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Register a new userNonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups
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Consider the stochastic heat equation $partial_tu=mathscr{L}u+lambdasigma(u)xi$, where $mathscr{L}$ denotes the generator of a L{e}vy process on a locally compact Hausdorff Abelian group $G$, $sigma:mathbf{R}tomathbf{R}$ is Lipschitz continuous, $lambdagg1$ is a large parameter, and $xi$ denotes space-time white noise on $mathbf{R}_+times G$. The main result of this paper contains a near-dichotomy for the (expected squared) energy $mathrm{E}(|u_t|_{L^2(G)}^2)$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $u$ is intermittent, the energy of the solution behaves generically as $exp{operatorname {const}cdot,lambda^2}$ when $G$ is discrete and $geexp{operatorname {const}cdot,lambda^4}$ when $G$ is connected.
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In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic control problem for an evolution system in an Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.
It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $dot{u}=frac12 u+uxi$, where $xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are macroscopically multifractal. See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as stretch factors. A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.
Let $mathscr{T}$ be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the $sf Pi$ map provides a linear parametrization of the nonlinear space of admissible models $sf M=(g,Pi)$ on $mathscr{T}$, in terms of the family of para-remainders used in the representation. We give an explicit description of the action of the most general class of renormalization schemes presently available on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps; the BHZ renormalization scheme has that property.
We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Phi^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
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