Topic
Stochastic process
About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.
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TL;DR: In this paper, it was shown that an Ito's stochastic equation with discontinuous coefficients can be constructed on any probability space by using Euler's polygonal approximations.
Abstract: Given strong uniqueness for an Ito's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ℝd and in domains in ℝd are considered.
466 citations
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TL;DR: In this article, an uncertainty quantification scheme based on generalized polynomial chaos (PC) representations is constructed, which is applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Benard instability.
463 citations
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TL;DR: This work identifies different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics in large populations, because differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process.
Abstract: Traditionally, frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes. Only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations. However, the relationship between deterministic and stochastic approaches remained unclear. Here we solve this problem by explicitly considering large populations. In particular, we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics. Moreover, differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process.
462 citations
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TL;DR: In this paper, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories.
Abstract: Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.
462 citations