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Register a new userGeneralized PDE estimates for KPZ equations through Hamilton-Jacobi-Bellman formalism
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J. Unterberger
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We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${cal W}$-spaces of functions with locally bounded averages, as in [24], implying in particular global existence and uniqueness of solutions.
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This paper is a review of results on Optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation is considered with respect to a partial ordering generated by a given cone. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. Our main objective is the derivation of the model PDEs which turn out to be parametric families of HJB single equations instead of systems of PDEs. However, this allows the use of the theory of Viscosity Solutions.
We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article is dedicated to existence results and PDE estimates for the solution. Our results extend in a non-trivial way those previously obtained for the noiseless equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in new functional spaces containing unbounded functions, implying existence and uniqueness. These new functional spaces made up of functions with locally bounded averages, generically called ${cal W}$-spaces thereafter, and which may be of interest for the study of parabolic equations in general, allow local or pointwise estimates. The comparison to the linear heat equation through a Cole-Hopf transform is an essential ingredient in the proofs, and our results are accordingly valid only for a function $V$ with at most quadratic growth at infinity.
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