粗路径理论概览：附带规则结构介绍

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"A Course on Rough Paths" 是一本由 Peter K. Friz 和 Martin Hairer 合著的书籍，旨在补充现有粗糙路径理论的文献。该书不试图全面涵盖所有内容，而是关注某些特定方面。它没有深入探讨非常低规则性的粗糙路径，而是提供了关于一般理论的提示，这些提示可以独立阅读，不会影响对书中其他内容的理解。另外，书中没有系统地研究签名（signatures），即与足够规则路径相关的所有固定区间上的迭代积分，这些积分提供了非线性特征的有趣描述。粗糙路径理论是数学中的一个分支，主要涉及处理低光滑度的路径，这些路径在传统的微分和积分理论中可能难以处理。该书可能适合研究生或更高层次的学习者，提供了一种更加灵活和适应性强的方法来分析不规则数据的动态行为，尤其在金融工程、物理和随机动力系统等领域有广泛应用。书中的“Regularity Structures”部分可能介绍了一种新的框架，用于结构化和分析低规则性路径的复杂行为。Regularity Structures 是 Martin Hairer 的创新性工作，它提供了一种工具，能够形式化和理解具有不同光滑度成分的函数空间中的方程。 Universitext 系列是由一系列数学教科书组成，覆盖了广泛的数学领域，适合硕士及以上水平的学习。这些书籍通常经过作者的精心教学检验，可能采用较为个人化或实验性的方法来阐述主题。随着教学课程的演变，许多成功的书籍经历了多次修订，发展成为成熟的教材。通过阅读 "A Course on Rough Paths"，读者可以预期学习到粗糙路径的基本概念、如何构建和分析它们，以及它们在签名理论中的角色。此外，书中可能还涵盖了与正则性结构相关的现代数学理论，这对于理解非线性问题和高度不规则数据的处理至关重要。对于那些对随机分析、偏微分方程或者金融数学感兴趣的人来说，这本书将是一个宝贵的资源。

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1.1 Controlled differential equations 3

B(ω)

7→(B, B)(ω)

7→Y (ω). (1.5)

The map

is universal in the sense that it depends neither on the initial condition, nor

on the vector ﬁelds driving the stochastic differential equation, but merely consists

of enhancing Brownian motion with iterated integrals of the form

i,j

(s, t) =



(r) −B

(s)



(r) . (1.6)

At this stage, the choice of stochastic integration in (1.6) (e.g. It

o or Stratonovich)

does matter and probabilistic techniques are required for the construction of

Indeed, the map

is only measurable and usually requires the use of some sort

of stochastic integration theory (or some equivalent construction, see for example

Section 10 below for a general construction in a Gaussian, non-semimartingale

context).

The solution map

on the other hand, the solution map to a rough differential

equation (RDE), also known as It

o–Lyons map and discussed in Chapter 8.1, is

purely deterministic and only makes use of analytical constructions. More precisely,

it allows input signals to be arbitrary rough paths which, as discussed in Chapter 2,

are objects (thought of as enhanced paths) of the form

(X, X)

, deﬁned via certain

algebraic properties (which mimic the interplay between a path and its iterated

integrals) and certain analytical, H

older-type regularity conditions. In Chapter 3 these

conditions will be seen to hold true a.s. for

(B, B)

; a typical realisation is thus called

Brownian rough path.

The It

o–Lyons map turns out, cf. Section 8.6, to be “nice” in the sense that it is a

continuous map of both its initial condition and the driving noise

(X, X)

, provided

that the dependency on the latter is measured in a suitable “rough path” metric. In

other words, rough path analysis allows for a pathwise solution theory for SDEs i.e.

for a ﬁxed realisation of the Brownian rough path. The solution map

is however

a much richer object than the original It

o map, since its construction is completely

independent of the choice of stochastic integral and even of the knowledge that the

driving path is Brownian. For example, if we denote by

(resp.

) the maps

B 7→ (B, B)

obtained by It

o (resp. Stratonovich) integration, then we have the almost

sure identities

S ◦Ψ

, S

S ◦Ψ

where

(resp.

) denotes the solution to (1.4) interpreted in the It

o (resp.

Stratonovich) sense. Returning to Theorem 1.2, we see that the convergence there

is really a deterministic consequence of the probabilistic question whether or not

) → Ψ

(B)

in probability and rough path topology, with

= ξ



. This

can be shown to hold in the case of molliﬁer, piecewise linear, and many other

approximations.

So how is this It

o–Lyons map

built? In order to solve (1.1), we need to be able

to make sense of the expression

4 1 Introduction

f(Y

) dX

, (1.7)

where

is itself the as yet unknown solution. Here is where the usual pathwise

approach breaks down: as we have seen in Proposition 1.1 it is in general impossible,

even in the simplest cases, to ﬁnd a Banach space of functions containing Brownian

sample paths and in which (1.7) makes sense. Actually, if we measure regularity

in terms of H

older exponents, then (1.7) makes sense as a limit of Riemann sums

for

and

that are arbitrary

-H

older continuous functions if and only if

α >

The keyword here is arbitrary: in our case the function

is anything but arbitrary!

