The Experts below are selected from a list of 7551 Experts worldwide ranked by ideXlab platform
Lauri Viitasaari - One of the best experts on this subject based on the ideXlab platform.
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representation of stationary and stationary increment processes via langevin equation and self similar processes
Statistics & Probability Letters, 2016Co-Authors: Lauri ViitasaariAbstract:Let Wt be a Standard Brownian Motion. It is well-known that the Langevin equation dUt=−θUtdt+dWt defines a stationary process called Ornstein–Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian Motion Wt with some other process G with stationary increments. In this article we prove that the converse also holds and all continuous stationary processes arise from a Langevin equation with certain noise G=Gθ. Discrete analogies of our results are given and applications are discussed.
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representation of stationary and stationary increment processes via langevin equation and self similar processes
arXiv: Probability, 2014Co-Authors: Lauri ViitasaariAbstract:Let $W_t$ be a Standard Brownian Motion. It is well-known that the Langevin equation $d U_t = -\theta U_td t + d W_t$ defines a stationary process called Ornstein-Uhlenbeck process. Furthermore, Langevin equation can be used to construct other stationary processes by replacing Brownian Motion $W_t$ with some other process $G$ with stationary increments. In this article we prove that the converse also holds and all continuous stationary processes arise from a Langevin equation with certain noise $G=G_\theta$. Discrete analogies of our results are given and applications are discussed.
Abraham Nitzan - One of the best experts on this subject based on the ideXlab platform.
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upside downside statistical mechanics of nonequilibrium Brownian Motion i distributions moments and correlation functions of a free particle
arXiv: Statistical Mechanics, 2019Co-Authors: Galen T Craven, Abraham NitzanAbstract:Statistical properties of Brownian Motion that arise by analyzing, separately, trajectories over which the system energy increases (upside) or decreases (downside) with respect to a threshold energy level, are derived. This selective analysis is applied to examine transport properties of a nonequilibrium Brownian process that is coupled to multiple thermal sources characterized by different temperatures. Distributions, moments, and correlation functions of a free particle that occur during upside and downside events are investigated for energy activation and energy relaxation processes, and also for positive and negative energy fluctuations from the average energy. The presented results are sufficiently general and can be applied without modification to Standard Brownian Motion. This article focuses on the mathematical basis of this selective analysis. In subsequent articles in this series we apply this general formalism to processes in which heat transfer between thermal reservoirs is mediated by activated rate processes that take place in a system bridging them.
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upside downside statistical mechanics of nonequilibrium Brownian Motion i distributions moments and correlation functions of a free particle
Journal of Chemical Physics, 2018Co-Authors: Galen T Craven, Abraham NitzanAbstract:Statistical properties of Brownian Motion that arise by analyzing, separately, trajectories over which the system energy increases (upside) or decreases (downside) with respect to a threshold energy level are derived. This selective analysis is applied to examine transport properties of a nonequilibrium Brownian process that is coupled to multiple thermal sources characterized by different temperatures. Distributions, moments, and correlation functions of a free particle that occur during upside and downside events are investigated for energy activation and energy relaxation processes and also for positive and negative energy fluctuations from the average energy. The presented results are sufficiently general and can be applied without modification to the Standard Brownian Motion. This article focuses on the mathematical basis of this selective analysis. In subsequent articles in this series, we apply this general formalism to processes in which heat transfer between thermal reservoirs is mediated by activated rate processes that take place in a system bridging them.
Oscar Peralta - One of the best experts on this subject based on the ideXlab platform.
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An explicit solution to the Skorokhod embedding problem for double exponential increments
arXiv: Probability, 2020Co-Authors: Giang T. Nguyen, Oscar PeraltaAbstract:Strong approximations of uniform transport processes to the Standard Brownian Motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego et al. (1971)) and improves their rate of strong convergence (Gorostiza et al. (1980)). We finalise by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
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An explicit solution to the Skorokhod embedding problem for double exponential increments
Statistics & Probability Letters, 2020Co-Authors: Giang T. Nguyen, Oscar PeraltaAbstract:Strong approximations of uniform transport processes to the Standard Brownian Motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego, 1971) and improves their rate of strong convergence (Gorostiza and Griego, 1980). We finalize by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
Maziar Raissi - One of the best experts on this subject based on the ideXlab platform.
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forward backward stochastic neural networks deep learning of high dimensional partial differential equations
arXiv: Machine Learning, 2018Co-Authors: Maziar RaissiAbstract:Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a Standard Brownian Motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.
Giang T. Nguyen - One of the best experts on this subject based on the ideXlab platform.
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An explicit solution to the Skorokhod embedding problem for double exponential increments
arXiv: Probability, 2020Co-Authors: Giang T. Nguyen, Oscar PeraltaAbstract:Strong approximations of uniform transport processes to the Standard Brownian Motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego et al. (1971)) and improves their rate of strong convergence (Gorostiza et al. (1980)). We finalise by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
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An explicit solution to the Skorokhod embedding problem for double exponential increments
Statistics & Probability Letters, 2020Co-Authors: Giang T. Nguyen, Oscar PeraltaAbstract:Strong approximations of uniform transport processes to the Standard Brownian Motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego, 1971) and improves their rate of strong convergence (Gorostiza and Griego, 1980). We finalize by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
