# The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation

## Abstract

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no global Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain L{sup 1} convergence in the white-noise limit.

- Authors:

- Tulane University, Department of Mathematics (United States)

- Publication Date:

- OSTI Identifier:
- 22787929

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Statistical Physics

- Additional Journal Information:
- Journal Volume: 173; Journal Issue: 2; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-4715

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CONVERGENCE; KERNELS; LANGEVIN EQUATION; MARKOV PROCESS; MASS; MATHEMATICAL SPACE; NOISE; PROBABILITY

### Citation Formats

```
Nguyen, Hung D., E-mail: hnguye25@tulane.edu.
```*The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation*. United States: N. p., 2018.
Web. doi:10.1007/S10955-018-2139-1.

```
Nguyen, Hung D., E-mail: hnguye25@tulane.edu.
```*The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation*. United States. https://doi.org/10.1007/S10955-018-2139-1

```
Nguyen, Hung D., E-mail: hnguye25@tulane.edu. Mon .
"The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation". United States. https://doi.org/10.1007/S10955-018-2139-1.
```

```
@article{osti_22787929,
```

title = {The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation},

author = {Nguyen, Hung D., E-mail: hnguye25@tulane.edu},

abstractNote = {We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no global Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain L{sup 1} convergence in the white-noise limit.},

doi = {10.1007/S10955-018-2139-1},

url = {https://www.osti.gov/biblio/22787929},
journal = {Journal of Statistical Physics},

issn = {0022-4715},

number = 2,

volume = 173,

place = {United States},

year = {2018},

month = {10}

}

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