Self-similar solutions of kinetic-type equations Kamil Bogus Wrocaw - PowerPoint PPT Presentation

Self-similar solutions of kinetic-type equations Kamil Bogus Wrocław University of Science and Technology (POLAND) Będlewo, 23rd May 2019 Kamil Bogus Self-similar solutions of kinetic-type equations 1 / 11

Article The talk is based on joint work with Dariusz Buraczewski and Alexander Marynych: K. Bogus, D. Buraczewski, A. Marynych, Self-similar solutions of kinetic-type equations: The boundary case , Stochastic Processes and their Applications, p. 18 (2019) https://doi.org/10.1016/j.spa.2019.03.005. Kamil Bogus Self-similar solutions of kinetic-type equations 2 / 11

Fourier–Stieltjes transform ( ρ t ) t ≥ 0 − time dependent family of probability measures, φ ( t , ξ ) − Fourier–Stieltjes transform (the characteristic function) of ρ t , � e i ξ v ρ t ( d v ) , φ ( t , ξ ) = t � 0 , ξ ∈ R , R Kamil Bogus Self-similar solutions of kinetic-type equations 3 / 11

Introduction - Smoothing transform � Q – smoothing transform � Q ( φ 1 , . . . , φ N )( ξ ) := E ( φ 1 ( A 1 ξ ) · . . . · φ N ( A N ξ )) , ξ ∈ R , where φ 1 , . . . , φ N − characteristic functions, N − fixed positive integer, A = ( A 1 , . . . , A N ) − vector of real-valued random variables. Example: N = 2 and A = ( sin θ, cos θ ) , where θ is a random angle uniformly distributed on [ 0 , 2 π ) Kamil Bogus Self-similar solutions of kinetic-type equations 4 / 11

Introduction - Smoothing transform � Q – smoothing transform � Q ( φ 1 , . . . , φ N )( ξ ) := E ( φ 1 ( A 1 ξ ) · . . . · φ N ( A N ξ )) , ξ ∈ R , where φ 1 , . . . , φ N − characteristic functions, N − fixed positive integer, A = ( A 1 , . . . , A N ) − vector of real-valued random variables. Example: N = 2 and A = ( sin θ, cos θ ) , where θ is a random angle uniformly distributed on [ 0 , 2 π ) Kamil Bogus Self-similar solutions of kinetic-type equations 4 / 11

Cauchy problem We consider the following Cauchy problem � ∂ � ∂ t φ ( t , ξ ) + φ ( t , ξ ) = Q ( φ ( t , · ) , . . . , φ ( t , · ))( ξ ) , t > 0 , φ ( 0 , ξ ) = φ 0 ( ξ ) , ξ ∈ R , The initial condition φ 0 is the characteristic function of some random variable X 0 defined on (Ω , F , P ) . Main Goal: Study asymptotic behavior of the solution φ . Kamil Bogus Self-similar solutions of kinetic-type equations 5 / 11

Cauchy problem We consider the following Cauchy problem � ∂ � ∂ t φ ( t , ξ ) + φ ( t , ξ ) = Q ( φ ( t , · ) , . . . , φ ( t , · ))( ξ ) , t > 0 , φ ( 0 , ξ ) = φ 0 ( ξ ) , ξ ∈ R , The initial condition φ 0 is the characteristic function of some random variable X 0 defined on (Ω , F , P ) . Main Goal: Study asymptotic behavior of the solution φ . Kamil Bogus Self-similar solutions of kinetic-type equations 5 / 11

Assumptions We assume that ( H γ ) The law of X 0 is centered and belongs to the domain of normal attraction of a γ -stable law with the characteristic function ˆ g γ ( A ) Weights ( A i ) i = 1 ,..., N are a.s. positive (Φ) For the function Φ : [ 0 , ∞ ) �→ R ∪ { + ∞} defined via � � N � A s Φ( s ) = E − 1 , s � 0 , i i = 1 we assume that s ∞ > 0 where s ∞ := sup { s � 0 : Φ( s ) < ∞} Kamil Bogus Self-similar solutions of kinetic-type equations 6 / 11

Assumptions We assume that ( H γ ) The law of X 0 is centered and belongs to the domain of normal attraction of a γ -stable law with the characteristic function ˆ g γ ( A ) Weights ( A i ) i = 1 ,..., N are a.s. positive (Φ) For the function Φ : [ 0 , ∞ ) �→ R ∪ { + ∞} defined via � � N � A s Φ( s ) = E − 1 , s � 0 , i i = 1 we assume that s ∞ > 0 where s ∞ := sup { s � 0 : Φ( s ) < ∞} Kamil Bogus Self-similar solutions of kinetic-type equations 6 / 11

Assumptions We assume that ( H γ ) The law of X 0 is centered and belongs to the domain of normal attraction of a γ -stable law with the characteristic function ˆ g γ ( A ) Weights ( A i ) i = 1 ,..., N are a.s. positive (Φ) For the function Φ : [ 0 , ∞ ) �→ R ∪ { + ∞} defined via � � N � A s Φ( s ) = E − 1 , s � 0 , i i = 1 we assume that s ∞ > 0 where s ∞ := sup { s � 0 : Φ( s ) < ∞} Kamil Bogus Self-similar solutions of kinetic-type equations 6 / 11

