Framework for studying stochastic partial differential equations
Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory.[1] The framework covers the Kardar–Parisi–Zhang equation , the
equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.
Hairer won the 2021 Breakthrough Prize in mathematics for introducing regularity structures.[2]
Definition
A regularity structure is a triple
consisting of:
- a subset
(index set) of
that is bounded from below and has no accumulation points;
- the model space: a graded vector space
, where each
is a Banach space; and
- the structure group: a group
of continuous linear operators
such that, for each
and each
, we have
.
A further key notion in the theory of regularity structures is that of a model for a regularity structure, which is a concrete way of associating to any
and
a "Taylor polynomial" based at
and represented by
, subject to some consistency requirements.
More precisely, a model for
on
, with
consists of two maps
,
.
Thus,
assigns to each point
a linear map
, which is a linear map from
into the space of distributions on
;
assigns to any two points
and
a bounded operator
, which has the role of converting an expansion based at
into one based at
. These maps
and
are required to satisfy the algebraic conditions
,
,
and the analytic conditions that, given any
, any compact set
, and any
, there exists a constant
such that the bounds
,
,
hold uniformly for all
-times continuously differentiable test functions
with unit
norm, supported in the unit ball about the origin in
, for all points
, all
, and all
with
. Here
denotes the shifted and scaled version of
given by
.
References