Recent advances in Mandelbrot martingales theory Julien Barral, - PowerPoint PPT Presentation

Recent advances in Mandelbrot martingales theory Julien Barral, Universit´ e Paris Nord Advances in Fractals and Related Topics, CUHK, September 2012 J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Σ = { 0 , 1 } N + and for n ≥ 1, Σ n = { 0 , 1 } n . W : positive rv. with E W = 1 / 2. { W ( w ) } σ ∈ � n ≥ 1 Σ n independent rv’s equidistributed with W . � Y n = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | n ) is a positive martingale σ ∈ Σ n J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Σ = { 0 , 1 } N + and for n ≥ 1, Σ n = { 0 , 1 } n . W : positive rv. with E W = 1 / 2. { W ( w ) } σ ∈ � n ≥ 1 Σ n independent rv’s equidistributed with W . � Y n = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | n ) is a positive martingale σ ∈ Σ n J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Y = lim n →∞ Y n . Writing � � Y n +1 = W ( j ) × W ( j · σ | 1 ) W ( j · σ | 2 ) · · · W ( j · σ | n ) j ∈{ 0 , 1 } σ ∈ Σ n = W (0) Y n (0) + W (1) Y n (1) yields Y = W (0) Y (0) + W (1) Y (1) , where { W ( j ) , Y ( j ) } j ∈{ 0 , 1 } are independent, W ( j ) ∼ W , Y ( j ) ∼ Y . Moreover, P ( Y > 0) ∈ { 0 , 1 } . Using this recursively yields the Mandelbrot random measure on [0 , 1] µ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ( σ ) . Theorem (Kahane (1976)) The following assertions are equivalent: (1) P ( Y > 0) = 1 ; (2) ( Y k ) k ≥ 1 is uniformly integrable; (3) E W log W < 0 . J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Y = lim n →∞ Y n . Writing � � Y n +1 = W ( j ) × W ( j · σ | 1 ) W ( j · σ | 2 ) · · · W ( j · σ | n ) j ∈{ 0 , 1 } σ ∈ Σ n = W (0) Y n (0) + W (1) Y n (1) yields Y = W (0) Y (0) + W (1) Y (1) , where { W ( j ) , Y ( j ) } j ∈{ 0 , 1 } are independent, W ( j ) ∼ W , Y ( j ) ∼ Y . Moreover, P ( Y > 0) ∈ { 0 , 1 } . Using this recursively yields the Mandelbrot random measure on [0 , 1] µ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ( σ ) . Theorem (Kahane (1976)) The following assertions are equivalent: (1) P ( Y > 0) = 1 ; (2) ( Y k ) k ≥ 1 is uniformly integrable; (3) E W log W < 0 . J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Y = lim n →∞ Y n . Writing � � Y n +1 = W ( j ) × W ( j · σ | 1 ) W ( j · σ | 2 ) · · · W ( j · σ | n ) j ∈{ 0 , 1 } σ ∈ Σ n = W (0) Y n (0) + W (1) Y n (1) yields Y = W (0) Y (0) + W (1) Y (1) , where { W ( j ) , Y ( j ) } j ∈{ 0 , 1 } are independent, W ( j ) ∼ W , Y ( j ) ∼ Y . Moreover, P ( Y > 0) ∈ { 0 , 1 } . Using this recursively yields the Mandelbrot random measure on [0 , 1] µ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ( σ ) . Theorem (Kahane (1976)) The following assertions are equivalent: (1) P ( Y > 0) = 1 ; (2) ( Y k ) k ≥ 1 is uniformly integrable; (3) E W log W < 0 . J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales Let Y = lim n →∞ Y n . Writing � � Y n +1 = W ( j ) × W ( j · σ | 1 ) W ( j · σ | 2 ) · · · W ( j · σ | n ) j ∈{ 0 , 1 } σ ∈ Σ n = W (0) Y n (0) + W (1) Y n (1) yields Y = W (0) Y (0) + W (1) Y (1) , where { W ( j ) , Y ( j ) } j ∈{ 0 , 1 } are independent, W ( j ) ∼ W , Y ( j ) ∼ Y . Moreover, P ( Y > 0) ∈ { 0 , 1 } . Using this recursively yields the Mandelbrot random measure on [0 , 1] µ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ( σ ) . Theorem (Kahane (1976)) The following assertions are equivalent: (1) P ( Y > 0) = 1 ; (2) ( Y k ) k ≥ 1 is uniformly integrable; (3) E W log W < 0 . J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales. Normalization and related equation Natural questions arise: (Mandelbrot, 1974) When E W log W ≥ 0, does there exist A n > 0 1 such that ( Y n / A n ) converges to a non-trivial limit Z , at least in distribution? If so A n / A n +1 converges to A , 0 < A < ∞ , and the limit satisfies d Z = A W (0) Z (0) + 1 W (1) Z (1) (Durrett and Liggett, 1983) In general, what are the non-trivial 2 solutions to d ( E ) Z = W (0) Z (0) + W (1) Z (1) ? Are there natural multifractal measures associated with solutions 3 of ( E )? J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales. Normalization and related equation Natural questions arise: (Mandelbrot, 1974) When E W log W ≥ 0, does there exist A n > 0 1 such that ( Y n / A n ) converges to a non-trivial limit Z , at least in distribution? If so A n / A n +1 converges to A , 0 < A < ∞ , and the limit satisfies d Z = A W (0) Z (0) + 1 W (1) Z (1) (Durrett and Liggett, 1983) In general, what are the non-trivial 2 solutions to d ( E ) Z = W (0) Z (0) + W (1) Z (1) ? Are there natural multifractal measures associated with solutions 3 of ( E )? J. Barral Recent advances in Mandelbrot martingales theory

