**Important note:** As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.

In these notes, random variables will be denoted in boldface.

Definition 1A real random variable is said to be normally distributed with mean and variance if one hasfor all test functions . Similarly, a complex random variable is said to be normally distributed with mean and variance if one has

for all test functions , where is the area element on .

A

real Brownian motionwith base point is a random, almost surely continuous function (using the locally uniform topology on continuous functions) with the property that (almost surely) , and for any sequence of times , the increments for are independent real random variables that are normally distributed with mean zero and variance . Similarly, acomplex Brownian motionwith base point is a random, almost surely continuous function with the property that and for any sequence of times , the increments for are independent complex random variables that are normally distributed with mean zero and variance .

Remark 2Thanks to the central limit theorem, the hypothesis that the increments be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).

Real and complex Brownian motions exist from any base point or ; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

- (i) (Translation invariance) If is a real Brownian motion with base point , and , show that is a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and , show that is a complex Brownian motion with base point .
- (ii) (Dilation invariance) If is a real Brownian motion with base point , and is non-zero, show that is also a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and is non-zero, show that is also a complex Brownian motion with base point .
- (iii) (Real and imaginary parts) If is a complex Brownian motion with base point , show that and are independent real Brownian motions with base point . Conversely, if are independent real Brownian motions of base point , show that is a complex Brownian motion with base point .

The next lemma is a special case of the optional stopping theorem.

Lemma 4 (Optional stopping identities)

- (i) (Real case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to (or more precisely, this event is measurable with respect to the algebra generated by this proprtion of the trajectory). Then
and

and

- (ii) (Complex case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to . Then

*Proof:* (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take . Let be an upper bound for . Since is a real normally distributed variable with mean zero and variance , we have

and

and

By the law of total expectation, we thus have

and

and

where the inner conditional expectations are with respect to the event that attains a particular point in . However, from the independent increment nature of Brownian motion, once one conditions to a fixed point , the random variable becomes a real normally distributed variable with mean and variance . Thus we have

and

and

which give the first two claims, and (after some algebra) the identity

which then also gives the third claim.

Exercise 5Prove the second part of Lemma 4.

** — 1. Conformal invariance of Brownian motion — **

Let be an open subset of , and a point in . We can define the *complex Brownian motion with base point restricted to * to be the restriction of a complex Brownian motion with base point to the first time in which the Brownian motion exits (or if no such time exists). We have a fundamental conformal invariance theorem of Lévy:

Theorem 6 (Lévy’s theorem on conformal invariance of Brownian motion)Let be a conformal map between two open subsets of , and let be a complex Brownian motion with base point restricted to . Define a rescaling byNote that this is almost surely a continuous strictly monotone increasing function. Set (so that is a homeomorphism from to ), and let be the function defined by the formula

Then is a complex Brownian motion with base point restricted to .

Note that this significantly generalises the translation and dilation invariance of complex Brownian motion.

*Proof:* (Somewhat informal – to do things properly one should first set up Ito calculus) To avoid technicalities we will assume that is bounded above and below on , so that the map is uniformly bilipschitz; the general case can be obtained from this case by a limiting argument that is not detailed here. With this assumption, we see that almost surely extends continuously to the endpoint time if this time is finite. Once one conditions on the value of and up to this time , we then extend this motion further (if ) by declaring for to be a complex Brownian motion with base point , translated in time by . Now is defined on all of , and it will suffice to show that this is a complex Brownian motion based at . The basing is clear, so it suffices to show for all times , the random variable is normally distributed with mean and variance .

Let be a test function. It will suffice to show that

If we define the field

for and , with , then it will suffice to prove the more general claim

for all and (with the convention that is just Brownian motion based at if lies outside of ), where

As is well known, is smooth on and solves the backwards heat equation

on this domain. The strategy will be to show that also solves this equation.

