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Register a new userThe scaling limit of the KPZ equation in space dimension 3 and higher
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Added by
Jeremie Unterberger M
Publication date
2017
fields
Physics
and research's language is
English
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We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda | abla h(t,x)|^2 +sqrt{D}, eta(t,x), qquad (t,x)inmathbb{R}_+timesmathbb{R}^d $$ in $dge 3$ dimensions in the perturbative regime, i.e. for $lambda>0$ small enough and a smooth, bounded, integrable initial condition $h_0=h(t=0,cdot)$. The forcing term $eta$ in the right-hand side is a regularized space-time white noise. The exponential of $h$ -- its so-called Cole-Hopf transform -- is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilsons renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer-Magnen. Standard large deviation estimates for $eta$ make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution $h$ may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards-Wilkinson model ($lambda=0$) with renormalized coefficients $ u_{eff}= u+O(lambda^2),D_{eff}=D+O(lambda^2)$.
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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 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.
We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behavior.
The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.
The direct method based on the definition of conserved currents of a system of differential equations is applied to compute the space of conservation laws of the (1+1)-dimensional wave equation in the light-cone coordinates. Then Noethers theorem yields the space of variational symmetries of the corresponding functional. The results are also presented for the standard space-time form of the wave equation.
We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution {`a} la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharpe estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekelands variational principle on the Wasserstein space.
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