Mandelbrot martingales. Normalization and related equation Natural questions arise: (Mandelbrot, 1974) When E W log W ≥ 0, does there exist A n > 0 1 such that ( Y n / A n ) converges to a non-trivial limit Z , at least in distribution? If so A n / A n +1 converges to A , 0 < A < ∞ , and the limit satisfies d Z = A W (0) Z (0) + 1 W (1) Z (1) (Durrett and Liggett, 1983) In general, what are the non-trivial 2 solutions to d ( E ) Z = W (0) Z (0) + W (1) Z (1) ? Are there natural multifractal measures associated with solutions 3 of ( E )? J. Barral Recent advances in Mandelbrot martingales theory

The derivative martingale in the critical case. Suppose that E W log W = 0. For all β ∈ [0 , 1), set � W β E W β = 1 / 2 , W β = 2 E W β . It satisfies . E W β log W β < 0 Define � n = − d W β ( σ | 1 ) W β ( σ | 2 ) · · · W β ( σ | n ) and Y ′ Y n ( β ) = d β Y n ( β ) . σ ∈ Σ n Theorem (Biggins-Kyprianou (1997), Liu (2000)) If E W 1+ ǫ < ∞ for some ǫ > 0 , then ( Y ′ n ) converges almost surely to Y ′ , Y ′ = W (0) Y ′ (0) + W (1) Y ′ (1) , E Y ′ = ∞ . This yields a.s. on [0 , 1] the “critical” measure µ ′ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ′ ( σ ) . J. Barral Recent advances in Mandelbrot martingales theory

The derivative martingale in the critical case. Suppose that E W log W = 0. For all β ∈ [0 , 1), set � W β E W β = 1 / 2 , W β = 2 E W β . It satisfies . E W β log W β < 0 Define � n = − d W β ( σ | 1 ) W β ( σ | 2 ) · · · W β ( σ | n ) and Y ′ Y n ( β ) = d β Y n ( β ) . σ ∈ Σ n Theorem (Biggins-Kyprianou (1997), Liu (2000)) If E W 1+ ǫ < ∞ for some ǫ > 0 , then ( Y ′ n ) converges almost surely to Y ′ , Y ′ = W (0) Y ′ (0) + W (1) Y ′ (1) , E Y ′ = ∞ . This yields a.s. on [0 , 1] the “critical” measure µ ′ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ′ ( σ ) . J. Barral Recent advances in Mandelbrot martingales theory

The derivative martingale in the critical case. Suppose that E W log W = 0. For all β ∈ [0 , 1), set � W β E W β = 1 / 2 , W β = 2 E W β . It satisfies . E W β log W β < 0 Define � n = − d W β ( σ | 1 ) W β ( σ | 2 ) · · · W β ( σ | n ) and Y ′ Y n ( β ) = d β Y n ( β ) . σ ∈ Σ n Theorem (Biggins-Kyprianou (1997), Liu (2000)) If E W 1+ ǫ < ∞ for some ǫ > 0 , then ( Y ′ n ) converges almost surely to Y ′ , Y ′ = W (0) Y ′ (0) + W (1) Y ′ (1) , E Y ′ = ∞ . This yields a.s. on [0 , 1] the “critical” measure µ ′ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ′ ( σ ) . J. Barral Recent advances in Mandelbrot martingales theory

The derivative martingale in the critical case. Suppose that E W log W = 0. For all β ∈ [0 , 1), set � W β E W β = 1 / 2 , W β = 2 E W β . It satisfies . E W β log W β < 0 Define � n = − d W β ( σ | 1 ) W β ( σ | 2 ) · · · W β ( σ | n ) and Y ′ Y n ( β ) = d β Y n ( β ) . σ ∈ Σ n Theorem (Biggins-Kyprianou (1997), Liu (2000)) If E W 1+ ǫ < ∞ for some ǫ > 0 , then ( Y ′ n ) converges almost surely to Y ′ , Y ′ = W (0) Y ′ (0) + W (1) Y ′ (1) , E Y ′ = ∞ . This yields a.s. on [0 , 1] the “critical” measure µ ′ ( I σ ) = W ( σ | 1 ) W ( σ | 2 ) · · · W ( σ | | σ | ) Y ′ ( σ ) . J. Barral Recent advances in Mandelbrot martingales theory