Heat semigroup and singular PDEs

Abstract

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anderson model equation and prove, under mild geometric conditions, its well-posed character in Hölders spaces, in small time on a potentially unbounded 2-dimensional Riemannian manifold, for an equation driven by a weighted noise, and for all times for the linear parabolic Anderson model equation in 2-dimensional unbounded manifolds. This machinery can be extended to an even more singular setting and deal with Sobolev scales of spaces rather than Hölder spaces. Contents 1 2 B.4 Resolution of PAM in such a 2-dimensional setting 79 of paracontrolled calculus described in a second. Recall that if M and N are two continuous martingales one has d(M N ) = M dN + N dM + d M, N .The above space of functions u = Π v (X) + u ♯ , can be turned into a Banach space. Once this ansatz for the solution space has been chosen, remark that the product u ξ can formally be written asSince u ♯ has Hölder regularity strictly bigger than 1 and ξ is −1 − -regular, the sum of their regularity indices is positive, and the term Π(u ♯ , ξ) is perfectly well-defined. This lives us with Π Π v (X), ξ as the only undefined term. The following fact is the workhorse of paracontrolled calculus. The trilinear maphappens to depend continuously on a, b and c provided they are Hölder distributions, with the sum of their Hölder exponents positive. Note the paralell between the continuity of this 'commutator' and the rule for stochastic differentials, for which, given another continuous martingale P , we haveThe formal product u ξ can thus be written as a sum of well-defined terms plus the formal product v Π(X, ξ), with a diagonal term Π(X, ξ) still undefined on a purely analytic basis. This is where probability comes into play. If one regularizes ξ into ξ ε , with X ε defined accordingly, one can prove that there exists a function/constant C ε such that the renormalized quantity ξ (2),ε := Π X ε , ξ ε − C ε converges in probability to some limit distribution ξ (2) of Hölder regularity 0 − = 1 − + (−1 − ); this is enough to make sense of the product v ξ (2) on an analytical basis; but replacing Π X ε , ξ ε by Π X ε , ξ ε − C ε in the decomposition of u ξ ε amounts to looking at the product u ξ ε − C ε . The enhancement ξ := ξ, ξ (2) of ξ is called a rough, or enhanced, distribution, and one can use it to define the product u ξ from the above formulas. At that point, it does not come as a surprise that one can then set (PAM) equation as a fixed point problem in the ansatz...

High Order Paracontrolled Calculus

“…This norm does not depend on the integer b ą |α| 2 , and the two spaces H α and C α coincide and have equivalent norms when 0 ă α ă 1 -see for instance Proposition 2.5 in [1]. These notions have parabolic counterparts which we now introduce.…”

“…as sums of smooth functions with localized frequencies; the paraproduct of g by f is defined as 1) and the resonant part as Π 0 pf, gq " ÿ |i´j|ď1 ∆ i pf q∆ j pgq, so we have the product decomposition…”

Quasilinear generalized parabolic Anderson model equation

“…Mention here that different choices can be done for the norm on the space of controlled functions; different purposes may lead to different choices -see for instance the study of the 2-dimensional generalised (PAM) equation done in [5]. Given a positive time horizon T , set u T 0 :" P T u 0 , to shorten notations, and recall for future use the bounds…”

“…(It can be seen to hold as follows. Writing L for the operator divpg∇¨q and setting R s :" psLqe´s L , we know that R s u is bounded in L 8 by s´a {2 if u is C´a -this semigroup picture of Hölder spaces is explained and used for instance in [5]. The above continuity estimate comes then from the integral representation Q t " ş 8 t R s ds s .)…”

“…Given a real regularity exponent α, we denote by C α the spatial Hölder space and by C α the parabolic Hölder space, both defined for instance in terms of Besov spaces built from the parabolic operator L -see e.g. [5]. For α P p0, 2q, the parabolic space C α , or C α T , coincides as a set with C T C α X C α{2 T L 8 , and the Besov norm on C α is equivalent to the elementary norm…”