A Sublinear Variance Bound for Solutions of a Random Hamilton–Jacobi Equation

Abstract

We estimate the variance of the value function for a random optimal control problem. The value function is the solution w ^ϵ of a Hamilton–Jacobi equation with random Hamiltonian H(p,x,ω)=K(p)−V(x/ϵ,ω) in dimension d≥2. It is known that homogenization occurs as ϵ→0, but little is known about the statistical fluctuations of w ^ϵ. Our main result shows that the variance of the solution w ^ϵ is bounded by O(ϵ/|logϵ|). The proof relies on a modified Poincaré inequality of Talagrand.

Appendix

Proof of Lemma 3.6

The bounds in (3.37) follow from the fact that ℙ is the product measure on $\varOmega= \{a,b\}^{\mathbb{Z}^{d}_{n}}$, with ℙ(ω(j)=a)=α and ℙ(ω(j)=b)=β. For every nonnegative integrable ψ, (3.37) implies

Similarly $\mathbb{E}(\psi)\geq C'\mathbb{E}(\psi\circ\phi_{j})$ for all such ψ. □

Proof of Theorem 1.2

Let us define

Then Theorem 1.2 is a slight modification of the following

Theorem A.1

([20], Theorem 1.5)

There is a constant C>0, such that

$$ \operatorname{var}(f)\leq C\cdot\sum_{j\in J} \frac{\|\Delta_j f\|_2^2}{1+\log\frac{\|\Delta_jf\|_2}{\|\Delta_jf\|_1}} $$

(A.58)

holds for all f∈L ²(Ω _J ).

To derive Theorem 1.2 from this, we start with elementary observation

$$C'\bigl|\rho_j f(\omega)\bigr|\leq\bigl|\Delta_j f( \omega)\bigr|\leq C''\bigl|\rho_j f(\omega)\bigr| $$

for C′=min{2α,2β} and C″=max{2α,2β}. Let κ=log(C″/C′)≥0. If $\log\frac{\|\rho_{j}f\|_{2}}{\|\rho_{j}f\|_{1}} \geq2\kappa$, then

$$\log\frac{\|\Delta_j f\|_2}{\|\Delta_jf\|_1} \geq\log\frac {C'}{C''}+\log\frac{\|\rho_jf\|_2}{\|\rho_jf\|_1} \geq \frac{1}{2} \log \frac{\|\rho_jf\|_2}{\|\rho_jf\|_1}. $$

Consequently, Theorem A.1 implies

$$\operatorname{var}(f)\leq C\cdot\sum_{j\in J} \frac{\|\Delta_j f\|_2^2}{1+\log\frac{\|\Delta_jf\|_2}{\|\Delta_jf\|_1}} \leq2\bigl(C''\bigr)^2 C\sum_{j\in J}\frac{\|\rho_jf\|_2^2}{1+\log\frac{\|\rho_jf\|_2}{\|\rho_jf\|_1}}. $$

On the other hand, if $\log\frac{\|\rho_{j}f\|_{2}}{\|\rho_{j}f\|_{1}} \in[0, 2\kappa)$, then Theorem A.1 implies

$$\operatorname{var}(f)\leq C\cdot\sum_{j\in J}\| \Delta_j f\|_2^2 \leq(1 + 2\kappa)2 \bigl(C''\bigr)^2 C\sum _{j\in J}\frac{\|\rho_jf\|_2^2}{1+\log\frac{\|\rho_jf\|_2}{\|\rho_jf\|_1}}. $$