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Hao Wang University of Oregon

A Class of Interacting Superprocesses. and Their Associated SPDEs. Hao Wang University of Oregon. Outline. What is Super-Brownian motion?. From branching particle systems to S uper-Brownia n motion. A model for interacting branching particle systems.

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Hao Wang University of Oregon

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  1. A Class of Interacting Superprocesses and Their Associated SPDEs Hao Wang University of Oregon

  2. Outline What is Super-Brownian motion? From branching particle systems to Super-Brownian motion A model for interacting branching particle systems State classification, a new class of SPDEs for density processes Purely-atomic superprocess and a degenerate SPDE A generalized new model and SDSM Location dependent state classification Singular and degenerate SPDE and coalescing Brownian motion On-going development of this model References

  3. Classical Model for Super-Brownian Motion Starting from a simplest branching particle system What is the super-Brownian motion? Intuitive ideas and graphs Rigorously mathematical construction

  4. Brownian branching particle systems Brownian binary branching particle trees

  5. Spatial motion assumption Independent Brownian motions

  6. Branching mechanism Exponential lifetime and binary branching independently

  7. Proportional rescaling limit convergence of mean lifetime, particle mass and initial distribution

  8. Empirical measure-valued processes

  9. Multiplicative property and infinite divisibility If we denote the particles’ empirical measures by The following multiplicative property holds:

  10. MultiP is equivalent to infinite divisibility Multiplicative property (MuP) is the fact that a measure-valued Markov process has MP if two such processes start at and , respectively, then their sum is equal in law to the same process starting at . This is just the infinite divisibility

  11. Log-Laplace functional and evolution equation Based on the MultiP, we have following log-Laplace Functional equation: Where u(t) is the solution of following nonlinear evolution equation See M. Jirina58; S. Watanabe68; D. Dawson75; M.L. Silverstein68.

  12. Structural properties of Super-Brownian motion Suppose that the initial measure is the Lebesgue measure denoted by According to Dawson-Hochberg and Roelly-Coppoletta

  13. SPDE for Super-Brownian motion according to Konno-Shiga88, the density process For is continuous in t and x and satisfies following SPDE

  14. Space-time Brownian motion For with finite Lebesgue measures, is Gaussian rv with mean zero and variance with finite Lebesgue measure,it is a square-integrable martingale

  15. New Model, spatial motion assumption For d =1, between branchings, the motion of each particle is driven by following SDE: Where is the space-time white noise.

  16. New Model, assumptions Assumption SS: is square integrable and has continuous second derivative. According as or , the condition will be referred as degenerate case or no-degenerate case, respectively.

  17. New Model, branching mechanism When a particle dies, it produces j particles with probability which satisfies

  18. Dependency of the particles in the new model The quadratic variational process for any two particles is

  19. Loss of the multiplicative property The log-Laplace functional does not hold for the new or interacting model. (See Wang98 for a counter example) How to construct this class of interacting superprocesses?

  20. Construction : pregenerator By Ito’s formula, we can formally find out the pregenerator of the limiting interacting superprocess of the branching particle systems as follows. where

  21. Construction : existence of solution of MP where Then, the existence of the MP can be obtained by tightness argument from finite branching particle system. Uniqueness is a difficult problem.

  22. Construction : uniqueness of solution of MP By a theorem of Stroock-Varadhan, if there exists a bounded measurable function which is independent of and is only function of such that holds.Then, the MP is well-posed. How to find the function ?

  23. Change the form of the generator The motivation to find out the dual process comes from the following observation of the generator. For monomial function We have

  24. Change the form of the generator where

  25. Dual generator has the structure of generator of function-valued Markov process described as follows: (1) Random jump-mechanism: (2) Deterministic spatial motion between jumps:

  26. Duality Define Stroock-Varadhan function as follows: where if Y(t) is a n-dimensional function. Then, by Feynman-Kac formula we get the duality and the uniqueness follows.

  27. Existence of the density process According to Wang97, in SS non-degenerate case, (i.e. if is a square integrable, has continuous second derivative, and ), then The ideato prove this result is estimating the moments of the dual process since the log-Laplace functional does not exist in the interacting case.

  28. Purely-atomic measure state According to Wang97, in SS degenerate case, (i.e. if is a square-integrable, has continuous second derivative, and ), then The proof of this result is based on following facts: (1) The generator drives the superprocess immediately entering into the space of purely-atomic measures.

  29. Degenerate and coalescence (2) In the degenerate case, by considering the distance process of any two particles and using Feller’s criterion of accessibility, we can get the coalescence property: Any two particles either never separate or never meet according as they have same initial states or not.

  30. Feller’s criterion of accessibility In SS degenerate case, define According to Feller’s criterion of accessibility

  31. Inaccessibility of zero Since is non-negative definite, by Bochner-Khinchin Theorem, there exists a F(.) such that Hence state 0 is inaccessible.

  32. Commutativity of semigroups (3) In the SS degenerate case, the semigroup generated by commutes with the semigroup generated by . Intuitive explanation:

  33. Derivations of SPDEs for density processes SPDE for Super-Brownian motion(space-time martingale Representation theorem, Konno-Shiga1988) SPDE for the interacting superprocess(decomposition Theorem, Dawson-Vaillancourt-Wang2000)

  34. Degenerate SPDE for purely-atomic superprocess Dawson-Li-Wang2002 has studied a degenerate SPDE for a purely-atomic measure valued superprocess and proved the existence and uniqueness of its strong solution.

  35. Simple facts Under SS assumption, following equation has unique strong solution. According to Yamada-Watanabe, following equation has unique strong solution.

  36. One dimensional case We can use stopping time technique and choice of test function to decompose following one-dimensional SPDE

  37. Multidimensional case We can use stopping time technique and choice of test function to decompose the degenerate SPDE into a sequence of one-dimensional SPDEs

  38. Singular and degenerate case Li-Wang-Xiong(2003) considered following case. For d =1, between branchings, the motion of each particle is driven by following SDE:

  39. Singular and degenerate SDSM The singular degenerate SDSM is constructed by scaling Limit technique: We replace by The tightness gives the existence of the solution and Duality gives the uniqueness. The generator of the dual Semigroup is the generator of coalescing Brownian motion

  40. Coalescing Brownian motion m-dimensional Coalescing Brownian motion can be intuitively described as follows: Before meeting, they are independent m Brownian particles. After first meeting of two of them, they become m-1 independent Brownian particles and so on. The generator and the domain can be specified as follows:

  41. Singular and degenerate SPDE Li-Wang-Xiong2003 has studied a SDSPDE and proved the existence and uniqueness of strong solution. Where are killing Brownian motions modified from coalescing Brownian motion.

  42. Conditional log-Laplace functional Li-Wang-Xiong2003 has studied the conditional log-Laplace for SDSM and proved the existence and uniqueness of strong continuous solution of following backward SPDE. where

  43. Conditional log-Laplace equation Then, following conditional log-Laplace equation holds.

  44. This is all for this presentation Thanks

  45. Conditional entrance law Then, we can construct the conditional entrance law for SDSM such that following equation holds.

  46. Excursion paths starting from x { Measure valued excursion paths starting from x } Then, we can construct the conditional excursion law for SDSM As follows. x

  47. Inhomogeneous excursion paths { Measure valued excursion paths starting from x } Define

  48. Conditional excursion law Define

  49. Conditional Poisson random measures Let be the conditional Poisson random measure with intensity measure Let be the conditional Poisson random Measure with intensity measure Define

  50. Conditional excursion representation We have Combining with existing results, this is what we want.

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