Brownian Motion and PDEs (and Fourier series) Stefan Steinerberger - PowerPoint PPT Presentation

Brownian Motion and PDEs (and Fourier series) Stefan Steinerberger Peter W. Jones Birthday Conference, KIAS

Undergraduate Seminar (Paul M¨ uller, JKU Linz, 2009)

The picture on the Yale homepage (still!)

x 0 ∆ u = f

x 0 ∆ u = f Brownian PDEs Motion

Philosophical Overview ◮ parabolic PDEs make things nice and smooth (easy) ◮ elliptic PDEs minimize some energy functional (hard) Alternatively: any solution of − div( a ( x ) ∇ u ) + ∇ V ∇ u + cu = 0 gives rise to a solution of a heat/diffusion equation u t + ( − div( a ( x ) ∇ u ) + ∇ V ∇ u + cu ) = 0 . Use Brownian motion to study parabolic (=elliptic) problems!

Quantilized Donsker-Varadhan estimates

M. Donsker, S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, PNAS 1975

Donsker-Varadhan Inequality Ω ⊂ R n (but also works on graphs) and Lu = − div( a ( x ) ∇ u ) + ∇ V ( x ) ∇ u . Question. What is the smallest λ > 0 for which � Lu = λ u has a solution with u ∂ Ω = 0? � Example: � 1 � 2 x 2 L = − ∆ + ∇ on [0 , 1] . � Lu , u � ≥ ? · � u � 2

Donsker-Varadhan Inequality Lu = − div( a ( x ) ∇ u ) + ∇ V ( x ) ∇ u . Question. What is the smallest λ > 0 for which � Lu = λ u has a solution with u ∂ Ω = 0? � Donsker-Varadhan: associate a drift diffusion process (wiggle with a ( x ), drift towards ∇ V ) and maximize the expected exit time. Donsker-Varadhan Inequality 1 λ 1 ≥ sup x ∈ Ω E x τ Ω c .

Instead of looking at the mean of the first exist time, we study quantiles: let : Ω → R ≥ 0 be d p ,∂ Ω the smallest time t such that the likelihood of exit- ing within that time is p . Figure: Jianfeng Lu (Duke) J. Lu and S., 2016 log (1 / p ) λ 1 ≥ sup x ∈ Ω d p ,∂ Ω ( x ) . Moreover, as p → 0, the lower bound converges to λ 1 .

Proof. Start drift-diffusion in the point, where the solution assumes its maximum. have (Feynman-Kac) � u � L ∞ = u ( x ) = e λ t E ω ( u ( ω ( t ))) with the convention that u ( ω ( t )) is 0 if the drift-diffusion processes leaves Ω at some point in the interval [0 , t ]. Let now t = d p ,∂ Ω ( x ), in which case we see that E ω ( u ( ω ( t ))) ≤ p � u � L ∞ + (1 − p )0 . Altogether, we obtain � u � L ∞ = e λ d p ,∂ Ω ( x ) E ω ( u ( ω ( t ))) ≤ e λ d ∂ Ω ( x ) p � u � L ∞ from which the statement follows.

Example 1 Let us consider L = − ∆ on [0 , 1] . Then λ 1 = π 2 . 10 − 1 10 − 2 10 − 8 p 1 / 2 1 / 4 Donsker-Varadhan lower bound 7 . 28 8 . 40 8 . 92 9 . 39 9 . 74 8

Example 2 Let us consider � 1 � 2 x 2 L = − ∆ + ∇ on [0 , 1] . Then λ 1 = 2. 0 . 5 0 . 3 0 . 2 0 . 1 0 . 05 Donsker-Varadhan p lower bound 1 . 52 1 . 67 1 . 74 1 . 79 1 . 83 1 . 678

Lieb’s inradius result and the Polya-Szeg˝ o conjecture Polya

In two dimensions, we have (Osserman, Makai, Hayman, Polya-Szeg˝ o, . . . ) Ω |∇ u | 2 dx � 1 λ 1 (Ω) = inf ∼ inradius 2 � Ω | u | 2 dx f � =0 One direction ( � ) is trivial. The other direction ( � ) was posed as a conjecture by Polya & Szeg˝ o in 1951 (proven by Makai (1965) and, independently, Hayman (1978)).

Theorem (M. Rachh and S, CPAM 2017) Let Ω ⊂ R 2 be simply connected and u : Ω → R 2 vanish on ∂ Ω. If u assumes a global extremum in x 0 ∈ Ω, then − 1 / 2 � � ∆ u � � y ∈ ∂ Ω � x 0 − y � ≥ c inf . � � u � � L ∞ (Ω) y x Ω

Proof. � t � 0 V ( ω ( z )) dz � � u � L ∞ = u ( x 0 ) = E x 0 u ( ω ( t )) e � t � 0 V ( ω ( z )) dz � ≤ (1 − p x 0 ( t )) � u � L ∞ (Ω) E x 0 e ≤ (1 − p x 0 ( t )) � u � L ∞ e t � V � L ∞ , Therefore (1 − p x 0 ( t )) e t � V � L ∞ ≥ 1 . x 2 ∂ Ω ∂ Ω ∂ Ω x 0 x 1 x 0 x 0 x 1 x 1 x 2 ∂ Ω x 2

Lieb’s theorem Such results are impossible in dimensions ≥ 3: one can take a ball and remove one-dimensional lines without affecting the PDE.

