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...

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm some singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators. We illustrate the efficiency of this elementary approach on the example of the 3-dimensional generalised parabolic Anderson model equation pBt`Lqu " f puqζ, and the generalised KPZ equation pBt`Lqu " f puqζ`gpuqpBuq 2 , driven by a p1`1q-dimensional space/time white noise.