Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory

  • Sergio Albeverio
  • Barbara Rüdiger
  • Jiang-Lun Wu


A review of work on the description of generators and processes associated with stochastic (partial or pseudo-) differential equations driven by general white noises (including jump as well as diffusion parts) is given. Processes with finite- or infinite-dimensional state space are described in a unified way using the theory of Dirichlet forms, combined with the technique of subordination of processes. In particular the analytic problems arising from subordinating sub-Markov semigroups are described. As examples the subordination of stochastic quantization processes is presented. It is also described how stochastic partial differential or pseudodifferential equations are used to construct relativistic quantum fields in indefinite metric with nontrivial scattering in four space- time dimensions.


Stochastic Differential Equation Pseudodifferential Operator Dirichlet Form Stochastic Partial Differential Equation Martingale Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Albeverio, Some recent developments and applications of path integrals, in M. C. Gutzwiller et al., eds., Path Integrals from meV to MeV, World Scientific, Singapore, 1986, 3–32.Google Scholar
  2. [2]
    S. Albeverio, Ph. Blanchard, Ph. Combe, R. Høegh-Krohn, and M. Sirugue, Local relativistic invariant flow for quantum fields, Comm. Math. Phys., 90 (1983), 329–351.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    S. Albeverio, V. Bogachev, and M. Röckner, On uniqueness and ergodicity of invariant measures for finite and infinite dimensional diffusions, Comm. Pure Appl. Math., 52 (1999), 325–362.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orlando, 1986 (in Russian).MATHGoogle Scholar
  5. [4a]
    S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, MIR Moscow, 1988 (in Russian).Google Scholar
  6. [5]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer Verlag, Berlin, 1988 (in Russian).MATHCrossRefGoogle Scholar
  7. [5a]
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, MIR, Moscow, 1990 (in Russian).Google Scholar
  8. [6]
    S. Albeverio and H. Gottschalk, Scattering theory for local relativistic QFT with indefinite metric, Comm. Math. Phys., 2001.Google Scholar
  9. [7]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Euclidean random fields, pseudodifferential operators, and Wightman functions, in I. M. Davies, A. Truman and K. D. Elworthy, eds., Proceedings of the Gregynog Symposium “Stochastic Analysis and Applications, “World Scientific, Singapore, 1996, 20–37.Google Scholar
  10. [8]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions, Rev. Math. Phys., 8 (1996), 763–817.MathSciNetMATHCrossRefGoogle Scholar
  11. [9]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Comm. Math. Phys., 184 (1997), 509–531.MathSciNetMATHCrossRefGoogle Scholar
  12. [10]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Remarks on some new models of interacting quantum fields with indefinite metric, Rep. Math. Phys., 40 (1997), 385–394.MathSciNetMATHCrossRefGoogle Scholar
  13. [11]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric, Phys. Lett. B, 405 (1997),243–248.MathSciNetCrossRefGoogle Scholar
  14. [12]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Partly divisible probability distributions, Forum Math., 10 (1998), 687–697.MathSciNetMATHCrossRefGoogle Scholar
  15. [13]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Scattering behaviour of relativistic quantum vector fields obtained from Euclidean covariant SPDEs, Rep. Math. Phys., 44 (1999), 21–28.MathSciNetMATHCrossRefGoogle Scholar
  16. [14]
    S. Albeverio, H. Gottschalk, and J.-L. Wu, Partly divisible probability measures on locally compact Abelian groups, Math. Nachr., 213 (2000), 5–15.MathSciNetMATHCrossRefGoogle Scholar
  17. [15]
