One more square root law for Brownian motion and its application to SPDEs

N. V. Krylov

doi:10.1007/s00440-003-0301-3

One more square root law for Brownian motion and its application to SPDEs

N. V. Krylov

School of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

It is proved that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for any t ∈ [0, 1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0, 2^-k] × [0, c2^-k/2] for all k belonging to a set with lower density ≥ p(c). This law of square root helps show that solutions of one-dimensional SPDEs are Hölder continuous up to the boundary.

Original language	English (US)
Pages (from-to)	496-512
Number of pages	17
Journal	Probability Theory and Related Fields
Volume	127
Issue number	4
DOIs	https://doi.org/10.1007/s00440-003-0301-3
State	Published - Dec 2003

Keywords

Brownian motion
Square root law
Stochastic partial differential equations

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10.1007/s00440-003-0301-3

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abstract = "It is proved that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for any t ∈ [0, 1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0, 2-k] × [0, c2-k/2] for all k belonging to a set with lower density ≥ p(c). This law of square root helps show that solutions of one-dimensional SPDEs are H{\"o}lder continuous up to the boundary.",

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AU - Krylov, N. V.

PY - 2003/12

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N2 - It is proved that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for any t ∈ [0, 1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0, 2-k] × [0, c2-k/2] for all k belonging to a set with lower density ≥ p(c). This law of square root helps show that solutions of one-dimensional SPDEs are Hölder continuous up to the boundary.

AB - It is proved that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for any t ∈ [0, 1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0, 2-k] × [0, c2-k/2] for all k belonging to a set with lower density ≥ p(c). This law of square root helps show that solutions of one-dimensional SPDEs are Hölder continuous up to the boundary.

KW - Brownian motion

KW - Square root law

KW - Stochastic partial differential equations

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AN - SCOPUS:0347532764

SN - 0178-8051

VL - 127

SP - 496

EP - 512

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 4

ER -

One more square root law for Brownian motion and its application to SPDEs

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