Diffusions, Markov Processes, and Martingales

Rogers, L. C. G.; Williams, David

doi:10.1017/cbo9781107590120

Cited by 517 publications

(235 citation statements)

References 204 publications

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“…In physics, they are also used as a general framework for modeling systems driven in nonequilibrium steady states by noise and external forces [52][53][54], such as interacting particle systems coupled to different particle and energy reservoirs, which have been studied actively in the mathematics and physics literature recently [75][76][77][78][79]. For general introductions to Markov processes and their applications in physics, see [52][53][54][80][81][82]; for references on the mathematics of these processes, see [3,38,39,83,84].…”

Section: Notations and Definitionsmentioning

confidence: 99%

“…In continuous time and continuous space, all Markov processes consist of a superposition of these two processes, combined possibly with deterministic motion [3,85,86]. The case of discrete-time Markov chains is discussed in Appendix E. A homogeneous Markov process X t is a pure jump process if the probability that X t undergoes one jump during the time interval [t, t + dt] is proportional to dt.…”

Section: Pure Jump Processes and Diffusionsmentioning

confidence: 99%

“…In solving this problem, he introduced a transformation of the Wiener process, now referred to as Doob's h-transform, which was later adapted under the same name to deal with other conditionings of stochastic processes, including the Brownian bridge [3], Gaussian bridges [4][5][6], and the Schrödinger bridge [7][8][9][10][11], obtained by conditioning a process on reaching a certain target distribution in time as opposed to a target point. Doob's transform also appears prominently in the theory of quasi-stationary distributions [12][13][14][15][16], which describes in the simplest case the conditioning of a process never to reach an absorbing state.…”

Section: Introductionmentioning

confidence: 99%

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Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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International audienceWe consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large-deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures, and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned

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Section: Notations and Definitionsmentioning

confidence: 99%

Section: Pure Jump Processes and Diffusionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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“…In the time-inhomogeneous setting, this can be replaced, e.g. by [RW87] Thm. V 23.5, guaranteeing a solution to the corresponding martingale problem and hence also a weak solution to the SDE (cf.…”

Section: The Models and Basic Toolsmentioning

confidence: 99%

“…V 23.5, guaranteeing a solution to the corresponding martingale problem and hence also a weak solution to the SDE (cf. [RW87] Thm. V 20.1).…”

Section: The Models and Basic Toolsmentioning

confidence: 99%

Interacting Fisher–Wright diffusions in a catalytic medium

Greven¹,

Klenke²,

Wakolbinger

2001

Probab Theory Relat Fields

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We study the longtime behaviour of interacting systems in a randomly fluctuating (space-time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher-Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium).Here we introduce a model of interacting Fisher-Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice.While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium.

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Brownian Motion

Riedi¹

2012

Encyclopedia of Environmetrics

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This article on Brownian motion starts with a short historical survey on its discovery and importance and continues with its most relevant properties. Mathematically detailed formulas are given and connections between the properties are indicated, however no derivations or proofs are provided. The properties covered start with the mathematical modeling of Brownian motion via the Wiener process; equivalent definitions and constructions such as the Lévy characterization, the midpoint displacement, and the rescaled random walks are discussed. Basic properties given include uniqueness, symmetry, restarted Brownian motion, and time inversion. More advanced topics treated cover stopping times, the strong Markov property, the reflection principle, extreme values, escape times from a strip, and last visits. Path oscillations, stationarity, Gaussian white noise, and Karhunen‐Löwe expansions are also discussed. Brownian motion with drift, the Brownian bridge, as well as intersections, recurrence, and transience of Brownian motion in higher dimensions, are mentioned briefly.

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Diffusions, Markov Processes, and Martingales

Cited by 517 publications

References 204 publications

Nonequilibrium Markov Processes Conditioned on Large Deviations

Nonequilibrium Markov Processes Conditioned on Large Deviations

Interacting Fisher–Wright diffusions in a catalytic medium

Brownian Motion

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