1,720,971 research outputs found
A Note on Supersymmetry and Stochastic Differential Equations
No full textWe obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas
Some Recent Developments on Lie Symmetry Analysis of Stochastic Differential Equations
No full textWe present a brief review of recent progresses on Lie symmetry analysis of stochastic differential equations (SDEs). In particular, we consider some general definitions of symmetries for Brownian motion driven SDEs, as well as of weak and gauge symmetries of SDEs driven by discrete-time semimartingales. Some applications of Lie symmetry analysis to reduction and reconstruction of SDEs, Kolmogorov equation and numerical schemes for SDEs are discussed. Studies on random symmetries of SDEs, as well as extension of Noether theorem on invariants to stochastic systems and the finding of finite-dimensional solutions to SPDEs are also briefly reviewed
The elliptic stochastic quantization of some two dimensional Euclidean QFTs
No full textWe study a class of elliptic SPDEs with additive Gaussian noise on R2 × M, with M a d-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential V, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper (Ann. Probab. 48 (2020) 1693–1741), the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on an abstract Wiener space over M. The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over T2, and with exponential interaction over R2 (known also as Høegh-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over R2+2 is derived as well as the dimensional reduction for the values α of the “charge parameter” (formula persented) for which the model has an Euclidean invariant, reflection positive probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space)
Random transformations and invariance of semimartingales on Lie groups
No full textInvariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non-trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed
Weak symmetries of stochastic differential equations driven by semimartingales with jumps
No full textStochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as (weak) symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs
Entropy Chaos and Bose-Einstein Condensation
No full textWe prove the entropy-chaos property for the system of N indistinguishable interacting diffusions rigorously associated with the ground state of N trapped Bose particles in the Gross–Pitaevskii scaling limit of infinitely many particles. On the path-space we show that the sequence of probability measures of the one-particle interacting diffusion weakly converges to a limit probability measure, uniquely associated with the minimizer of the Gross-Pitaevskii functional
Strong Kac's chaos in the mean-field Bose-Einstein Condensation
No full textA stochastic approach to the (generic) mean-field limit in Bose-Einstein Condensation is described and the convergence of the ground-state energy as well as of its components are established. For the one-particle process on the path space, a total variation convergence result is proved. A strong form of Kac's chaos on path-space for the k-particles probability measures is derived from the previous energy convergence by purely probabilistic techniques notably using a simple chain-rule of the relative entropy. Fisher's information chaos of the fixed-time marginal probability density under the generic mean-field scaling limit and the related entropy chaos result are also deduced
A symmetry-adapted numerical scheme for SDEs
No full textWe propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE puts in evidence notable invariant properties. An approximation scheme preserving the symmetry properties of the equation is introduced. Our algorithmic procedure is applied to the family of general linear SDEs for which two theoretical estimates of the numerical forward error are established
Elliptic stochastic quantization
No full textWe prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in R2. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (Phys. Rev. Lett. 43 (1979) 744-745), which links the law of an elliptic SPDE in d +2 dimension with a Gibbs measure in d dimensions. This phenomenon is similar to the relation between a Rd+1 dimensional parabolic SPDE and its Rd dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (Phys. Rev. 150 (1966) 1079-1085) and Parisi and Wu (Sci. Sin. 24 (1981) 483-496). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (Comm. Math. Phys. 94 (1984) 459-482). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our d = 0 context the arguments are nontrivial and a nonsupersymmetric, elementary proof seems only to be available in the Gaussian case
Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions
Full text linkWe introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C∗-algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model
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