Actually, since the function

solves (1.1), one would expect the small-scale ﬂuctua-

tions of

to look exactly like the small-scale ﬂuctuations of

in the sense that one

would expect that

s,t

= f(Y

s,t

+ R

s,t

where, for any path

with values in a linear space, we set

s,t

= F

− F

, and

where R

s,t

is some remainder that one would expect to be “of higher order”.

Suppose now that

is a “rough path”, which is to say that it has been “enhanced”

with a two-parameter function

which should be interpreted as giving the values for

i,j

(s, t) =

s,r

. (1.8)

Note here that this identity should be read in the reverse order from what one may be

used to: it is the right hand side that is deﬁned by the left hand side and not the other

way around! The idea here is that if

is too rough, then we do not a priori know

how to deﬁne the integral of

against itself, so we simply postulate its values. Of

course,

cannot just be anything, but should satisfy a number of natural algebraic

identities and analytical bounds, see Chapter 2 below.

Anyway, assuming that we are provided with the data

(X, X)

, then we know how

to give meaning to the integral of components of

against other components of

this is precisely what

encodes. Intuitively, this suggests that if we similarly encode

the fact that

“looks like

at small scales”, then one should be able to extend

the deﬁnition of (1.7) to a large enough class of integrands to include solutions to

(1.1), even when

α <

. One of the achievements of rough path theory is to make

this intuition precise. Indeed, in the framework of rough integration sketched here

and made precise in Chapter 4, the barrier

α =

can be lowered to

α =

. In

principle, this can be lowered further by further enhancing

with iterated integrals

of higher-order, but we decided to focus on the ﬁrst non-trivial case for the sake of

simplicity and because it already covers the most important case when

is given

by a Brownian motion, or a stochastic process with properties similar to those of

Brownian motion. We do however indicate very brieﬂy in Sections 2.4, 4.5 and 7.6

how the theory can be modiﬁed to cover the case

α ≤

, at least in the “geometric”

case when X is a limit of smooth paths.

The simplest way for

to “look like

” is when

Y = G(X)

for some sufﬁciently

regular function

. Despite what one might guess, it turns out that this particular

1.1 Controlled differential equations 5

class of functions

is already sufﬁciently rich so that knowing how to deﬁne

integrals of the form

G(X

) dX

for (non-gradient) functions

allows to give a

meaning to equations of the type (1.1), which is the approach originally developed

in [

Lyo98

]. More recently, Gubinelli realised in [

Gub04

] that, in order to be able to

give a meaning to

given the data

(X, X)

, it is sufﬁcient that

admits a

“derivative” Y

such that

s,t

= Y

s,t

+ R

s,t

with a remainder satisfying

s,t

(|t−s|

2α

)

. This extension of the original theory

turns out to be quite convenient, especially when applying it to problems other than

the resolution of evolution equations of the type (1.1).

An intriguing question is to what extent rough path theory, essentially a theory

of controlled ordinary differential equations, can be extended to partial differential

equations. In the case of ﬁnite-dimensional noise, and very loosely stated, one has

for instance a statement of the following type. (See [

CF09

CFO11

FO14

GT10

Tei11, DGT12] as well as Section 12.1 below.)

Theorem 1.3.

Classes of SPDEs of the form

du = F [u] dt + H[u] ◦ dB

, with

second and ﬁrst order differential operators

and

, respectively, and driven

by ﬁnite-dimensional noise, with the Zakai equation from ﬁltering and stochastic

Hamilton–Jacobi–Bellman (HJB) equations as examples, can be solved pathwise, i.e.

for a ﬁxed realisation of the Brownian rough path. As in the SDE case, the SPDE

solution map factorises as

S ◦Ψ

where

, the solution map to a rough partial

differential equation (RPDE) is continuous in the rough path topology.

As a consequence, if

= δ

∗ ξ

denotes the regularisation of white noise in

time with a compactly supported smooth molliﬁer

that is scaled by

, and if

denotes the random PDE solutions driven by

(instead of

◦dB

) then

converges

in probability. The limiting process does not depend on the choice of molliﬁer

and is viewed as Stratonovich SPDE solution. The same conclusion holds whenever

) → Ψ

(B) in probability and rough path topology.

The case of SPDEs driven by inﬁnite-dimensional noise poses entirely different

problems. Already the stochastic heat equation in space dimension one has not

enough spatial regularity for additional nonlinearities of the type

g(u)∂

(which

arises in applications from path sampling [

Hai11b

HW13

]) or

(∂

(the Kardar–

Parisi–Zhang equation) to be well-deﬁned. In space dimension one, “spatial” rough

paths indexed by

, rather than

, have proved useful here and the quest to handle

dimension larger than one led to the general theory of regularity structures, see

Section 1.3 below.

