OBLIQUELY REFLECTED BROWNIAN MOTION IN FRACTAL DOMAINS Krzysztof - PowerPoint PPT Presentation

OBLIQUELY REFLECTED BROWNIAN MOTION IN FRACTAL DOMAINS Krzysztof Burdzy University of Washington Krzysztof Burdzy Reflected Brownian motion 1 / 28

Reference KB, Zhenqing Chen, Donald Marshall, Kavita Ramanan “Obliquely reflected Brownian motion in non-smooth planar domains” Ann. Probab. 45 (2017) 2971–3037 Krzysztof Burdzy Reflected Brownian motion 2 / 28

Obliquely reflected Brownian motion Krzysztof Burdzy Reflected Brownian motion 3 / 28

Obliquely reflected Brownian motion in fractal domains Krzysztof Burdzy Reflected Brownian motion 4 / 28

Technical challenges Krzysztof Burdzy Reflected Brownian motion 5 / 28

Technical challenges Krzysztof Burdzy Reflected Brownian motion 5 / 28

Obliquely reflected Brownian motion – technical remarks Classical Dirichlet form approach to Markov processes is limited to symmetric processes. Obliquely reflected Brownian motion is not symmetric. Krzysztof Burdzy Reflected Brownian motion 6 / 28

Obliquely reflected Brownian motion – technical remarks Classical Dirichlet form approach to Markov processes is limited to symmetric processes. Obliquely reflected Brownian motion is not symmetric. Non-symmetric Dirichlet form approach to obliquely reflected Brownian motion had limited success (Kim, Kim and Yun (1998) and Duarte (2012)). Krzysztof Burdzy Reflected Brownian motion 6 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D Krzysztof Burdzy Reflected Brownian motion 7 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D THEOREM (B and Marshall; 1993) For an arbitrary measurable θ , obliquely reflected Brownian motion X in D with the oblique angle of reflection θ exists. Krzysztof Burdzy Reflected Brownian motion 7 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D THEOREM (B and Marshall; 1993) For an arbitrary measurable θ , obliquely reflected Brownian motion X in D with the oblique angle of reflection θ exists. Lions and Sznitman (1984), Harrison, Landau and Shepp (1985), Varadhan and Williams (1985) Krzysztof Burdzy Reflected Brownian motion 7 / 28

Jumps on the boundary Krzysztof Burdzy Reflected Brownian motion 8 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D Krzysztof Burdzy Reflected Brownian motion 9 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D θ ↔ ( h , µ ) Krzysztof Burdzy Reflected Brownian motion 9 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D θ ↔ ( h , µ ) h ( x ) dx – stationary distribution Krzysztof Burdzy Reflected Brownian motion 9 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D θ ↔ ( h , µ ) h ( x ) dx – stationary distribution µ – rate of rotation Krzysztof Burdzy Reflected Brownian motion 9 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 , θ ↔ ( h , µ ) THEOREM (B, Chen, Marshall, Ramanan; 2017) h = e − i ( θ + i � θ ) π cos θ (0) + i tan θ (0) h + i � , and µ = tan θ (0) , π ��� � � θ + i � θ = i log( h + i � 1 + µ 2 h − i µ/π ) − i log /π . Krzysztof Burdzy Reflected Brownian motion 10 / 28

Parametrization of obliquely reflected Brownian motions D – unit disc in R 2 , θ ↔ ( h , µ ) THEOREM (B, Chen, Marshall, Ramanan; 2017) h = e − i ( θ + i � θ ) π cos θ (0) + i tan θ (0) h + i � , and µ = tan θ (0) , π ��� � � θ + i � θ = i log( h + i � 1 + µ 2 h − i µ/π ) − i log /π . Harrison, Landau and Shepp (1985): smooth θ Krzysztof Burdzy Reflected Brownian motion 10 / 28

Rate of rotation of obliquely reflected Brownian motion THEOREM (B, Chen, Marshall, Ramanan; 2017) (i) lim t →∞ arg X t / t = µ ? Krzysztof Burdzy Reflected Brownian motion 11 / 28

Rate of rotation of obliquely reflected Brownian motion THEOREM (B, Chen, Marshall, Ramanan; 2017) (i) lim t →∞ arg X t / t = µ ? No. Krzysztof Burdzy Reflected Brownian motion 11 / 28

Rate of rotation of obliquely reflected Brownian motion THEOREM (B, Chen, Marshall, Ramanan; 2017) (i) lim t →∞ arg X t / t = µ ? No. (ii) Assume that sup x | θ ( x ) | < π/ 2. Then, a.s., X is continuous and, therefore, arg X t is well defined for t > 0. The distributions of 1 t arg X t − µ converge to the Cauchy distribution when t → ∞ . Krzysztof Burdzy Reflected Brownian motion 11 / 28