    S. Albeverio, Z. Haba, and F. Russo, On non-linear two-space-dimensional wave equation perturbed by space-time white noise, in Stochastic Analysis, Random Fields and Measure-Valued Processes, Ramat Gar, 1996, 1–15.Google Scholar
  18. [16]
    S. Albeverio, Z. Haba, and F. Russo, Trivial solution for a nonlinear two-space dimensional wave equation perturbated by space-time white noise, Stochastics Stochastics Rep. 56 (1996), 127–160.MathSciNetMATHGoogle Scholar
  19. [17]
    S. Albeverio and R. Høegh-Krohn, Quasi-invariant measures, symmetric diffusion processes and quantum fields, in Les méthodes mathématiques de la théorie quantique des champs, Colloques Internationaux du CNRS, Marseille, CNRS, 1976, 11–59.Google Scholar
  20. [18]
    S. Albeverio and R. Høegh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahr. verw. Geb., 40 (1977), 1–57.MATHCrossRefGoogle Scholar
  21. [19]
    S. Albeverio R. Høegh-Krohn, Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations, Phys. Lett. B, 177 (1986), 175–179.MathSciNetCrossRefGoogle Scholar
  22. [20]
    S. Albeverio and R. Høegh-Krohn, Quaternionic non-abelian relativistic quantum fields in four space-time dimensions, Phys. Lett, B, 189 (1987), 329–336.MathSciNetCrossRefGoogle Scholar
  23. [21]
    S. Albeverio and R. Høegh-Krohn, Construction of interacting local relativistic quantum fields in four space-time dimensions, Phys. Lett. B, 200 (1988), 108–114.MathSciNetMATHCrossRefGoogle Scholar
  24. [21a]
    S. Albeverio and R. Høegh-Krohn, erratum, Phys. Lett. B, 202 (1988), 621.CrossRefGoogle Scholar
  25. [22]
    S. Albeverio, R. Høegh-Krohn, and H. Holden, Stochastic multiplicative measures, generalized Markov semigroups and group valued stochastic processes and fields, J. Funct. Anal, 78 (1988), 154–184.MathSciNetMATHCrossRefGoogle Scholar
  26. [23]
    S. Albeverio, R. Høegh-Krohn, H. Holden, and T. Kolsrud, Representation and construction of multiplicative noise, J. Funct. Anal., 87 (1989), 250–272.MathSciNetMATHCrossRefGoogle Scholar
  27. [24]
    S. Albeverio, R. Høegh-Krohn, and L. Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys., 18 (1977), 907–917.MATHCrossRefGoogle Scholar
  28. [25]
    S. Albeverio, K. Iwata, and T. Kolsrud, Random fields as solutions of the inhomogeneous quaternionic Cauchy-Riemann equation I: In variance and analytic continuation, Comm. Math. Phys., 132 (1990), 555–580.MathSciNetMATHCrossRefGoogle Scholar
  29. [26]
    S. Albeverio and P. Kurasov, Pseudo-Differential operators with point interaction, Lett. Math. Phys., 41 (1997), 79–92.MathSciNetMATHCrossRefGoogle Scholar
  30. [27]
    S. Albeverio, Z.-M. Ma, and M. Röckner, A Beurling-Deny type structure theorem for Dirichlet forms on general state space, in S. Albeverio, J. E. Fenstad, H. Holden, and T. Lindstrøm, eds., Ideas and Methods in Mathematical Analysis, Stochastics and Applications: Memorial Volume for R. Høegh-Krohn, Vol. I, Cambridge University Press, Cambridge, UK, 1992.Google Scholar
  31. [28]
    S. Albeverio, Z.-M. Ma, M. Röckner, Quasi-regular Dirichlet forms and Markov processes, J. Funct. Anal., 111 (1993), 118–154.MathSciNetMATHCrossRefGoogle Scholar
  32. [29]
    S. Albeverio and K. Makarov, in preparation.Google Scholar
  33. [30]
    S. Albeverio, L. M. Morato, and S. Ugolini, Non-symmetric diffusions and related Hamiltonians, Potential Anal., 8 (1998), 195–204.MathSciNetMATHCrossRefGoogle Scholar
  34. [31]
    S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimension: Solutions via Dirichlet forms, Probab. Theory Related Fields, 89 (1992), 347–386.CrossRefGoogle Scholar
  35. [32]
    S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces: The construction of the associated diffusion process, Probab. Theory Related Fields, 83 (1989), 405–434.MathSciNetMATHCrossRefGoogle Scholar
  36. [33]
    S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces: Closability and a Cameron-Martin formula, J. Funct. Anal., 88 (1990), 395–436.MathSciNetMATHCrossRefGoogle Scholar
