Stochastic Equations of Super-L evy Process with General Branching - PowerPoint PPT Presentation

Stochastic Equations of Super-L´ evy Process with General Branching Mechanism Xu Yang (Joint work with Hui He and Zenghu Li) Beijing Normal University June 18, 2012 Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 1 / 11

Content Introduction Main results Proof of Theorem 2 Further result: SPDE driven by α -stable noise Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 2 / 11

Introduction Let { X t : t ≥ 0 } be a binary branching super-Brownian motion (SBM). Then X t ( dx ) = X t ( x ) dx and the density is the unique positive weak solution to (Konno-Shiga (1988) and Reimers (1989)): ∂ X t ( x ) = 1 X t ( x ) ˙ � 2 ∆ X t ( x ) + W t ( x ) , t ≥ 0 , x ∈ R , (1) ∂ t where ˙ W t ( x ) is the derivative of a space-time Gaussian white noise (GWN). • The pathwise uniqueness for (1) is unknown. Progress: Perkins, Sturm, Mytnik, etc. • Xiong (2012) studied the pathwise uniqueness to SPDE for the distribution function process of the SBM. Pathwise uniqueness to similar equation see Dawson and Li (2012). • This talk is to generalize the result of Xiong (2012) to the super-L´ evy process with general branching mechanism. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 3 / 11

Introduction Let { X t : t ≥ 0 } be a binary branching super-Brownian motion (SBM). Then X t ( dx ) = X t ( x ) dx and the density is the unique positive weak solution to (Konno-Shiga (1988) and Reimers (1989)): ∂ X t ( x ) = 1 X t ( x ) ˙ � 2 ∆ X t ( x ) + W t ( x ) , t ≥ 0 , x ∈ R , (1) ∂ t where ˙ W t ( x ) is the derivative of a space-time Gaussian white noise (GWN). • The pathwise uniqueness for (1) is unknown. Progress: Perkins, Sturm, Mytnik, etc. • Xiong (2012) studied the pathwise uniqueness to SPDE for the distribution function process of the SBM. Pathwise uniqueness to similar equation see Dawson and Li (2012). • This talk is to generalize the result of Xiong (2012) to the super-L´ evy process with general branching mechanism. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 3 / 11

Introduction Let { X t : t ≥ 0 } be a binary branching super-Brownian motion (SBM). Then X t ( dx ) = X t ( x ) dx and the density is the unique positive weak solution to (Konno-Shiga (1988) and Reimers (1989)): ∂ X t ( x ) = 1 X t ( x ) ˙ � 2 ∆ X t ( x ) + W t ( x ) , t ≥ 0 , x ∈ R , (1) ∂ t where ˙ W t ( x ) is the derivative of a space-time Gaussian white noise (GWN). • The pathwise uniqueness for (1) is unknown. Progress: Perkins, Sturm, Mytnik, etc. • Xiong (2012) studied the pathwise uniqueness to SPDE for the distribution function process of the SBM. Pathwise uniqueness to similar equation see Dawson and Li (2012). • This talk is to generalize the result of Xiong (2012) to the super-L´ evy process with general branching mechanism. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 3 / 11

Introduction Let { X t : t ≥ 0 } be a binary branching super-Brownian motion (SBM). Then X t ( dx ) = X t ( x ) dx and the density is the unique positive weak solution to (Konno-Shiga (1988) and Reimers (1989)): ∂ X t ( x ) = 1 X t ( x ) ˙ � 2 ∆ X t ( x ) + W t ( x ) , t ≥ 0 , x ∈ R , (1) ∂ t where ˙ W t ( x ) is the derivative of a space-time Gaussian white noise (GWN). • The pathwise uniqueness for (1) is unknown. Progress: Perkins, Sturm, Mytnik, etc. • Xiong (2012) studied the pathwise uniqueness to SPDE for the distribution function process of the SBM. Pathwise uniqueness to similar equation see Dawson and Li (2012). • This talk is to generalize the result of Xiong (2012) to the super-L´ evy process with general branching mechanism. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 3 / 11

• D ( R ) := { f : f is bounded right continuous increasing and f ( −∞ ) = 0 } . M ( R ) := { finite Borel measures on R } . There is a 1-1 correspondence between D ( R ) and M ( R ) assigning a measure to its distribution function. We endow D ( R ) with the topology induced by this correspondence from the weak convergence topology of M ( R ) . • The branching mechanism φ : � ∞ ( e − z λ − 1 + z λ ) m ( dz ) . φ ( λ ) = b λ + c λ 2 / 2 + 0 • M ( R ) -valued { X t } process is called a super-L´ evy process if � E µ � � exp [ −� X t , f � ] = exp {−� µ, v t �} , ∂ ∂ t v t ( x ) = Av t ( x ) + φ ( v t ( x )) , v 0 ( x ) = f ( x ) . Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 4 / 11