Assumptions We assume that ( H γ ) The law of X 0 is centered and belongs to the domain of normal attraction of a γ -stable law with the characteristic function ˆ g γ ( A ) Weights ( A i ) i = 1 ,..., N are a.s. positive (Φ) For the function Φ : [ 0 , ∞ ) �→ R ∪ { + ∞} defined via � � N � A s Φ( s ) = E − 1 , s � 0 , i i = 1 we assume that s ∞ > 0 where s ∞ := sup { s � 0 : Φ( s ) < ∞} Kamil Bogus Self-similar solutions of kinetic-type equations 6 / 11

Spectral function The function µ ( s ) = Φ( s ) , s > 0 , s is called spectral function. We denote by γ ∗ point minimizing this function. Φ( s ) N − 1 γ ∗ s α ( γ ∗ , Φ( γ ∗ )) − 1 Figure: Plot of the function s �→ Φ( s ) (solid red) with tan α = µ ( γ ∗ ) = Φ ′ ( γ ∗ ) = Φ( γ ∗ ) /γ ∗ . Kamil Bogus Self-similar solutions of kinetic-type equations 7 / 11

Spectral function The function µ ( s ) = Φ( s ) , s > 0 , s is called spectral function. We denote by γ ∗ point minimizing this function. Φ( s ) N − 1 γ ∗ s α ( γ ∗ , Φ( γ ∗ )) − 1 Figure: Plot of the function s �→ Φ( s ) (solid red) with tan α = µ ( γ ∗ ) = Φ ′ ( γ ∗ ) = Φ( γ ∗ ) /γ ∗ . Kamil Bogus Self-similar solutions of kinetic-type equations 7 / 11

Known results The solution to the considered equation can be derived analytically in terms of the Wild series, Results on probabilistic interpretation of the solution can be found in works of Gabetta, Regazzini, Carlen and Carvalho, Based on McKean’s ideas, Bassetti, Ladelli and Matthes expressed the solution in a convenient probabilistic way, Assuming that ( H γ ) holds for some γ ∈ ( 0 , 2 ] and there exists δ > γ such that µ ( δ ) < µ ( γ ) < ∞ , they showed that φ ( t , e − µ ( γ ) t ξ ) converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Kamil Bogus Self-similar solutions of kinetic-type equations 8 / 11

Known results The solution to the considered equation can be derived analytically in terms of the Wild series, Results on probabilistic interpretation of the solution can be found in works of Gabetta, Regazzini, Carlen and Carvalho, Based on McKean’s ideas, Bassetti, Ladelli and Matthes expressed the solution in a convenient probabilistic way, Assuming that ( H γ ) holds for some γ ∈ ( 0 , 2 ] and there exists δ > γ such that µ ( δ ) < µ ( γ ) < ∞ , they showed that φ ( t , e − µ ( γ ) t ξ ) converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Kamil Bogus Self-similar solutions of kinetic-type equations 8 / 11

Known results The solution to the considered equation can be derived analytically in terms of the Wild series, Results on probabilistic interpretation of the solution can be found in works of Gabetta, Regazzini, Carlen and Carvalho, Based on McKean’s ideas, Bassetti, Ladelli and Matthes expressed the solution in a convenient probabilistic way, Assuming that ( H γ ) holds for some γ ∈ ( 0 , 2 ] and there exists δ > γ such that µ ( δ ) < µ ( γ ) < ∞ , they showed that φ ( t , e − µ ( γ ) t ξ ) converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Kamil Bogus Self-similar solutions of kinetic-type equations 8 / 11

Known results The solution to the considered equation can be derived analytically in terms of the Wild series, Results on probabilistic interpretation of the solution can be found in works of Gabetta, Regazzini, Carlen and Carvalho, Based on McKean’s ideas, Bassetti, Ladelli and Matthes expressed the solution in a convenient probabilistic way, Assuming that ( H γ ) holds for some γ ∈ ( 0 , 2 ] and there exists δ > γ such that µ ( δ ) < µ ( γ ) < ∞ , they showed that φ ( t , e − µ ( γ ) t ξ ) converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Kamil Bogus Self-similar solutions of kinetic-type equations 8 / 11

Known results The solution to the considered equation can be derived analytically in terms of the Wild series, Results on probabilistic interpretation of the solution can be found in works of Gabetta, Regazzini, Carlen and Carvalho, Based on McKean’s ideas, Bassetti, Ladelli and Matthes expressed the solution in a convenient probabilistic way, Assuming that ( H γ ) holds for some γ ∈ ( 0 , 2 ] and there exists δ > γ such that µ ( δ ) < µ ( γ ) < ∞ , they showed that φ ( t , e − µ ( γ ) t ξ ) converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Kamil Bogus Self-similar solutions of kinetic-type equations 8 / 11