Theorem (Elliott Lieb, 1984, Inventiones) Ω contains a (1 − ε )-fraction of a ball with radius c ε r ∼ � λ 1 (Ω)

x 2 x 1 Lemma (S, 2014, Comm. PDE) If you start Brownian motion in the maximum of the eigenfunction − ∆ u = λ u , then the likelihood of it impacting the nodal set within time t = λ − 1 is less than 64%. This means that not ’much’ boundary can be close to the maximum.

Theorem (Rachh and S, 2017, CPAM) If, with Dirichlet conditions, − ∆ u = Vu in Ω then Ω contains a (1 − ε )-fraction of a ball with radius c ε r ∼ � � V � L ∞ centered around the maximum of u .

A ’Real Life’ Application(?)

Anomaly detection Fiddling with results of this type suggest that for − ∆ φ λ = λφ λ , the quantity 1 | φ λ ( x ) | √ � φ λ � L ∞ λ is a decent proxy for the distance to the nearest nodal set. How about summing over distances to nodal lines 1 | φ λ ( x ) | � √ ? � φ λ � L ∞ λ λ ≤ N

Anomaly detection ;

Anomaly detection On the torus T , the quantity 1 | φ λ ( x ) | � √ � φ λ � L ∞ λ λ ≤ N simplifies to n | sin k π x | � . k k =1

Anomaly detection n | sin k π x | � . k k =1 1.0 0.9 0.8 0.7 0.6 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 1 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.30 1.25 1.20 1.15 1.10 1.05 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 2 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.45 1.40 1.35 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 3 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.65 1.60 1.55 1.50 1.45 1.40 1.35 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 4 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.70 1.65 1.60 1.55 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 5 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.9 1.8 1.7 1.6 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 6 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 1.95 1.90 1.85 1.80 1.75 1.70 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 7 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 2.0 1.9 1.8 1.7 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 8 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 2.10 2.05 2.00 1.95 1.90 1.85 1.80 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 9 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 2.2 2.1 2.0 1.9 1.8 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 10 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 2.60 2.55 2.50 2.45 2.40 2.35 2.30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 20 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 2.85 2.80 2.75 2.70 2.65 2.60 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 30 on [0 . 2 , 0 . 8]

Anomaly detection n | sin k π x | � . k k =1 3.60 3.55 3.50 3.45 3.40 3.35 0.3 0.4 0.5 0.6 0.7 0.8 ; Figure: n = 100 on [0 . 2 , 0 . 8]

n | sin ( k π x ) | � f n ( x ) = . k k =1 0 . 1 0 . 9 0 . 38 0 . 39 Figure: Right: the big cusp in the right picture is located at x = 5 / 13, the two smaller cusps are at x = 8 / 21 and x = 7 / 18.

0 . 42 0 . 425 Theorem (S. 2016) f n has a strict local minimum in x = p / q ∈ Q as soon as n ≥ (1 + o (1)) q 2 π .

Handwritten digits (ongoing w/ X. Cheng/Gal Mishne)

Seamine (ongoing w/ X. Cheng/Gal Mishne)

Seamines (ongoing w/ X. Cheng/Gal Mishne)

Seamines (ongoing w/ X. Cheng/Gal Mishne)

Homer (ongoing w/ X. Cheng/Gal Mishne)

Strict local maxima, elliptic PDEs, lifetime of Brownian motion and topological bounds on Fourier coefficients

Level sets of elliptic PDEs Generally tricky. Maybe (P.-L. Lions) convex Ω and − ∆ u = f ( u ) implies convex level sets?

Level sets of elliptic PDEs Maybe (P.-L. Lions) convex Ω and − ∆ u = f ( u ) implies convex level sets? Yes for − ∆ u = 1 (Makar-Limanov, 70s) Yes for − ∆ u = λ 1 u (Brascamp-Lieb, 70s). Yes, for some other f (various). No: Hamel, Nadirashvili & Sire (2016).

Level sets of elliptic PDEs Can level sets ever be fundamentally more eccentric than the domain?

Let Ω ⊂ R 2 be convex and consider − ∆ u = 1 with Dirichlet boundary conditions. This is the expected lifetime of Brownian motion. It also has some meaning in mechanics (St. Venant torsion).

The three basic questions in hiking 1. How big is the mountain? ( � u ( x 0 ) � L ∞ ∼ inrad(Ω) 2 ) 2. Where is the maximum? ( x 0 , maximal lifetime) 3. What’s the view from the top? ( D 2 u ( x 0 )?)