Rather than trying to survey all applications to date of rough paths to stochastics,

let us say that the past few years have seen an explosion of results made possible by

the use of rough paths theory. New stimulus to the ﬁeld was given by its use in rather

diverse mathematical ﬁelds, including for example quantum ﬁeld theory [

GL09

nonlinear PDEs [

Gub12

], Malliavin calculus [

CFV09

], non-Markovian H

ormander

and ergodic theory, [

CF10

HP13

CHLT12

] and the analysis of chaotic behaviour in

fast-slow systems [KM14].

6 1 Introduction

In view of these developments, we believe that it is an opportune time to try to

summarise some of the main results of the theory in a way that is as elementary as

possible, yet sufﬁciently precise to provide a technical working knowledge of the

theory. We therefore include elementary but essentially complete proofs of several

of the main results, including the continuity and deﬁnition of the It

o–Lyons map,

the lifting of a class of Gaussian processes to the space of rough paths, etc. In

contrast to the available textbook literature [

LQ02

LCL07

FV10b

], we emphasize

Gubinelli’s view on rough integration [

Gub04

Gub10

] which allows to linearise

many considerations and to simplify the exposition. That said, the resulting theory

of rough differential equations is (immediately) seen to be equivalent to Davie’s

deﬁnition [

Dav08

] and, generally, we have tried to give a good idea what other

perspectives one can take on what amounts to essentially the same objects.

1.2 Analogies with other branches of mathematics

As we have just seen, the main idea of the theory of rough paths is to “enhance”

a path

with some additional data

, namely the integral of

against itself, in

order to restore continuity of the It

o map. The general idea of building a larger

object containing additional information in order to restore the continuity of some

nonlinear transformation is of course very old and there are several other theories

that have a similar “ﬂavour” to the theory of rough paths, one of them being the

theory of Young measures (see for example the notes [

Bal00

]) where the value of

a function is replaced by a probability measure, thus allowing to describe limits of

highly oscillatory functions.

Nevertheless, when ﬁrst confronted with some of the notions just outlined, the

ﬁrst reaction of the reader might be that simply postulating the values for the right

hand side of (1.8) makes no sense. Indeed, if

is smooth, then we “know” that there

is only one “reasonable” choice for the integral

against itself, and this is the

Riemann integral. How could this be replaced by something else and how can one

expect to still get a consistent theory with a natural interpretation? These questions

will of course be fully answered in these notes.

For the moment, let us draw an analogy with a very well established branch of

geometric measure theory, namely the theory of varifolds [Alm66, LY02].

Varifolds arise as natural extensions of submanifolds in the context of certain

variational problems. We are not going into details here, but loosely speaking a

-dimensional varifold in

is a (Radon) measure

× G(k, n)

, where

G(k, n)

denotes the space of all

-dimensional subspaces of

. Here, one should

interpret

G(k, n)

as the space of all possible tangent spaces at any given point for

-dimensional submanifold of

. The projection of

onto

should then be

interpreted as a generalisation of the natural “surface measure” of a submanifold,

while the conditional (probability) measure on

G(k, n)

induced at almost every point

by disintegration should be interpreted as selecting a (possibly random) tangent

space at each point. Why is this a reasonable extension of the notion of submanifold?

1.2 Analogies with other branches of mathematics 7

Consider the following sequence M

of one-dimensional submanifolds of R

⇒

It is intuitively clear that, as

ε → 0

, this converges to a circle, but the right half has

twice as much “weight” as the left half so that, if we were to describe the limit

simply as a manifold, we would have lost some information about the convergence of

the surface measures in the process. More dramatically, there are situations where one

has a sequence of smooth manifolds such that the limit is again a smooth manifold,

but with a limiting “tangent space” which has nothing to do with the actual tangent

space of the limit! Indeed, consider the sequence of one-dimensional submanifolds

of R

given by

This time, the limit is a piece of straight line, which is in principle a perfectly nice

smooth submanifold, but the limiting tangent space is deterministic and makes a 45

◦

angle with the canonical tangent space associated to the limit.

The situation here is philosophically very similar to that of the theory of rough

paths: a subset

M ⊂ R

may be sufﬁciently “rough” so that there is no way of

canonically associating to it either a

-dimensional Riemannian volume element,

or a

-dimensional tangent space, so we simply postulate them. The two examples

given above show that even in situations where

is a nice smooth manifold, it

still makes sense to associate to it a volume element and / or tangent space that are

different from the ones that one would construct canonically. A similar situation

arises in the theory of rough paths. Indeed, it may so happen that

is actually

given by a smooth function. Even so, this does not automatically mean that the right

hand side of (1.8) is given by the usual Riemann integral of

against itself. An

explicit example illustrating this fact is given in Exercise 2.17 below. Similarly to

the examples of “non-canonical” varifolds given above, “non-canonical” rough paths

can also be constructed as limits of ordinary smooth paths (with the second-order

term

deﬁned by (1.8) where the integral is the usual Riemann integral), provided

that one takes limits in a suitably weak topology.

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