Rate of rotation of obliquely reflected Brownian motion THEOREM (B, Chen, Marshall, Ramanan; 2017) (i) lim t →∞ arg X t / t = µ ? No. (ii) Assume that sup x | θ ( x ) | < π/ 2. Then, a.s., X is continuous and, therefore, arg X t is well defined for t > 0. The distributions of 1 t arg X t − µ converge to the Cauchy distribution when t → ∞ . (iii) Let arg ∗ X t be arg X t “without large excursions from ∂ D .” Then, a.s., t →∞ arg ∗ X t / t = µ. lim Krzysztof Burdzy Reflected Brownian motion 11 / 28

Obliquely reflected Brownian motion in fractal domains D – simply connected bounded open set in R 2 THEOREM (B, Chen, Marshall, Ramanan; 2017) For every positive harmonic function h in D with L 1 norm equal to 1 and every real number µ , there exists a (unique in distribution) obliquely reflected Brownian motion in D with the stationary distribution h ( x ) dx and rate of rotation µ . Krzysztof Burdzy Reflected Brownian motion 12 / 28

Rotation rate field Let D be the unit disc and µ ( z ) be the rate of rotation around z ∈ D . In other words, the distributions of 1 t arg( X t − z ) − µ ( z ) converge to the Cauchy distribution when t → ∞ . Krzysztof Burdzy Reflected Brownian motion 13 / 28

Rotation rate field Let D be the unit disc and µ ( z ) be the rate of rotation around z ∈ D . In other words, the distributions of 1 t arg( X t − z ) − µ ( z ) converge to the Cauchy distribution when t → ∞ . THEOREM (B, Chen, Marshall, Ramanan; 2017) The function µ ( z ) is harmonic in D . Krzysztof Burdzy Reflected Brownian motion 13 / 28

Rotation rate field D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D h ( x ) dx – stationary distribution µ ( z ) – rate of rotation around z Krzysztof Burdzy Reflected Brownian motion 14 / 28

Rotation rate field D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D h ( x ) dx – stationary distribution µ ( z ) – rate of rotation around z θ ↔ ( h , µ (0)) ↔ { µ ( z ) } z ∈ D Arrows indicate one to one mappings. Are the mappings surjective? Krzysztof Burdzy Reflected Brownian motion 14 / 28

Rotation rate field D – unit disc in R 2 θ ( x ) – angle of reflection at x ∈ ∂ D h ( x ) dx – stationary distribution µ ( z ) – rate of rotation around z θ ↔ ( h , µ (0)) ↔ { µ ( z ) } z ∈ D Arrows indicate one to one mappings. Are the mappings surjective? In the first case, yes. Krzysztof Burdzy Reflected Brownian motion 14 / 28

Rotation rate field – limitations Suppose that φ ( z ) is harmonic in the unit disc D . φ ( z ) does not have to be positive. Krzysztof Burdzy Reflected Brownian motion 15 / 28

Rotation rate field – limitations Suppose that φ ( z ) is harmonic in the unit disc D . φ ( z ) does not have to be positive. Set K − = inf z ∈ D � φ ( z ) and K + = sup z ∈ D � φ ( z ). Krzysztof Burdzy Reflected Brownian motion 15 / 28

Rotation rate field – limitations Suppose that φ ( z ) is harmonic in the unit disc D . φ ( z ) does not have to be positive. Set K − = inf z ∈ D � φ ( z ) and K + = sup z ∈ D � φ ( z ). If a , b ∈ R with − 1 / | K + | ≤ b ≤ 1 / | K − | , then the function µ ( z ) = a + b φ ( z ) is the rotation field of an obliquely reflected Brownian motion in D . Krzysztof Burdzy Reflected Brownian motion 15 / 28

Rotation rate field – limitations Suppose that φ ( z ) is harmonic in the unit disc D . φ ( z ) does not have to be positive. Set K − = inf z ∈ D � φ ( z ) and K + = sup z ∈ D � φ ( z ). If a , b ∈ R with − 1 / | K + | ≤ b ≤ 1 / | K − | , then the function µ ( z ) = a + b φ ( z ) is the rotation field of an obliquely reflected Brownian motion in D . For b > 1 / | K − | and b < − 1 / | K + | , the function µ ( z ) = a + b φ ( z ) does not represent the rotation field of an obliquely reflected Brownian motion in D . Krzysztof Burdzy Reflected Brownian motion 15 / 28

Obliquely reflected Brownian motion in fractal domains Krzysztof Burdzy Reflected Brownian motion 16 / 28

Smooth domain approximation D ⊂ R 2 – open bounded simply connected set D k ⊂ D k +1 , � k D k = D , D k have smooth boundaries Krzysztof Burdzy Reflected Brownian motion 17 / 28