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The infinitesimal generator of the stochastic Burgers equation
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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 [GP18a] to a wider class of equations.
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Cited by 2 Pith papers
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Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes
Under a frame condition on the stationary density, scaling limits of multi-species fluctuation fields solve a coupled Burgers SPDE, the formal gradient of a coupled KPZ equation.
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Stationary Directed Polymers and Energy Solutions of the Burgers Equation
The stationary O'Connell-Yor polymer increments converge to energy solutions of the stochastic Burgers equation without using the Cole-Hopf transform.
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