We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator L has a generalized drift. We investigate existence and uniqueness of generalized solutions of class C 1 . The generator L is associated with a Markov process X which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is X. Since X is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.KEY WORDS AND PHRASES: Backward stochastic differential equations; random terminal time; martingale problem; distributional drift; elliptic partial differential equations.MSC 2010: 60H10; 60H30; 35H99.
Bell's theorem prevents local Kolmogorov-simulations of the singlet state of two spin-1/2 particles. We derive a positive lower bound for the L 2 -distance between the quantum mechanical spin singlet anticorrelation function cos and any of its classical approximants C formed by the stationary autocorrelation functions of mean-squarecontinuous, 2π-periodic, ±1-valued, stochastic processes. This bound is given by C − cos ≥ 1 − 8 π 2 / √ 2 ≈ 0.133 95.
This paper is devoted to three topics. First, proving a measurability theorem for multifunctions with values in non-metrizable spaces, which is required to show that solutions to stochastic wave equations with interval parameters are random sets; second, to apply the theorem to wave equations in arbitrary space dimensions; and third, to computing upper and lower probabilities of the values of the solution in the case of one space dimension.
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