Tightness of Probabilities On $C(\lbrack 0, 1 \rbrack; \mathscr{Y}')$ and $D(\lbrack 0, 1 \rbrack; \mathscr{Y}')$

Mitoma, Itaru

doi:10.1214/aop/1176993447

Cited by 192 publications

(21 citation statements)

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“…Recall that {φ w j } is an orthonormal basis for S w . We can choose this basis such that for each j ∈ N, φ w j ∈ S. It follows from (4.31) that ξ(t), φ Therefore, 33) where the last inequality follows from (4.32). Thus ξ(t) = 0∀ t ∈ [0, T ] and uniqueness follows.…”

Section: Laplace Lower Boundmentioning

confidence: 99%

Moderate deviation principles for weakly interacting particle systems

Budhiraja

2016

Probab. Theory Relat. Fields

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Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of l2 (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.AMS 2000 subject classifications: 60F10, 60K35, 60J75, 60J60.

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Section: Laplace Lower Boundmentioning

confidence: 99%

Moderate deviation principles for weakly interacting particle systems

Budhiraja

2016

Probab. Theory Relat. Fields

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“…It follows from the results of Fouque in [17] and Mitoma in [21] that in order to verify the tightness of the family of distributions W h (z) it is sufficient to establish tightness of the processes…”

Section: Tightnessmentioning

confidence: 99%

Self-Averaging from Lateral Diversity in the Itô–Schrödinger Equation

Papanicolaou¹,

Ryzhik²,

Sølna³

2007

Multiscale Model. Simul.

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We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Itô-Schrödinger stochastic partial differential equation (SPDE) which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.

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“…Based on Theorem 3.1 of [17] we conclude the tightness of { Z ǫ t , f } in C ([0, 1]; S ′ (R)). Now we prove the convergence of the centered process of the general SPDE to a Gaussian process to obtain the CLT for the general SPDE.…”

Section: Clt For the General Spdementioning

confidence: 58%