Brownian Motion and the Dirichlet Problem

Transcription

1 Brownian Motion and the Dirichlet Problem Mario Teixeira Parente August 29, /22

2 Topics for the talk 1. Solving the Dirichlet problem on bounded domains 2. Application: Recurrence/Transience of Brownian motion 2/22

3 Topics for the talk 1. Solving the Dirichlet problem on bounded domains 2. Application: Recurrence/Transience of Brownian motion 3/22

4 What is Brownian motion? Definition A stochastic process (B t ) t 0 is called a standard Brownian motion on a probability space (Ω, F, P), if it satisfies the following properties: 1. B 0 = 0 P-a.s. 2. t B t (ω) is continuous for every ω Ω. 3. If 0 s < t, then B t B s N (0, t s). 4. For 0 t 0 < t 1 <... < t k, the increments ( B ti+1 B ti )i=0,...,k 1 are independent. 4/22

5 What is Brownian motion? (cont.) Figure: Two sample paths of one-dimensional Brownian motion 5/22

6 What is the Dirichlet problem? Assume D is a connected open subset of R n. Find h C 2 (D) C(D) with { h = 0 in D h = f on D. Remark: Continuity up to the boundary in this case means lim h(x) = f (z), x z x D z D. 6/22

7 Our target Consider Brownian motion (B t ) t 0 in n dimensions. Define τ D := inf {t > 0 : B t D c } as the exit time of D. (τ D is a stopping time!) Theorem! Suppose D is bounded, every point on D is regular and f : D R is continuous. Then the unique solution to the Dirichlet problem is given by h(x) := E x f (B τd ). Blackboard 7/22

8 What does Itô s formula say? Assume h C 2 (R n ) is a solution to the Dirichlet problem on D with boundary values f. Itô formula for Brownian motion: h(b t ) = h(b 0 ) + t 0 h(b s ) db s with stopping time τ D : τd t h(b τd t) = h(b 0 ) + h(b s ) db s }{{} Martingale in t t 0 h(b s ) ds }{{} =! τd t 0 h(b s ) ds }{{} =0 8/22

9 What does Itô s formula say? (cont.) Let x D and take expectations: E x h(b τd t) = E x h(b 0 ) = h(x) Since τ D is finite and h is bounded on D, letting t gives h(x) = E x h(b τd ) = E x f (B τd ). 9/22

10 Steps for the theorem We have to show: 1. h(x) := E x f (B τd ) is harmonic in D (i.e. h = 0 in D) 2. Continuity of h up to the boundary D: lim h(x) = f (z), x z x D z D 3. Uniqueness 10/22

11 Harmonicity of h From PDE: h is harmonic in D h satisfies the mean value property in D Mean value property: h(x) = B(x,r) h(y) σ x,r (dy) for every ball B(x, r) D Blackboard 11/22

12 Harmonicity of h (cont.) We will need the strong Markov property. Our context is the canonical model, i.e. Ω = C 0 [0, ) := {ω : [0, ) R n : ω is continuous and ω(0) = 0} and P is chosen to be the probability measure such that B t (ω) := ω(t) becomes a Brownian motion. Necessary ingredient: Time-shift-operator θ s : Ω Ω, ω ω( + s) This means θ s (ω)(t) = ω(t + s) = B t+s (ω). 12/22

13 Harmonicity of h (cont.) We have a family of probability measures (P x ) x R n with P x (B 0 = x) = 1. Strong Markov property (our version): Let X be a bounded random variable, τ be a stopping time and x D. Then E x [X θ τ F τ ] = E Bτ X P x -a.s. on {τ < }. Blackboard 13/22

14 Harmonicity of h (cont.) Take a ball B = B(x, r) D and note that B τd = B τd θ τb. Proving the mean value property for h(x) := E x f (B τd ): h(x) = E x [E x [f (B τd ) θ τb }{{} SMP =f (B τd ) F τb ]] [ ] = E x E Bτ B [f (BτD )] = E x h(b τb ) = h(y) σ x,r (dy) B(x,r) 14/22

15 Topics for the talk 1. Solving the Dirichlet problem on bounded domains 2. Application: Recurrence/Transience of Brownian motion 15/22

16 Example Let n 2, take 0 < r 1 < r 2 and consider D := {x R n : r 1 < x < r 2 } with boundary values f (z) := { 0 if z = r 1, 1 if z = r 2. 16/22

17 Example (cont.) = h 1 (x) := E x f (B τd ) = P x (B t exits D through the outer boundary) solves the corresponding Dirichlet problem. 17/22

18 Example (cont.) Typical examples of harmonic functions are also h 2 (x) := { a log x + b if n = 2, a + b x n 2 if n 3. If we choose a, b correctly, h 2 also attains boundary values f. By uniqueness, h 1 = h 2 on D. 18/22

19 Example (cont.) We get P x (B t exits D through the outer boundary) log x log r 1 log r 2 log r 1 if n = 2, = ( ) n 2 r2 x n 2 r n 2 1 if n 3 x = P x (τ r2 < τ r1 ), r n 2 2 r n 2 1 where τ r denotes the hitting time of B(0, r). 19/22

20 Example (cont.) For (point) recurrence, let r 1 0 first: lim r 1 0 Px (τ r2 < τ r1 ) = 1 for 0 < x < r 2 Since lim r1 0 τ r1 = τ 0, which is the hitting time of 0, it follows Finally, lim r2 τ r2 = implies P x (τ r2 < τ 0 ) = 1 for 0 < x < r 2. P x (τ 0 = ) = 1 for x 0. 20/22

21 Example (cont.) For neighborhood recurrence/transience, only let r 2 and get for x > r 1 0 if n = 2, P x (τ r1 = ) = lim r2 Px (τ r2 < τ r1 ) = ( n 2 1 r1 x ) if n 3. = Brownian motion is neighborhood recurrent if n = 2, but is neighborhood transient for n 3. 21/22

22 References Mario Teixeira Parente. Master thesis: Brownian Motion and the Dirichlet Problem. Thomas M. Liggett. Continuous Time Markov Processes: An Introduction, volume v. 113 of Graduate studies in mathematics. American Mathematical Society, Providence, R.I., Lawrence C. Evans. Partial Differential Equations, volume 19 of Graduate studies in mathematics. American Math. Soc, Providence, RI, 2. ed. edition, Lothar Partzsch, René L. Schilling, and Björn Böttcher. Brownian Motion: An Introduction to Stochastic Processes. De Gruyter Textbook. De Gruyter, München, 2. aufl., 2nd revised and extended edition edition, Richard Durrett. Probability: Theory and Examples, volume 31 of Cambridge series in statistical and probabilistic mathematics. Cambridge Univ. Press, Cambridge, 4. ed. edition, /22