Let and . If then clearly . If instead and , then is a Brownian motion and then we have . Now suppose that be small enough that , where is an upper bound for on . Let be the first time such that either or

Then if we let be the quantity

then and . Let us now condition on a specific value of , and on the trajectory up to time . Then the (conditional) distribution of is that of , and hence the conditional expectation is . By the law of total expectation, we conclude the identity

Next, we obtain the analogous estimate

From Taylor expansion we have

Taking expectations and applying Lemma 4, (2) and Hölder’s inequality (which can interpolate between the bounds and to conclude ), we obtain the desired claim (3). Subtracting, we now have

The expression in the expectation vanishes unless , hence by the triangle inequality

Iterating this using the fact that vanishes at , and sending to zero (noting that the cumulative error term will go to zero since ), we conclude that for all , giving the claim.

One can use Lévy’s theorem (or variants of this theorem) to prove various results in complex analysis rather efficiently. As a quick example, we sketch a Brownian motion-based proof of Liouville’s theorem (omitting some technical steps). Suppose for contradiction that we have a nonconstant bounded entire function . If is a complex Brownian motion based at , then a variant of Levy’s theorem can be used to show that the image is a time parameterisation of Brownian motion. But it is easy to show that Brownian motion is almost surely unbounded, so the image cannot be bounded.

If is an open subset of whose complement contains an arc, then one can show that for any , the complex Brownian motion based at will hit the boundary of in a finite time . The location where this motion first hits the boundary is then a random variable in ; the law of this variable is called the *harmonic measure* of with base point , and we will denote it by ; it is a probability measure on . The reason for the terminology “harmonic measure” comes from the following:

Theorem 7Let be a bounded open subset of , and let be a harmonic (or holomorphic) function that extends continuously to . Then for any , one has the representation formula

*Proof:* (Informal) For simplicity let us assume that extends smoothly to some open neighbourhood of . Let be the motion that is equal to up to time , and then is constant at for all later times. A variant of the Taylor expansion argument used to prove Lévy’s theorem shows that

for any , which on iterating and sending to zero implies that is independent of time. Since this quantity converges to as and to as , the claim follows.

This theorem can also extend to unbounded domains provided that does not grow too fast at infinity (for instance if is bounded, basically thanks to the neighbourhood recurrent properties of complex Brownian motion); we do not give a precise statement here. Among other things, this theorem gives an immediate proof of the maximum principle for harmonic functions, since if on the boundary then from the triangle inequality one has for all . It also gives an alternate route to Liouville’s theorem: if is entire and bounded, then applying the maximum principle to the complement of a small disk we see that for all distinct .

When the boundary is sufficiently nice (e.g. analytic), the harmonic measure becomes absolutely continuous with respect to one-dimensional Lebesgue measure; however, we will not pay too much attention to these sorts of regularity issues in this set of notes.

From Levy’s theorem on the conformal invariance of Brownian motion we deduce the conformal invariance of harmonic measure, thus for any conformal map that extends continuously to the boundaries and any , the harmonic measure of with base point is the pushforward of the harmonic measure of with base point , thus

for any continuous compactly supported test function , and also

for any (Borel) measurable .

- (i) If and , show that the measure on the unit circle is given by
where is arclength measure. In particular, when , then is the uniform measure on the unit circle.

- (ii) If and , show that the measure on the real line is given by
(For this exercise one can assume that harmonic measure is well defined for unbounded domains, and that the representation formula (4) continues to hold for bounded harmonic or holomorphic functions.)

Exercise 9 (Brownian motion description of conformal mapping)Let be the region enclosed by a Jordan curve , and let be three distinct points on in anticlockwise order. Let be three distinct points on the boundary of the unit disk , again traversed in anticlockwise order. Let be the conformal map that takes to for (the existence and uniqueness of this map follows from the Riemann mapping theorem). Let , and for , let be the probability that the terminal point of Brownian motion at with base point lies in the arc between and (here we use the fact that the endpoints are hit with probability zero, or in other words that the harmonic measure is continuous; see Exercise 15 below). Thus are non-negative and sum to . Let be the complex numbers , , . Show the crossratio identityIn principle, this allows one to describe conformal maps purely in terms of Brownian motion.

