Lagrangian for pinned diffusion process

Hara, Keisuke; Takahashi, Yōichirō

doi:10.1007/978-4-431-68532-6_7

Cited by 6 publications

(9 citation statements)

References 8 publications

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“…x(t) is differentiable Any can work x(t) is a Langevin process, equation (2) Stratonovich, equation ( 5) x(t) is a path in the covariant action, equations (19) or (21) Covariant, equation ( 9) x(t) is a path in the standard action, equations (20) or (22) None works…”

Section: Situation Required Discretizationmentioning

confidence: 99%

“…Related works in mathematics have made such an approach rigorous. Either using changes of path probability [67,68] or more direct techniques (see the work of Takahashi and collaborators [21,22] and of Capitaine [23]), the idea is to determine the most probable path 7 going from one point to an other as extremizing an Onsager-Machlup covariant action. Such constructions are possible but do not provide the path probability of an arbitrary non-differentiable trajectories (which is the aim of our theoretical physicist's construction).…”

Section: Covariant Approachesmentioning

confidence: 99%

See 1 more Smart Citation

Building a path-integral calculus: a covariant discretization approach

Cugliandolo¹,

Lecomte²,

Wijland³

2019

J. Phys. A: Math. Theor.

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Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem, through a direct time-discretization procedure.

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Section: Situation Required Discretizationmentioning

confidence: 99%

Section: Covariant Approachesmentioning

confidence: 99%

Building a path-integral calculus: a covariant discretization approach

Cugliandolo¹,

Lecomte²,

Wijland³

2019

J. Phys. A: Math. Theor.

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“…where we have used in the second inequality the fact that the O(ǫ) is uniform in t according to the uniform equivalence of the family of metrics {g(t)} t∈[0,T ] . In order to control the first term in (3.4) we will use the following Theorem [4]. 1 ([4]).…”

Section: Proof Of the Theoremmentioning

confidence: 99%

“…We will also use the non singular drift introduced by Hara. Using this drift, Hara and Takahashi in [4] have made a substantial simplification of the previous proof of Onsager-Machlup functional. We propose a time dependent parallel transport along a curve according to a family of metrics and use it to make the computation of the Onsager-Machlup functional in the inhomogeneous case.…”

Section: Introductionmentioning

confidence: 99%

Onsager-Machlup Functional for Uniformly Elliptic Time-Inhomogeneous Diffusion

Coulibaly-Pasquier

2014

Lecture Notes in Mathematics

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In this paper we will make the computation of the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional will have formally the same picture as in the homogeneous case, the only difference come from the infinitesimal variation of the volume. For example in the Ricci flow case, we find some functional which is not so far to the L0 distance used by Lott to study this flow [6]. We finish by a application to small ball probability for weighted sup norm, for inhomogeneous diffusion.

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“…where, for each k Di erentiating twice it is easy to check t h a t g satis es ( 2 ; 1)g(s) = g 00 (s) 0 s 1 (7) with initial conditions g( 0 ) = 0 a n d g(1) = ;g 0 (1). Notice that equation (7) clearly implies that 6 = 0 . Set = 2 ;1 .…”

Section: The Karhunen-lo Eve Expansionmentioning

confidence: 99%