The infinitesimal generator of the stochastic Burgers equation
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The infinitesimal generator of the stochastic Burgers equation Massimiliano Gubinelli1 · Nicolas Perkowski2 Received: 29 October 2018 / Revised: 13 August 2020 © The Author(s) 2020
Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic. Mathematics Subject Classification 60H17
Financial support by DFG via the CRC 1060 and partially by EPSRC Grant Number EP/R014604/1 is gratefully acknowledged. Financial support by DFG via the Heisenberg program and via Research Unit FOR 2402 is gratefully acknowledged.
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Nicolas Perkowski [email protected] Massimiliano Gubinelli [email protected]
1
Hausdorff Center for Mathematics and Institute for Applied Mathematics, Universität Bonn, Bonn, Germany
2
Institut für Mathematik, Freie Universität Berlin, Berlin, Germany
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M. Gubinelli, N. Perkowski
1 Introduction The (conservative) stochastic Burgers equation u : R+ ×T → R (or u : R+ ×R → R) ∂t u = u + ∂x u 2 +
√
2∂x ξ,
(1)
where ξ is a space-time white noise, is one of the most prominent singular stochastic PDEs, a class of equations that are ill posed due to the interplay of very irregular noise and nonlinearities. The difficulty is that u has only distributional regularity (under the stationary measure it is a white noise in space for all times), and therefore the meaning of the nonlinearity ∂x u 2 is dubious. In recent years, new solution theories like regularity structures [20,40] or paracontrolled distributions [26,33] were developed for singular SPDEs, see [38] for an up-to-date and fairly exhaustive review. These theories are based on analytic (as opposed to probabilistic) tools. In the example of the stochastic Burgers equation we roughly speaking use that u is not a generic distribution, but it is a local perturbation of a Gaussian (obtained from ξ ). We construct the nonlinearity and some higher order terms of the Gaussian by explicit computation, and then we freeze the realization of ξ and of the nonlinear terms we just constructed and use pathwise and analytic tools to control the nonlinearity for the (better behaved) remainder. This requires the
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