We remark that the link between Brownian motion and conformal mapping can help gain an intuitive understanding of the Carathéodory kernel theorem (Theorem 12 from Notes 3). Consider for instance the example in Exercise 13 from those notes. It is intuitively clear that a Brownian motion based at the origin will very rarely pass through the slit beween and , instead hitting the right side of the boundary of first. As such, the harmonic measure of the left side of the bounadry should be very small, and in fact one can use this to show that the preimage under of the region to the left of the boundary goes to zero in diameter as , which helps explain why the limiting function does not map to this region at all.

Exercise 10 (Brownian motion description of conformal radius)

- (i) Let and with . Show that the probability that the Brownian motion hits the circle before it hits is equal to . (
Hint:is harmonic away from the origin.)- (ii) Let be a simply connected proper subset of , let be a point in , and let be the conformal radius of around . Show that for small , the probability that a Brownian motion based at a point with will hit the circle before it hits the boundary is equal to , where denotes a quantity that goes to zero as .

Exercise 11Let be a connected subset of , let be a Brownian motion based at the origin, and let be the first time this motion exits . Show that the probability that hits is at least for some absolute constant . (Hint:one can control the event that makes a “loop” around a point in at radius less than , which is enough to force intersection with , at least if one works some distance away from the boundary of the disk.)

We now sketch the proof of a basic Brownian motion estimate that is useful in applications. We begin with a lemma that says, roughly speaking, that “folding” a set reduces the probability of it being hit by Brownian motion.

Lemma 12Let , and let be a closed subset of the unit disk . Write and , and write (i.e. reflected onto the upper half-plane). Let be a complex Brownian motion based at , and let be the first time this motion hits the boundary of . Then

*Proof:* (Informal) To illustrate the argument at a heuristic level, let us make the (almost surely false) assumption that the Brownian motion only crosses the real axis at a finite set of times before hitting the disk. Then the Brownian motion would split into subcurves for , with the convention that . Each subcurve would lie in either the upper half-plane or the lower half-plane, with equal probability of each; furthermore, one could arbitrarily apply complex conjugation to one or more of these subcurves and still obtain a motion with the same law. Observe that if one conditions on the Brownian motion up to time , and the subcurve has a probability of hitting when it lies in the upper half-plane, and a probability of hitting when it lies in the lower half-plane, then it will have a probability of at most of hitting when it lies in the upper half-plane, and probability of hitting when it lies in the lower half-plane; thus the probability of this subcurve hitting is less than or equal to that of it hitting . In principle, the lemma now follows from repeatedly applying the law of total expectation.

This naive argument does not quite work because a Brownian motion starting at a real number will in fact almost surely cross the real axis an infinite number of times. However it is possible to adapt this argument by redefining the so that after each time , the Brownian motion is forced to move some small distance before one starts looking for the next time it hits the real axis. See the proof of Lemma 6.1 of these notes of Lawler for a complete proof along these lines.

This gives an inequality similar in spirit to the Grötzsch modulus estimate from Notes 2:

Corollary 13 (Beurling projection theorem)Let , and let be a compact connected subset the annulus that intersects both boundary circles of the annulus. Let be a complex Brownian motion based at , and let be the first time this motion hits the outer boundary of the annulus. Then the probability that intersects is greater than or equal to the probability that intersects the interval .

*Proof:* (Sketch) One can use the above lemma to fold around the real axis without increasing the probability of being hit by Brownian motion. By rotation, one can similarly fold around any other line through the origin. By repeatedly folding in this fashion to reduce its angular variation, one can eventually replace with a set that lies inside the sector for any . However, by the monotone convergence theorem, the probability that intersects this sector converges to the probability that it intersects in the limit , and the claim follows.

Exercise 14With the notation as the above corollary, show that the probability that intersects the interval is . (Hint:apply a square root conformal map to the disk with removed, and then compare with the half-plane harmonic measure from Exercise 8(ii).)

