Multiplicative stochastic heat equations on the whole space

By Martin Hairer, Cyril Labbé. 2015

138 citations recorded by the source corpus.

We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson model on $\mathbf{R}^3$, and on the other hand the KPZ equation on $\mathbf{R}$ via the Cole-Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/25f2f0f242fe7bdf744b95fa2541fad0d7ce5fd8

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Citations in the corpus

A CLASS OF GROWTH MODELS RESCALING TO KPZ (2015, background): “not expect this to be the case [QV13]. 6. Unbounded space. Our results are on a ﬁnite interval with periodic boundary conditions, and extending them to the real line represents a challenge. Recently, [HL15] introduced weighted spaces allowing the extension of the results on convergence of smoothed n”
Discretisations of rough stochastic PDEs (2015, methodology): “This lays the foundations of a systematic approximation theory which can in principle be applied to many other singular SPDEs, for example, stochastic Burgers-type equations [17, 22, 23], the KPZ equation [3, 18, 29] or the continuous parabolic Anderson model [19, 21].”
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (2018, methodology): “ space of paths over a manifold [43], and other interesting models like those related to Abelian gauge theories [82] or some equations PARACONTROLLED APPROACH TO 3-dSTOCHASTIC NLW 3 in the full space [45]. On the other hand, the theory of paracontrolled distributions has revealed itself as an eectiv”
Heat semigroup and singular PDEs (2015, methodology): “us – see example 2 in section 2.1. (We are also able to deal with unbounded manifolds by working with weighted noises; which can also be done with regularity structures in Rd, as testiﬁed by the work [35] of Hairer and Labb´e on the linear PAM equation in R3, and their elementary approach [36] to th”