  37. [34]
    S. Albeverio and M. Röckner, Dirichlet form methods for uniqueness of martingale problems and applications, in M. C. Cranston and M. A. Pinsky, eds., Proceedings of the 1993 Summer Research Institute on Stochastic Analysis, AMS, Providence, 1995, 513–528.Google Scholar
  38. [35]
    S. Albeverio and B. Rüdiger, The perturbed fractional power of the Laplacian operator, analytic and probabilistic aspects, in preparation.Google Scholar
  39. [36]
    S. Albeverio and B. Rüdiger, The Lévy-Ito decomposition theorem on separable Banach spaces, in preparation.Google Scholar
  40. [37]
    S. Albeverio, B. Rüdiger, and J.-L. Wu, Processes with jumps on infinite dimensional state spaces obtained via subordination, in preparation.Google Scholar
  41. [38]
    S. Albeverio, B. Rüdiger, and J.-L. Wu, Invariant measures and symmetry property of Lévy type operators,Potential Anal, 2000, in press.MATHGoogle Scholar
  42. [39]
    S. Albeverio and F. Russo, Stochastic partial differential equations, infinite dimensional stochastic processes and random fields: A short introduction, in L. Vàzquez, L. Streit, and V. M. Pérez-Garcia, eds., Nonlinear Klein-Gordon and Schrödinger Systems: Theory and Applications, World Scientific, Singapore, 1996, 68–86.Google Scholar
  43. [40]
    S. Albeverio and J.-L. Wu, Euclidean random fields obtained by convolution from generalized white noise, J. Math. Phys., 36 (1995), 5217–5245.MathSciNetMATHCrossRefGoogle Scholar
  44. [41]
    S. Albeverio and J.-L. Wu, On the lattice approximation for certain generalized vector Markov fields in four space-time dimensions, Acta Appl Math., 47 (1997), 31–48.MathSciNetMATHCrossRefGoogle Scholar
  45. [42]
    S. Albeverio, J.-L. Wu, and T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastics Proc. Appl., 74 (1998), 21–36.MathSciNetMATHCrossRefGoogle Scholar
  46. [43]
    S. Albeverio, J.-L. Wu, and T.-S. Zhang, On parabolic SPDEs driven by higher dimensional space-time non Gaussian white noise, in preparation.Google Scholar
  47. [44]
    D. Applebaum and J.-L. Wu, Stochastic partial differential equations driven by Lévy space-time white noise, Random Operators Stochastic Equations, 8 (2000), 245–260.MathSciNetMATHGoogle Scholar
  48. [45]
    S. Assing, Infinite dimensional Langevin equations: Uniqueness and rate of convergence for finite dimensional approximations, BiBoS preprint 99–11–15, 1999.Google Scholar
  49. [46]
    O. Barndorff-Nielsen and V. Pérez-Abreu, Stationary and self similar processes driven by Lévy processes, MaPhySto preprint, 2000Google Scholar
  50. [47]
    C. Becker, Wilson loops in two-dimensional space-time regarded as white noise, J. Funct. Anal., 134 (1995), 321–349.MathSciNetMATHCrossRefGoogle Scholar
  51. [48]
    C. Becker, R. Gielerak, and P. Lugiewicz, Covariant SPDEs and quantum field structures, J. Phys. A, 31 (1998), 231–258.MathSciNetMATHCrossRefGoogle Scholar
  52. [49]
    C. Becker, H. Gottschalk, and J.-L. Wu, Generalized random vector fields and Euclidean quantum fields, in R. Dalang, M. Dozzi, and F. Russo, eds., Second Seminar on Stochastic Analysis, Random Fields and Applications, Progress in Probability 45, Birkhäuser, Basel, Boston, Berlin, 1999, 15–24.CrossRefGoogle Scholar
  53. [50]
    Yu. M. Berezanski and Yu. G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Vols. 1 and 2, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1995.CrossRefGoogle Scholar
  54. [51]
    Ch. Berg, Kh. Boyadzhiev, R. de Laubenfels, Generation of generators of holomorphic semigroups, J. Austral, Math. Soc. Ser. A, 55 (1993), 246–269.MathSciNetMATHCrossRefGoogle Scholar
  55. [52]
    L. Bertini, E. Presutti, B. Rüdiger, and E. Saada, Dynamical fluctuations at the critical point, Theory Probab. Appl., 38 (1993), 689–741.MathSciNetCrossRefGoogle Scholar
  56. [53]
    J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, UK, 1996.MATHGoogle Scholar
