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15
Properties of Localnondeterminism of Gaussian and Stable Random Fields and Their Applications
, 2005
"... In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show som ..."
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Cited by 20 (10 self)
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In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of (N, d)Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the Brownian sheet, fractional Brownian sheets and so on.
Hitting probabilities for systems of nonlinear stochastic heat equations with additive noise
, 2007
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Strong Local Nondeterminism and Sample Path Properties of Gaussian Random Fields
"... We provide sufficient conditions for a realvalued Gaussian random field X = {X(t), t ∈ RN} to be strongly locally nondeterministic. As applications, we establish small ball probability estimates, Hausdorff measure of the sample paths, sharp Hölder conditions and tail probability estimates for the ..."
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Cited by 16 (11 self)
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We provide sufficient conditions for a realvalued Gaussian random field X = {X(t), t ∈ RN} to be strongly locally nondeterministic. As applications, we establish small ball probability estimates, Hausdorff measure of the sample paths, sharp Hölder conditions and tail probability estimates for the local times of a large class of Gaussian random fields.
A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand
 COMMUN. PROBAB
, 2008
"... We give a new representation of fractional Brownian motion with Hurst parameter H ≤ 1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give si ..."
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Cited by 3 (1 self)
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We give a new representation of fractional Brownian motion with Hurst parameter H ≤ 1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Lévy.
Hitting properties of s.p.d.e.’s with reflection
, 2004
"... We study the hitting properties of the solutions u of a class of stochastic p.d.e.’s with singular drifts that prevent u from becoming negative. The drifts can be a reflecting term or a nonlinearity cu −3, with c> 0. We prove that almost surely, for all time t> 0, the solution ut hits the lev ..."
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Cited by 1 (1 self)
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We study the hitting properties of the solutions u of a class of stochastic p.d.e.’s with singular drifts that prevent u from becoming negative. The drifts can be a reflecting term or a nonlinearity cu −3, with c> 0. We prove that almost surely, for all time t> 0, the solution ut hits the level 0 only at a finite number of space points, which depends explicitly on c. In particular, this number of hits never exceeds 4, and if c> 15/8, then level 0 is not hit.
Criteria for hitting probabilities with applications to systems of stochastic wave equations
, 2009
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Elect. Comm. in Probab. 14 (2009), 55–65 ELECTRONIC COMMUNICATIONS in PROBABILITY A CONNECTION BETWEEN THE STOCHASTIC HEAT EQUATION AND FRACTIONAL BROWNIAN MOTION, AND A SIMPLE PROOF OF A RESULT OF TALAGRAND
, 2008
"... We give a new representation of fractional Brownian motion with Hurst parameter H ≤ 1 2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simp ..."
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We give a new representation of fractional Brownian motion with Hurst parameter H ≤ 1 2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Lévy. 1