The following consequence of the above estimate, giving a sort of Hölder regularity of Brownian measure, is particularly useful in applications.

Exercise 15 (Beurling estimate)Let be an open set not containing , with the property that the connected component of containing intersects the unit circle . Let be such that . Then for any , one has ; that is to say, the probability that a Brownian motion based at exits at a point within from the origin is . (Hint:one can use conformal mapping to show that the probability appearing at the end of Corollary 13 is .) Conclude in particular that harmonic measures are always continuous (they assign zero to any point).

Exercise 16Let be a region bounded by a Jordan curve, let , let be the Brownian motion based at , and let be the first time this motion exits . Then for any , show that the probability that the curve has diameter at least is at most .

Exercise 17Let be a conformal map with , and let be a curve with and for . Show that(Hint: use Exercise 11.)

** — 2. Half-plane capacity — **

One can use Brownian motion to construct other close relatives of harmonic measure, such Green’s functions, excursion measures. See for instance these lecture notes of Lawler for more details. We will focus on one such use of Brownian motion, to interpret the concept of *half-plane capacity*; this is a notion that is particularly well adapted to the study of chordal Loewner equations (it plays a role analogous to that of conformal radius for the radial Loewner equation).

Let be the upper half-plane. A subset of the upper half-plane is said to be a *compact hull* if it is bounded, closed in , and the complement is simply connected. By the Riemann mapping theorem, for any compact hull , there is a unique conformal map which is normalised at infinity in the sense that

for some complex numbers . The quantity is particularly important and will be called the *half-plane capacity* of and denoted .

In general, we have the following Brownian motion characterisation of half-plane capacity:

Proposition 19Let be a compact hull, with conformal map and half-plane capacity .

- (i) If is complex Brownian motion based at some point , and is the first time this motion exits , then
- (ii) We have

*Proof:* (Sketch) Part (i) follows from applying Theorem 7 to the bounded harmonic function . Part (ii) follows from part (i) by setting for a large , rearranging, and sending using (5).

Among other things, this proposition demonstrates that for all , and that the half-plane capacity is always non-negative (in fact it is not hard to show from the above proposition that it is strictly positive as long as is non-empty).

If are two compact hulls with , then will map conformally to the complement of in . Thus is also a convex hull, and by the uniqueness of Riemann maps we have the identity

which on comparing Laurent expansions leads to the further identity

In particular we have the monotonicity , with equality if and only if . One may verify that these claims are consistent with Exercise 18.

Exercise 20 (Submodularity of half-plane capacity)Let be two compact hulls.

- (i) If , show that
(

Hint:use Proposition 19, and consider how the times in which a Brownian motion exits , , , and are related.)- (ii) Show that

Exercise 21Let be a compact hull bounded in a disk . For any , show thatas , where is complex Brownian motion based at and is the first time it exits . Similarly, for any , show that

This formula gives a Brownian motion interpretation for on the portion of the boundary of . It can be used to give useful quantitative estimates for in this region; see Section 3.4 of Lawler’s book.

** — 3. The chordal Loewner equation — **

We now develop (in a rather informal fashion) the theory of the chordal Loewner equation, which roughly speaking is to conformal maps from the upper half-plane to the complement of complex hulls as the radial Loewner equation is to conformal maps from the unit disk to subsets of the complex plane. A more rigorous treatment can be found in Lawler’s book.

Suppose one has a simple curve such that and . There are important and delicate issues regarding the regularity hypotheses on this curve (which become particularly important in SLE, when the regularity is quite limited), but for this informal discussion we will ignore all of these issues.

For each time , the set forms a compact hull, and so has some half-plane capacity . From the monotonicity of capacity, this half-plane capacity is increasing in . It is traditional to normalise the curve so that

this is analogous to normalising the Loewner chains from Notes 3 to have conformal radius at time . A basic example of such normalised curves would be the curves for some fixed , since the normalisation follows from (6).