• D ( R ) := { f : f is bounded right continuous increasing and f ( −∞ ) = 0 } . M ( R ) := { finite Borel measures on R } . There is a 1-1 correspondence between D ( R ) and M ( R ) assigning a measure to its distribution function. We endow D ( R ) with the topology induced by this correspondence from the weak convergence topology of M ( R ) . • The branching mechanism φ : � ∞ ( e − z λ − 1 + z λ ) m ( dz ) . φ ( λ ) = b λ + c λ 2 / 2 + 0 • M ( R ) -valued { X t } process is called a super-L´ evy process if � E µ � � exp [ −� X t , f � ] = exp {−� µ, v t �} , ∂ ∂ t v t ( x ) = Av t ( x ) + φ ( v t ( x )) , v 0 ( x ) = f ( x ) . Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 4 / 11

• D ( R ) := { f : f is bounded right continuous increasing and f ( −∞ ) = 0 } . M ( R ) := { finite Borel measures on R } . There is a 1-1 correspondence between D ( R ) and M ( R ) assigning a measure to its distribution function. We endow D ( R ) with the topology induced by this correspondence from the weak convergence topology of M ( R ) . • The branching mechanism φ : � ∞ ( e − z λ − 1 + z λ ) m ( dz ) . φ ( λ ) = b λ + c λ 2 / 2 + 0 • M ( R ) -valued { X t } process is called a super-L´ evy process if � E µ � � exp [ −� X t , f � ] = exp {−� µ, v t �} , ∂ ∂ t v t ( x ) = Av t ( x ) + φ ( v t ( x )) , v 0 ( x ) = f ( x ) . Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 4 / 11

Our aim in this talk is that under a mild condition on A , { Y t } , defined by Y t ( x ) = X t ( −∞ , x ] , is the pathwise unique solution to � t � t � Y s ( x ) Y s ( x ) ds + √ c Y t ( x ) = Y 0 ( x ) − b W ( ds , du ) 0 0 0 � ∞ � t � Y s − ( x ) � t z ˜ A ∗ Y s ( x ) ds , + N 0 ( ds , dz , du ) + (2) 0 0 0 0 where W ( ds , du ) is a GWN and ˜ N 0 ( ds , dz , du ) compensated Poisson random measure (CPRM), A ∗ denotes the dual operator of A . • Xiong (2012): A = ∆ / 2 and b = ˜ N 0 = 0. • Key approach: connecting (2) with a backward doubly SDE. Xiong (2012) used an L 2 -argument. We use an L 1 -argument. • For M ( R ) -valued process { X t } , its distribution { Y t } is D ( R ) -valued. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 5 / 11

Our aim in this talk is that under a mild condition on A , { Y t } , defined by Y t ( x ) = X t ( −∞ , x ] , is the pathwise unique solution to � t � t � Y s ( x ) Y s ( x ) ds + √ c Y t ( x ) = Y 0 ( x ) − b W ( ds , du ) 0 0 0 � ∞ � t � Y s − ( x ) � t z ˜ A ∗ Y s ( x ) ds , + N 0 ( ds , dz , du ) + (2) 0 0 0 0 where W ( ds , du ) is a GWN and ˜ N 0 ( ds , dz , du ) compensated Poisson random measure (CPRM), A ∗ denotes the dual operator of A . • Xiong (2012): A = ∆ / 2 and b = ˜ N 0 = 0. • Key approach: connecting (2) with a backward doubly SDE. Xiong (2012) used an L 2 -argument. We use an L 1 -argument. • For M ( R ) -valued process { X t } , its distribution { Y t } is D ( R ) -valued. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 5 / 11

Our aim in this talk is that under a mild condition on A , { Y t } , defined by Y t ( x ) = X t ( −∞ , x ] , is the pathwise unique solution to � t � t � Y s ( x ) Y s ( x ) ds + √ c Y t ( x ) = Y 0 ( x ) − b W ( ds , du ) 0 0 0 � ∞ � t � Y s − ( x ) � t z ˜ A ∗ Y s ( x ) ds , + N 0 ( ds , dz , du ) + (2) 0 0 0 0 where W ( ds , du ) is a GWN and ˜ N 0 ( ds , dz , du ) compensated Poisson random measure (CPRM), A ∗ denotes the dual operator of A . • Xiong (2012): A = ∆ / 2 and b = ˜ N 0 = 0. • Key approach: connecting (2) with a backward doubly SDE. Xiong (2012) used an L 2 -argument. We use an L 1 -argument. • For M ( R ) -valued process { X t } , its distribution { Y t } is D ( R ) -valued. Xu Yang (BNU) Stochastic Equations of Super-L´ evy Process with General Branching Mechanism June 18, 2012 5 / 11