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Article overview
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Geometric stochastic heat equations | | Yvain Bruned
; Franck Gabriel
; Martin Hairer
; Lorenzo Zambotti
; | | Date: |
8 Feb 2019 | | Abstract: | We consider a natural class of $mathbf{R}^d$-valued one-dimensional
stochastic PDEs driven by space-time white noise that is formally invariant
under the action of the diffeomorphism group on $mathbf{R}^d$. This class
contains in particular the KPZ equation, the multiplicative stochastic heat
equation, the additive stochastic heat equation, and rough Burgers-type
equations. We exhibit a one-parameter 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 It^o calculus (additive and
multiplicative stochastic heat equation), rough path theory (rough Burgers-type
equations), or the Hopf-Cole 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 It^o’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 It^o
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 $L^2$-gradient flow for the Brownian bridge
measure. | | Source: | arXiv, 1902.2884 | | Services: | Forum | Review | PDF | Favorites | |
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