On the support of solutions to the heat equation with noise

Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

“…This concludes the proof because it is easy to see that (∇ ε F )(x) has the same distribution as F (ε), regardless of the value of x ∈ R, and the latter random variable has the same law as ε (α−1)/2 times a standard Gaussian random variable N , whence 24) uniformly in x ∈ R and for all λ > 0.…”

Malliavin calculus for regularity structures: The case of gPAM

“…In particular it holds for the linear multiplicative stochastic heat equation in dimension d = 1 (cf. [19]) where we recover Mueller's work, [25], and to the linear PAM equation in dimensions d = 2, 3 for which the result appears to be new.…”

“…Indeed, the argument (of Theorem 5.1), despite written in the context of gPAM, adapts immediately to other situations, such as the linear multiplicative stochastic heat equation in dimension d = 1 (cf. [19]) where we recover Mueller's work, [25], and to the linear PAM equation in dimensions d = 2, 3 for which the result appears to be new. Remark that maximum principles have played no role so far in the study of singular SPDEà la Hairer (or Gubinelli et al) -presumably for the simple reason that a maximum principle hings on the second order nature of a PDE, whereas the local solution theory of singular SPDEs is mainly concerned with the regularization properties of convolution with singular kernels (or Fourier multipliers) making no second order assumptions whatsoever.…”

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

“…By Jensen's inequality, it suffices to prove this in the case that ν = 2. We will borrow liberally several localization ideas from two related papers by Mueller [24] and Mueller and Perkins [25].…”