Invariant Measures and Evolution Equations for Markov Processes Characterized Via Martingale Problems

Bhatt, Abhay G.; Karandikar, Rajeeva L.

doi:10.1214/aop/1176989019

Cited by 55 publications

(28 citation statements)

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“…This is an improvement on several results on this theme (see [1,2,3,4,6,9,10]). In the last section, we give examples of operators (and their domains) satisfying the conditions of §3, and such that the corresponding martingale problems are well-posed in the class of solutions that are continuous in probability but for which no r.c.l.l.…”

Section: Introductionsupporting

confidence: 56%

On characterisation of Markov processes via martingale problems

Bhatt

Karandikar

Rao

2006

Proc. Indian Acad. Sci. (Math. Sci.)

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Abstract. It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probability measure to be invariant for the semigroup associated with the Markov process. We also give examples of martingale problems that are well-posed in the class of solutions which are continuous in probability but for which no r.c.l.l. solution exists.

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Section: Introductionsupporting

confidence: 56%

On characterisation of Markov processes via martingale problems

Bhatt

Karandikar

Rao

2006

Proc. Indian Acad. Sci. (Math. Sci.)

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“…Then, due to Bhatt-Karandikar [2, Theorem 5.2] (see also Remark 3.1 and Theorem 5.1 in [2] and Theorem B.1 in [12]), uniqueness for (4.33) holds.…”

Section: Thus (Vmentioning

confidence: 99%

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Fournier

Guérin

2008

J Stat Phys

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Abstract. We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka [15]: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other.

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“…For a general discussion of martingale problems see [StV06], [EK86], [MR92], [AR89a], [AR90b], [ARZ93b], [E99], [BhK93].…”

Section: Symmetric Quasi Regular Dirichlet Formsmentioning

confidence: 99%

Quasi regular Dirichlet forms and the stochastic quantization problem

Albeverio

Ming²,

Röckner³

2014

Festschrift Masatoshi Fukushima

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After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov uniqueness in infinite dimensions.

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Invariant Measures and Evolution Equations for Markov Processes Characterized Via Martingale Problems

Cited by 55 publications

References 0 publications

On characterisation of Markov processes via martingale problems

On characterisation of Markov processes via martingale problems

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Quasi regular Dirichlet forms and the stochastic quantization problem

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