  57. [54]
    O. Bilstein, Feyman-Kac Formel für Lévy-Prozesse, Diplomarbeit, Bochum, 1998.Google Scholar
  58. [55]
    R. M. Blumental and R. K. Getoor, Markov Processes and Potential Theory, Mathematics 29, Academic Press, New York, London, 1968.Google Scholar
  59. [56]
    S. Bochner, Harmonic Analysis and the Theory of Probability, California Monographs in Math. Sci., University of California Press, Berkeley, CA, 1955.MATHGoogle Scholar
  60. [57]
    V. I. Bogachev, P. Lescot, and M. Röckner, The martingale problem for pseudo-differential operators on infinite-dimensional spaces, preprint, 1997.Google Scholar
  61. [58]
    V. I. Bogachev and M. Röckner, Regularity of invariant measures on finite and infinite dimensional spaces and applications, J. Funct. Anal., 133 (1995), 168–223.MathSciNetMATHCrossRefGoogle Scholar
  62. [59]
    V. Bogachev, M. Röckner, and T.-S. Zhang, Existence and uniqueness of invariant measures: An approach via sectorial forms, Appl. Math. Optim., 41 (2000), 87–109.MathSciNetMATHCrossRefGoogle Scholar
  63. [60]
    V. S. Borkar, R. T. Chari, and S. K. Mitter, Stochastic quantization of field theory in finite and infinite volume, J. Funct. Anal., 81 (1988), 184–206.MathSciNetMATHCrossRefGoogle Scholar
  64. [61]
    N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, Studies in Mathematics 14, Walter de Gruyter, Berlin, 1991.CrossRefGoogle Scholar
  65. [62]
    R. Carmona, W. C. Masters, B. Simon, Relativistic Schroedinger operators: Asymptotic behaviour of the eigenfunctions, J. Funct. Anal., 91 (1990), 117–142.MathSciNetMATHCrossRefGoogle Scholar
  66. [63]
    J. A. van Casteren, Some problems in stochastic analysis and semigroup theory, preprint 99–10, Universitaire Instelling Antwerpen, 1999.Google Scholar
  67. [64]
    G. Da Prato and L. Tubaro, Introduction to stochastic quantization, preprint, Trento, 1996.Google Scholar
  68. [65]
    G. Da Prato and J. Zabczyck, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.MATHCrossRefGoogle Scholar
  69. [66]
    A. De Acosta, A. Araujo, and E. Giné, On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces, in Probability in Banach Spaces, Adv. Probab. Related Topics 4, Marcel Dekker, New York, 1978, 1–68.Google Scholar
  70. [67]
    G. Del Grosso and R. Marra, Girsanov and Feynman-Kac formulas in the discrete stochastic mechanics, in Stochastic Differential Systems (Bad Honnef, 1985), Lecture Notes in Control and Inform. Sci. 78, Springer, Berlin, New York, 1986, 163–170.Google Scholar
  71. [68]
    J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York, Berlin, 1984.MATHCrossRefGoogle Scholar
  72. [69]
    A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators, Lecture Notes in Mathematics 1718, Springer-Verlag, Berlin, Heidelberg, New York, 1999.MATHGoogle Scholar
  73. [70]
    W. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A, 15 (1982), 3025–3055.MathSciNetMATHCrossRefGoogle Scholar
  74. [71]
    W. Faris and B. Simon, Degenerate and non-degenerate ground states for Schroedinger operators, Duke Math. J., 42 (1975), 559–567.MathSciNetMATHCrossRefGoogle Scholar
  75. [72]
    J. Fritz and B. Rüdiger, Time dependent critical fluctuations of a one-dimensional local mean field model, Probab. Theory Related Fields, 103 (1995).Google Scholar
  76. [73]
    J. Fritz and B. Rüdiger, Approximation of a one-dimensional stochastic PDE by local mean field type lattice systems, in T. Funaki and W. Woycsynski, eds., Nonlinear Stochastic PDEs: Hydrodynamic Limit and Burger’s Turbolence, IMA Volumes in Mathematics and Its Applications.Google Scholar
  77. [74]
    T. Fujiwara and H.Kunita, Canonical SDE’s based on semimartingales with spatial parameters, part 1: Stochastic flows of diffeomorphisms, Kyushu J. Math., 53 (1999), 265–300.MathSciNetMATHCrossRefGoogle Scholar
  78. [75]
