[1] Xiong J, Yang X. Superprocesses with interaction and immigration. Stochastic Process Appl, 2016, 126:3377-3401 [2] Li Z. Measure-Valued Branching Markov Processes[M]. Berlin:Springer, 2011 [3] Watanabe S. A limit theorem of branching processes and continuous state branching processes. J Math Kyoto Univ, 1968, 8:141-167 [4] Dawson D A. Stochastic evolution equations and related measure processes. J Multivariate Anal, 1975, 5:1-52 [5] Fleming W H, Viot M. Some measure-valued Markov processes in population genetics theory. Indiana Univ Math J, 1979, 28:817-843 [6] Etheridge A. An introduction to superprocesses[M]. American Mathematical Society. University Lecture Series, 20:2000 [7] Li Z. Immigration processes associated with branching particle systems. Adv in Appl Probab, 1998, 30:657-675 [8] Pardox E, Peng S. Backward doubly stochastic differential equations and systems of quasilimear SPDEs. Probab. Theory Related Fields, 1994, 98:209-227 [9] He H, Li Z, Yang X. Stochastic equations of super-Lévy processes with general branching mechanism. Stochastic Process Appl, 2014, 124:1519-1565 [10] Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes[M]. 2nd Ed. Amsterdam:North-Holland; Tokyo:Kodansha, 1989 [11] Xu W. Backward doubly stochastic equations with jumps and comparison theorems. J Math Anal Appl, 2016, 443:596-624 [12] Konno N, Shiga T. Stochastic partial differential equations for some measure-valued diffusions. Probab Theory Related Fields, 1988, 79:201-225 [13] Reimers M. One dimensional stochastic differential equations and the branching measure diffusion[J]. Probab. Theory Related Fields, 1989, 81:319-340 [14] Mytnik L, Perkins E, Sturm A. On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients[J]. Ann Probab, 2006, 34:1910-1959 [15] Rippl T, Sturm A. New results on pathwise uniqueness for the heat equation with colored noise[J]. Electron J Probab, 2013, 18:1-46 [16] Watanabe S, Yamada T. On the uniqueness of solutions of stochastic differential equations[J]. J Math Kyoto Univ, 1971, 11:553-563 [17] Neuman E. Pathwise uniqueness of the stochastic heat equations with spatially inhomogeneous white noise[J]. Submitted. ArXiv:1403.4491, 2014 [18] Mytnik L, Perkins E. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients:the white noise case[J]. Probab Theory Related Fields, 2011, 149:1-96 [19] Mytnik L, Neuman E. Pathwise uniqueness for the stochastic heat equation with Hölder continuous drift and noise coefficients[J]. Stochastic Process Appl, 2015, 125:3355-3372 [20] Burdzy K, Mueller C, Perkins E A. Nonuniqueness for nonnegative solutions of parabolic stochastic partial differential equations[J]. Illinois J Math, 2011, 54:1481-1507 [21] Mueller C, Mytnik L, Perkins E. Nonuniqueness for a parabolic SPDE with 34-ε-Hölder diffusion coefficients[J]. Ann Probab, 2014, 42:2032-2112 [22] Chen Y. Pathwise uniqueness for the SPDE's of some super-Brownian motion with immigration[J]. Ann Probab, 2015, 43:3359-3467 [23] Mueller C, Mytnik L, Perkins E. On the boundary of the support of super-Brownian motion[J]. Ann Probab, 2017, 45:3481-3534 [24] Mytnik L. Stochastic partial differential equation driven by stable noise[J]. Probab Theory Related Fields, 2003, 123:157-201 [25] Mytnik L, Perkins E. Regularity and irregularity of (1+ β)-stable super-Brownian motion[J]. Ann Probab, 2003, 31:1413-1440 [26] Fleischmann K, Mytnik L, Wachtel V. Optimal local Hölder index for density states of superprocesses with 1+ β-branching mechanism[J]. Ann Probab, 2010, 38:1180-1220 [27] Fleischmann K, Mytnik L, Wachtel V. Hölder index at a given point for density states of super-α-stable motion of index 1+ β[J]. J Theoret Probab, 2011, 24:66-92 [28] Mytnik L, Wachtel V. Multifractal analysis of superprocesses with stable branching in dimension one[J]. Ann Probab, 2015, 43:2763-2809 [29] Mytnik L, Wachtel V. Regularity and Irregularity of Superprocesses with (1+ β)-stable Branching Mechanism[M]. Springer International Publishing, 2017 [30] Yang X, Zhou X. Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by α-stable noise[J]. Electron J Probab, 2017, 22:1-48 [31] Yang X, Zong G. Stable branching super-Lévy process as the pathwise unique solution to an SPDE (in Chinese)[J]. Sci Sin Math, 2019, 49:699-716 [32] Xiong J, Yang X. Existence and pathwise uniqueness to an SPDE driven by α-stable colored noise[J]. Stochastic Process Appl, 2019(to appear) [33] Mueller C. The heat equation with Lévy noise[J]. Stochastic Process Appl, 1998, 74:67-82 [34] Saint Loubert Bié E. řtude dune EDPS conduite par un bruit poissonnien[J]. Probab Theory Related Fields, 1998, 111:287-321 [35] Albeverio S, Wu J, Zhang T. Parabolic SPDEs driven by Poisson white noise[J]. Stochastic Process Appl, 1998, 74:21-36 [36] Applebaum D, Wu J. Stochastic partial differential equations driven by Lévy space-time white noise[J].Random Oper Stoch Equ, 2000, 8:245-259 [37] Chen Z, Zhang T. Stochastic evolution equations driven by Lévy processes[J]. Osaka J Math, 2011, 48:311-327 [38] Peszat S, Zabczyk J. Stochastic Partial Differential Equations with Lévy Noise[M]. Cambridge:Cambridge University Press, 2007 [39] Xiong J. Super-Brownian motion as the unique strong solution to an SPDE[J]. Ann Probab, 2013, 41:1030-1054 [40] Wang L, Yang X, Zhou X. A distribution-function-valued SPDE and its applications[J]. J Differential Equations, 2017, 262:1085-1118 [41] Li Z, Liu H, Xiong J, Zhou X. Some properties of the generalized Fleming-Viot processes[J]. Stochastic Process Appl, 2013, 123:4129-4155 [42] Li Z, Yang X, Zong G. General super-Lévy processes in random environments (in Chinese)[J]. Submitted, 2018 [43] Mytnik L, Xiong J. Well-posedness of the martingale problem for superprocess with interaction[J]. Illinois J Math, 2015, 59:485-497 [44] Dawson D A, Li Z. Stochastic equations, flows and measure-valued processes[J]. Ann Probab, 2012, 40:813-857 [45] Dawson D A, Li Z. Skew convolution semigroups and affine Markov processes[J]. Ann Probab, 2006, 34:1103-1142 [46] Xiong J. Three Classes of Nonlinear Stochastic Partial Differential Equations[M]. Singapore:World Scientific, 2013 [47] Gomez A, Lee K, Mueller C, Wei A, Xiong J. Strong uniqueness for an spde via backward doubly stochastic differential equations[J]. Statist Probab Lett, 2013, 83:2186-2190 |