Transformations of diffusion and Schrödinger processes

SummaryA transformation by means of a new type of multiplicative functionals is given, which is a generalization of Doob's space-time harmonic transformation, in the case of arbitrary non-harmonic function ϕ (t, x) which may vanish on a subset of [a, b]xℤd. The transformation induces an additional (singular) drift term ∇φ/φ, like in the case of Doob's space-time harmonic transformation. To handle the transformation, an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the Feynman-Kac theorem is established for a class of singular potentials. The transformation is applied to Schrödinger processes which are defined following an idea of E. Schrödinger (1931).

[1]  M. Nagasawa.  Markov process with creation and annihilation , 1969 .

[2]  Kiyosi Itô,et al.  Transformation of Markov processes by multiplicative functionals , 1965 .

[3]  An application of flows to time shift and time reversal in stochastic processes , 1985 .

[4]  E. B. Dynkin,et al.  Transformations of Markov Processes Connected with Additive Functionals , 1961 .

[5]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[6]  M. Nagasawa.  The adjoint process of a diffusion with reflecting barrier , 1961 .

[7]  Y. Oshima,et al.  On a transformation of symmetric markov process and recurrence property , 1987 .

[8]  A. A. Novikov.  On moment inequalities and identities for stochastic integrals , 1973 .

[9]  A. Beurling,et al.  An Automorphism of Product Measures , 1960 .

[10]  M. Nagasawa.  Segregation of a population in an environment , 1980 .

[11]  S. E. Kuznetsov.  Construction of Markov Processes with Random Times of Birth and Death , 1974 .

[12]  Joanna Mitro.  Dual Markov processes: Construction of a useful auxiliary process , 1979 .

[13]  丸山 儀四郎,et al.  On the Transition Probability Functions of the Markov Process , 1954 .

[14]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[15]  M. Nagasawa.  Time Reversions of Markov Processes , 1964, Nagoya Mathematical Journal.

[16]  A. Kolmogoroff,et al.  Zur Umkehrbarkeit der statistischen Naturgesetze , 1937 .

[17]  Edward Nelson.  Derivation of the Schrodinger equation from Newtonian mechanics , 1966 .

[18]  E. Schrödinger.  Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique , 1932 .

[19]  R. Getoor,et al.  Additive functionals and entrance laws , 1985 .

[20]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[21]  Hiroshi Tanaka,et al.  Diffusion with interactions and collisions between coloured particles and the propagation of chaos , 1987 .

[22]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[23]  B. Jamison.  The Markov processes of Schrödinger , 1975 .

[24]  Hiroshi Tanaka,et al.  A diffusion process in a singular mean-drift-field , 1985 .

[25]  Macroscopic, intermediate, microscopic and mesons , 1986 .

[26]  B. Simon,et al.  Brownian motion and harnack inequality for Schrödinger operators , 1982 .

[27]  H. Kunita,et al.  NOTES ON TRANSFORMATIONS OF MARKOV PROCESSES CONNECTED WITH MULTIPLICATIVE FUNCTIONALS , 1963 .

[28]  P. Blanchard,et al.  Diffusion processes with singular drift fields , 1987 .

[29]  Diffusion process corresponding to $$\frac{1}{2}\sum {\frac{{\partial ^2 }}{{\partial x^{i2} }}} + \sum {b^i (x)\frac{\partial }{{\partial x^i }}} $$ , 1960 .

[30]  I. V. Girsanov.  On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures , 1960 .

[31]  J. Zambrini.  Variational processes and stochastic versions of mechanics , 1986 .

[32]  A. Kolmogoroff.  Zur Theorie der Markoffschen Ketten , 1936 .

[33]  J. M. Clark,et al.  The Representation of Functionals of Brownian Motion by Stochastic Integrals , 1970 .

[34]  Hiroshi Tanaka,et al.  Propagation of chaos for diffusing particles of two types with singular mean field interaction , 1986 .

[35]  R. Ho,et al.  A remark on the connection between stochastic mechanics and the heat equation , 1974 .

[36]  J. Doob.  Classical potential theory and its probabilistic counterpart , 1984 .

[37]  J. Glover,et al.  Constructing Markov processes with random times of birth and death , 1986 .

[38]  Weian Zheng,et al.  Tightness results for laws of diffusion processes application to stochastic mechanics , 1985 .

[39]  Zambrini.  Stochastic mechanics according to E. Schrödinger. , 1986, Physical review. A, General physics.

[40]  R. Carmona.  Processus de diffusion gouverne par la forme de dirichlet de l'operateur de Schrödinger , 1979 .