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A solution to Schrödinger’s problem of nonlinear integral equations. (English) Zbl 0835.45002
Summary: A solution of Schrödinger’s system of nonlinear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.
Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable nonnegative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.
For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.

45G15 Systems of nonlinear integral equations
Full Text: DOI
[1] Aebi, R.,Diffusions with singular drift related to wave functions, Probab. Theory Relat. Fields96, 107-121 (1993). · Zbl 0791.60066 · doi:10.1007/BF01195885
[2] Aebi, R. and Nagasawa, M.,Large deviations and the propagation of chaos for Schrödinger processes, Probab. Theory Relat. Fields94, 53-68 (1992). · Zbl 0767.60056 · doi:10.1007/BF01222509
[3] Aebi, R.,Itô’ s formula for non-smooth functions, Publ. RIMS, Kyoto Univ.28, 595-602 (1992). · Zbl 0795.60075 · doi:10.2977/prims/1195168209
[4] Bernstein, S.,Sur les liaisons entre les grandeurs aléatoires, Verhandlungen des Internat. Math. Kongresses Zürich1, 288-309 (1932).
[5] Beurling, A.,An automorphism of product measures, Ann. Math.72, no. 1, 189-200 (1960). · Zbl 0091.13001 · doi:10.2307/1970151
[6] Csiszar, I.,I-Divergence geometry of probability distributions and minimization problems, Ann. Probab.3, 146-158 (1975). · Zbl 0318.60013 · doi:10.1214/aop/1176996454
[7] Dawson, D., Gorostiza, L. and Wakolbinger, A.,Schrödinger processes and large deviations, J. Math. Phys.31(10), 2385-2388 (1990). · Zbl 0736.60071 · doi:10.1063/1.528840
[8] Föllmer, H.,Random Fields and Diffusion Processes, École d’É’e de Saint Flour XV?XVII (1985-87), Lect. Notes Math. 1362 Springer-Verlag, Berlin 1988.
[9] Fortet, R.,Résolution dún systéme d’équation de M. Schrödinger, J. Math. Pures et Appl.IX, 83-95 (1940). · JFM 66.0498.01
[10] Nagasawa, M. and Tanaka, H.,Propagation of chaos for diffusing particles of two types with singular mean field interaction, Probab. Theory Relat. Fields71, 69-83 (1986). · Zbl 0604.60079 · doi:10.1007/BF00366273
[11] Nagasawa, M. and Tanaka, H.,Diffusion with interactions and collisions between coloured particles and the propagation of chaos, Probab. Theory Relat. Fields74, 161-198 (1987). · Zbl 0587.60095 · doi:10.1007/BF00569988
[12] Nagasawa, M. and Tanaka, H.,A proof of the propagation of chaos for diffusion processes with drift coefficients not of average form, Tokyo J. Math.10, 403-418 (1987). · Zbl 0637.60031 · doi:10.3836/tjm/1270134523
[13] Nagasawa, M.,Transformations of diffusion and Schrödinger processes, Probab. Theory Relat. Fields82, 109-136 (1989). · Zbl 0666.60073 · doi:10.1007/BF00340014
[14] Nagasawa, M.,Schrödinger Equations and Diffusion Theory, Monographs in Mathematics, vol. 86, Birkhäuser Verlag, Basel 1993. · Zbl 0780.60003
[15] Schrödinger, E., Überdie Umkehrung der Naturgesetze, Sitzungsberichte der Preussischen Akademie der Wissenschaften, phys-math. Klasse, 144-153 (1931). · Zbl 0001.37503
[16] Schrödinger, E.,Sur la theorie relativiste de l’électron et l’interp’etation de la mécanique quantique, Ann. Inst. Henri Poincaré, vol. II, Paris 1932. · JFM 58.0933.03
[17] Tanaka, H.,Limit Theorems for Certain Diffusion Processes with Interaction, Stochastic Analysis (ed. Itô, K.), Kinokuniya, Tokyo 1984, pp. 469-488.
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