    M. Fukushima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motion, Phys. Rep., 11 (1981), 255–262.MathSciNetCrossRefGoogle Scholar
  79. [76]
    M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980.MATHGoogle Scholar
  80. [77]
    M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994.MATHCrossRefGoogle Scholar
  81. [78]
    I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions IV: Some Applications of Harmonic Analysis, Academic Press, New York, London, 1964.Google Scholar
  82. [79]
    R. Gielerak and P. Ługiewicz, From stochastic differential equation to quantum field theory, Rep. Math. Phys., 44 (1999), 101–110.MathSciNetMATHCrossRefGoogle Scholar
  83. [80]
    J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd ed., Springer-Verlag, Berlin, New York, 1987.Google Scholar
  84. [81]
    H. Gottschalk, Die Momente gefalteten Gauss-Poissonschen Weißen Rauschens als Schwingerfunktionen, Diplomarbeit, Bochum, 1995.Google Scholar
  85. [82]
    H. Gottschalk, Green’s functions for scattering in local relativistic quantum field theory, Dissertation, Bochum, 1998.MATHGoogle Scholar
  86. [83]
    H. Gottschalk and J.-L. Wu, On the formulation of SPDEs leading to relativistic QFT with indefinite metric and non trivial S-matrix, preprint SFB 256, Bonn, 1999.Google Scholar
  87. [84]
    M. Grothaus and L. Streit, Construction of relativistic quantum fields in the framework of white noise analysis, J. Math. Phys., 40 (1999), 5387–5407.MathSciNetMATHCrossRefGoogle Scholar
  88. [85]
    R. Haag, Quantum fields with composite particles and asymptotic conditions, Phys. Rev., 112 (1958), 669–673.MathSciNetMATHCrossRefGoogle Scholar
  89. [86]
    G. G. Hamedani and V. Mandrekar, Lévy-Khinchine representation and Banach spaces of type and cotype, Stud. Math., 66 (1980), 299–306.MathSciNetMATHGoogle Scholar
  90. [87]
    K. Hepp, On the connection between the LSZ and Wightman quantum field theory, Comm. Math. Phys., 1 (1965), 95–111.MathSciNetMATHCrossRefGoogle Scholar
  91. [88]
    T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise: An Infinite Dimensional Calculus, Kluwer Academic Publishers, Dordrecht, 1993.MATHGoogle Scholar
  92. [89]
    W. Hoh, The martingale problem for a class of pseudo-differential operators, Math. Ann., 300 (1994), 121–147.MathSciNetMATHCrossRefGoogle Scholar
  93. [90]
    H. Holden, B. Øksendal, Ubøe, andT.-S. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise, Functional Approach, Birkhäuser, Basel, 1996.MATHGoogle Scholar
  94. [91]
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Kodansha, 1981.MATHGoogle Scholar
  95. [92]
    K. Iwata, The inverse of a local operator preserves the Markov property, Ann. Scuola Norm. Sup. Pisa., XIX (1992), 223–253.MathSciNetGoogle Scholar
  96. [93]
    K. Iwata and J. Schäfer, Markov property and cokernels of local operators, preprint, Bochum, 1994.Google Scholar
  97. [94]
    N. Jacob, Pseudo-Differential Operators and Markov Processes, Akademie Verlag, Berlin, 1996.MATHGoogle Scholar
  98. [95]
    N. Jacob, Further pseudodifferential operators generating Feller semigroups and Dirichlet forms, Rev. Mat. Iberoamericana, 9 (1993).Google Scholar
  99. [96]
    J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987.MATHGoogle Scholar
  100. [97]
    G. Jona-Lasinio, P. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys., 101 (1985), 409–436.MathSciNetMATHCrossRefGoogle Scholar
  101. [98]
    O. Kallenberg, Random Measures, Academie Verlag, Berlin, 1976.MATHGoogle Scholar
  102. [99]
    G. Kallianpur and V. Perez-Abreu, Stochastic evolution equations driven by nuclear-space-valued martingales, Appl. Math. Optim., 17 (1988), 237–272.MathSciNetMATHCrossRefGoogle Scholar
  103. [100]
    T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, Heidelberg, New York, 1980.Google Scholar
  104. [101]
    J. F. C. Kingman, Poisson Processes, Clarendon Press, Oxford, 1993.MATHGoogle Scholar
  105. [102]
