Abstract
We consider a natural class of Rdvalued onedimensional stochastic PDEs driven by spacetime white noise that is formally invariant under the action of the diffeomorphism group on Rd. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgerstype equations. We exhibit a oneparameter family of solution theories with the following properties:
 For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgerstype equations), or the HopfCole transform (KPZ equation).
 Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.
 Every solution theory satisfies an analogue of Itô's isometry.
 The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.
In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the L2gradient flow for the Brownian loop measure.
 For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgerstype equations), or the HopfCole transform (KPZ equation).
 Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.
 Every solution theory satisfies an analogue of Itô's isometry.
 The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.
In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the L2gradient flow for the Brownian loop measure.
Original language  English 

Publisher  ArXiv 
Publication status  Published  12 Feb 2019 
Keywords
 math.PR
 math.AP
 60H15, 60L30
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Profiles

Yvain Bruned
 School of Mathematics  Lecturer in Mathematical Sciences
Person: Academic: Research Active (Teaching)