega gdx\right)^{1/2}

= \left(\int_{f<g}|(1-f/g)^2g|dx\right)^{1/2}


(\int_{f<g}(g-f)dx)^2 \leq \int_{f<g}|(1-f/g)^2g|dx.

This completes the proof.

About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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