    V. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lecture Notes in Mathematics 1724, Springer-Verlag, Berlin, 2000.MATHGoogle Scholar
  106. [103]
    T. Komatsu, Markov processes associated with certain integro-differential operators, Osaka J. Math., 10 (1973), 271–303.MathSciNetMATHGoogle Scholar
  107. [104]
    H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, UK, 1988.Google Scholar
  108. [105]
    W. Linde, Probability in Banach Spaces: Stable and Infinitely Divisible Distributions, Wiley, Chichester, New York, 1983.Google Scholar
  109. [106]
    V. Liskevich and M. Röckner, Strong uniqueness for a class of infinite dimensional Dirich- let operators and applications to stochastic quantization, Ann. Scuola Norm. Sup. Pisa, XXVII (1998).Google Scholar
  110. [107]
    J. U. Löbus, Closability of positive symmetric bilinear forms under non-regularity assumptions and its probabilistic counterpart, preprint, 1996.Google Scholar
  111. [108]
    E. Lukacs, Characteristic Functions, 2nd ed. Griffin, London, 1970.MATHGoogle Scholar
  112. [109]
    Z.-M. Ma and M. Röckner, An Introduction on the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992.CrossRefGoogle Scholar
  113. [110]
    B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Springer-Verlag, Berlin, 1997.MATHGoogle Scholar
  114. [111]
    J. Mecke, R. Schneider, D. Stoyan, and W. Weil, Stochastische Geometrie, Birkhäuser, Basel, 1990.MATHGoogle Scholar
  115. [112]
    R. Mikulievikius and B. L. Rozovskii, Martingale problems for stochastic PDEs, in R. Carmona and B. L. Rozovskii Stochastic Partial Differential Equations: Six Perspectives, AMS, Providence, 1999, 251–333.Google Scholar
  116. [113]
    G. Morchio and F. Strocchi, Infrared singularities, vacuum structure and pure phases in local quantum field theory, Ann. Inst. H. Poincaré, 33 (1980), 251–282.MathSciNetGoogle Scholar
  117. [114]
    C. Mueller, The heat equation with Lévy noise, Stochastics Proc. Appl., 74 (1998), 67–82.MathSciNetMATHCrossRefGoogle Scholar
  118. [115]
    D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer-Verlag, Berlin, 1995.Google Scholar
  119. [116]
    E. Pardoux, Stochastic partial differential equations, a review, Bull. Sci. Math. 2, 117 (1993), 29–47.MathSciNetMATHGoogle Scholar
  120. [117]
    R. S. Philipps, On the generation of semigroups of linear operators, Pacific J. Math., 2 (1952), 343–369.MathSciNetGoogle Scholar
  121. [118]
    N. Privault and J.-L. Wu, Poisson stochastic integration in Hilbert spaces, Ann. Math. Blaise Pascal, 6-2 (1999), 41–61.MathSciNetMATHCrossRefGoogle Scholar
  122. [119]
    P. Protter, Stochastic Integration and Differential Equations: A New Approach, Applications of Mathematics 21, Springer-Verlag, Berlin, 1990.Google Scholar
  123. [120]
    M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged edition, Academic Press, London 1980.Google Scholar
  124. [121]
    M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.MATHGoogle Scholar
  125. [122]
    M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, London, 1979.MATHGoogle Scholar
  126. [123]
    M. Röckner and T.-S. Zhang, Uniqueness of generalized Schroedinger operators and applications, J. Fund. Anal., 105 (1992), 187–231.MATHCrossRefGoogle Scholar
  127. [124]
    Yu. A. Rozanov, Markov Random Fields, Springer-Verlag, New York, Heidelberg, Berlin, 1982.MATHCrossRefGoogle Scholar
  128. [125]
    B. Rüdiger, Glauber evolution for Kac potentials: Analysis of critical fluctuations: Derivation of a non linear SPDE, in A. Verbeure, ed., Micro, Meso, and Macroscopic Approaches in Physics, NATO/ASI Series, 1990.Google Scholar
  129. [126]
    B. Rüdiger, Processes with jumps properly associated to non local quasi-regular (non symmetric) Dirichlet forms obtained by subordination, in preparation.Google Scholar
  130. [127]