Let be the conformal maps associated to these compact hulls. From (8) we will have

for any and , where is the conformal map associated to the compact hull . From (9) this hull has half-plane capacity , thus we have the Laurent expansion

It can be shown (using the Beurling estimate) that extends continuously to the tip of the curve , and attains a real value at that point; furthermore, depends continuously on . See Lemma 4.2 of Lawler’s book. As such, should be a short arc (of length ) starting at . If , it is possible to use a quantitative version of Exercise 21 (again using the Beurling estimate) to obtain an estimate basically of the form

for any fixed . If is non-zero, we instead have

For instance, if , then for all , and from Exercise 18 we have the exact formula

Inserting (12) into (11) and using the chain rule, we obtain

and we then arrive at the *(chordal) Loewner equation*

for all and . This equation can be justified rigorously for any simple curve : see Proposition 4.4 of Lawler’s book. Note that the imaginary part of is negative, which is consistent with the observation made previously that the imaginary part of is decreasing in .

We have started with a chain of compact hulls associated to a simple curve, and shown that the resulting conformal maps obey the Loewner equation for some continuous driving term . Conversely, suppose one is given a continuous driving term . It follows from Picard existence and uniqueness theorem that for each there is a unique maximal time of existence such that the ODE (13) with initial data can be solved for time , one can show that for each time , is a conformal map from to with the Laurent expansion

hence the complement are an increasing sequence of compact hulls with half-plane capacity . Proving complex differentiability of can be done from first principles, and the Laurent expansion near infinity is also not hard; the main difficulty is to show that the map is surjective, which requires solving (13) backwards in time (and here one can do this indefinitely as now one is moving away from the real axis instead of towards it). See Theorem 4.6 of Lawler’s book for details (in fact a more general theorem is proven, in which the single point is replaced by a probability measure, analogously to how the radial Loewner equation uses Herglotz functions instead of a single driving function when not restricted to slit domains). However, there is a subtlety, in that the hulls are not necessarily the image of simple curves . This is often the case for short times if the driving function does not oscillate too wildly, but it can happen that the curve that one would expect to trace out eventually intersects itself, in which case the region it then encloses must be absorbed into the hull (cf. the “pinching off” phenomenon in the Carathéodory kernel theorem). Nevertheless, it is still possible to have Loewner chains that are “generated” by non-simple paths , in the sense that consists of the unbounded connected component of the complement .

There are some symmetries of the transform from the to the . If one translates by a constant, , then the resulting domains are also translated, , and . Slightly less trivially, for any , if one performs a rescaled dilation , then one can check using (13) that , and the corresponding conformal maps are given by . On the other hand, just performing a scalar multiple on the driving force can transform the behavior of dramatically; the transform from to is very definitely not linear!

** — 4. Schramm-Loewner evolution — **

In the previous section, we have indicated that every continuous driving function gives rise to a family of conformal maps obeying the Loewner equation (13). The (chordal) Schramm-Loewner evolution () with parameter is the special case in which the driving function takes the form for some real Brownian motion based at the origin. Thus is now a random conformal map from a random domain , defined by solving the Schramm-Loewner equation

with initial condition for , and with defined as the set of all for which the above ODE can be solved up to time taking values in . The parameter cannot be scaled away by simple renormalisations such as scaling, and in fact the behaviour of is rather sensitive to the value of , with special behaviour or significance at various values such as playing particularly special roles; there is also a duality relationship between and which we will not discuss here.

The case is rather boring, in which is deterministic, and is just with the line segment between and removed. The cases are substantially more interesting. It is a non-trivial theorem (particularly at the special value ) that is almost surely generated by some random path ; see Theorem 6.3 of Lawler’s book. The nature of this path is sensitive to the choice of parameter :

- For , the path is almost surely simple and goes to infinity as ; it also avoids the real line (except at time ).
- For ; it also has non-trivial intersection with the real line.
- For , the path is almost surely space-filling (which of course also implies that ), and also hits every point on .

