Informal course:

Paracontrolled distributions and singularstochastic PDEs

Tadahiro Oh(Notes taken by Justin Forlano)

Spring 2018

Contents:

• Lecture 1: Introduction, Littlewood-Paley theory, paraproducts,local well-posedness of the 2-d SQE 3

• Lecture 2: Local well-posedness of the 2-d SQE with rough initial data,heat semigroup estimate (Schauder estimate),Wiener chaos estimate 11

• Lecture 3: Wick renormalization 23

• Lecture 4: Reformulation of the 3-d SQE (a la Mourrat-Weber) 37

• Lecture 5: Local well-posedness of the 3-d SQE 52

• Lecture 6: Nonlinear estimates and commutator estimates 64

• Lecture 7: Commutator estimate,construction of stochastic objects I 74

• Lecture 8: Construction of stochastic objects II: , , and 84

• Lecture 9: Construction of stochastic objects III:On multiple Wiener-Ito integrals, reduction to the diagonal case,estimates on

=, and a renormalization for

=94

• Lecture 10: Construction of stochastic objects IV:=

and=

106

• Appendix: On the moments of a complex-valued Gaussian random variable 130

N

2

| n - n | ~ | n |2 2

| n |12

1)

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