    B. Rüdiger and J.-L. Wu, Construction by subordination of processes with jumps on infinite dimensional state spaces and corresponding non local Dirichlet formsn, Stochastic Processes, Physics and Geometry: New Interplays: A Volume in Honor of Sergio Albeverio, Proceedings of the International Conference on Infinite Dimensional (Stochastic) Analysis and Quantum Physics (Leipzig 1999), Canadian Mathematical Society Conference Proceedings Series, to appear.Google Scholar
  131. [128]
    E. Saint Loubert Bié, Etude d’une EDPS conduite par un bruit poissonnien, Probab. Theory Related Fields, 111 (1998), 287–321.MathSciNetMATHCrossRefGoogle Scholar
  132. [129]
    K. Sato, Lévy Processes and Infinitely Divisible Distributions, Studies in Advanced Mathematics 68, Cambridge University Press, Cambridge, UK, 1999.Google Scholar
  133. [130]
    J. Schäfer, White noise on vector bundles and local functionals, J. Funct. Anal., 134 (1995), 1–32.MathSciNetMATHCrossRefGoogle Scholar
  134. [131]
    J. Schäfer, Generalized random fields on manifolds and Markov properties, Dissertation, Bochum, 1993.MATHGoogle Scholar
  135. [132]
    R. Schilling, On the domain of the generator of a subordinate semigroup, in Kral, Lokes, Netuka, and Vesely, eds., Potential Theory: ICTP94, Walter de Gruyter, Berlin, New York, 1996.Google Scholar
  136. [133]
    R. Schilling, Subordination in the sense of Bochner and a related functional calculus, J. Austral Math. Soc. Ser. A, 64 (1998), 368–396.MathSciNetMATHCrossRefGoogle Scholar
  137. [ 134]
    I. Shigekawa, Existence of invariant measures of diffusions on an abstract Wiener space, Osaka J. Math., 24 (1987), 37–59.MathSciNetMATHGoogle Scholar
  138. [135]
    M. F. Schlesinger, G. M. Zavslavsky, and U. Frisch, eds., Lévy Flights and Related Topics in Physics, Springer-Verlag, Berlin, New York, Heidelberg, 1995.Google Scholar
  139. [136]
    B. Simon, The P(φ)]2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, NJ, 1975.Google Scholar
  140. [137]
    B. Simon and R. Høegh-Krohn, Hypercontractive semigroups and two-dimensional self- coupled Bose fields, J. Funct. Anal, 9 (1972), 121–180.MATHCrossRefGoogle Scholar
  141. [138]
    F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, Lecture Notes in Physics 51, World Scientific, Singapore, 1993.MATHGoogle Scholar
  142. [139]
    D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahr. verw. Geb., 32 (1975), 209–244.MathSciNetMATHCrossRefGoogle Scholar
  143. [140]
    D. Surgailis, On covariant stochastic differential equations and Markov property of their solutions, preprint, Department of Physics, Universitá di Roma, 1979.Google Scholar
  144. [141]
    A. S. Üstunel, Additive processes on nuclear spaces, Ann. Prob., 12 (1984), 858–868.CrossRefGoogle Scholar
  145. [142]
    J. B. Walsh, An introduction to stochastic partial differential equations, in Ecole d’ȁté de Probabilités de St. Flour XIV, Lecture Notes in Mathematics 1180, Springer-Verlag, Berlin, 1986, 266–439.Google Scholar
  146. [143]
    S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Bombay, 1984.MATHGoogle Scholar
  147. [144]
    B. J. West, An Essay of the Importance of Being Non Linear, Lecture Notes in Biomathematics 62, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
  148. [145]
    K. Yosida, Fractional powers of infinitesimal generators and the analyticity of the semi group generated by them, Proc. Japan Acad., 36 (1960), 86–89.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
    • 3
    • 4
    • 5
  • Barbara Rüdiger
    • 1
  • Jiang-Lun Wu
    • 1
    • 6
    • 7
  1. 1.Institut für Angewandte Mathematik, SFB 256Universität BonnBonnGermany
  2. 2.SFB 237Essen-Bochum-DüsseldorfGermany
  3. 3.BiBoS-Research CentreBielefeldGermany
  4. 4.CERFIMLocarnoSwitzerland
  5. 5.Acc. Arch.USISwitzerland
  6. 6.Department of MathematicsUniversity of Wales SwanseaSwanseaUK
  7. 7.Institute of Applied MathematicsAcademia SinicaBeijingPeople’s Republic of China

Personalised recommendations