See Section 6.2 of Lawler’s book. The path becomes increasingly fractal as increases: it is a result of Rohde and Schramm and Beffara that the image almost surely has Hausdorff dimension .

We have asserted that defines a random path in that starts at the origin and generally “wanders off” to infinity (though for it keeps recurring back to bounded sets infinitely often). By the Riemann mapping theorem, we can now extend this to other domains. Let be a simply connected open proper subset of whose boundary we will assume for simplicity to be a Jordan curve (this hypothesis can be relaxed). Let be two distinct points on the boundary . By the Riemann mapping theorem and Carathéodory’s theorem (Theorem 20 from Notes 2), there is a conformal map whose continuous extension maps and to and respectively; this map is unique up to rescalings for . One can then define the Schramm-Loewner evolution on from to to be the family of conformal maps for , where is the usual Schramm-Loewner evolution with parameter . The Schramm-Loewner evolution on is well defined up to a time reparameterisation . The Markovian and stationary nature of Brownian motion translates to an analogous Markovian and conformally invariant property of . Roughly speaking, it is the following: if is any reasonable domain with two boundary points , is on this domain from to with associated path , and is any time, then after conditioning on the path up to time , the remainder of the path has the same image as the path on the domain from to . Conversely, under suitable regularity hypotheses, the processes are the *only* random path processes on domains with this property (much as Brownian motion is the only Markovian stationary process, once one normalises the mean and variance). As a consequence, whenever one now a random path process that is known or suspected to enjoy some conformal invariance properties, it has become natural to conjecture that it obeys the law of (though in some cases it is more natural to work with other flavours of SLE than the chordal SLE discussed here, such as radial SLE or whole-plane SLE). For instance, in the pioneering work of Schramm, this line of reasoning was used to conjecture that the loop-erased random walk in a domain has the law of (radial) ; this conjecture was then established by Lawler, Schramm, and Werner. Many further processes have since been either proven or conjectured to be linked to one of the SLE processes, such as the limiting law of a uniform spanning tree (proven to be ), interfaces of the Ising model (proven to be ), or the scaling limit of self-avoiding random walks (conjectured to be ). Further discussion of these topics is beyond the scope of this course, and we refer the interested reader to Lawler’s book for more details.

## 6 comments

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29 May, 2018 at 12:06 pm

AnonymousWhat is known about the original motivation to introduce the SLE ?

30 May, 2018 at 12:04 pm

Terence TaoMy understanding is that Schramm introduced the (radial) SLE in his 2000 paper to study the scaling limit of loop-erased random walk. Schramm’s ICM lecture also gives some further motivation (though not the original one) coming from Smirnov’s proof of conformal invariance of hexagonal percolation.

4 June, 2018 at 7:38 am

AnonymousHi Terry, Rohde and Schramm’s paper proved that SLE is almost surely a curve with Hausdorff dimension at most {\min(1 + \frac{\kappa}{8}, 2)}. The exact Hausdorff dimension is established by Beffara in a separate paper: https://arxiv.org/pdf/math/0211322.pdf.

[Corrected, thanks – T.]5 June, 2018 at 2:04 am

AnonymousDear Terry,

I think the scaling limit of self-avoiding random walk is conjectured to be SLE_{8/3}, not SLE_{4/3}. See for example, https://en.wikipedia.org/wiki/Self-avoiding_walk

[Corrected, thanks – T.]5 June, 2018 at 9:15 am

John MangualI’ve always found the SLE curves exceedingly ugly insofar as I have read about them. Yet, these are the curves that provably are the limit shapes of the interfaces between regions that arise in statistical models such as Uniform Spanning Tree or Potts or Percolation (of various types and colors) and we are looking for the SLE constant.

Even more basic, is that the image Brownian motion under conformal transformation is a time change of Brownian motion. So this is like doing random walk in Polar Coodinates or some other very complicated grid network.

6 June, 2018 at 2:16 pm

Pablo LessaI think Corollary 13 should say that the probability of intersecting is greater than or equal to that of intersecting the segment .

[Corrected, thanks – T.]