{"id":357473,"date":"2017-01-25T10:30:14","date_gmt":"2017-01-25T18:30:14","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357473"},"modified":"2018-10-16T19:59:35","modified_gmt":"2018-10-17T02:59:35","slug":"brownian-motion-variable-drift-0-1-laws-hitting-probabilities-hausdorff-dimension","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/brownian-motion-variable-drift-0-1-laws-hitting-probabilities-hausdorff-dimension\/","title":{"rendered":"Brownian Motion With Variable Drift: 0-1 Laws, Hitting Probabilities And Hausdorff Dimension"},"content":{"rendered":"

By the Cameron\u2013Martin theorem, if a function f<\/span> is in the Dirichlet space D<\/span>, then B<\/span> + f<\/span> has the same a.s. properties as standard Brownian motion, B<\/span>. In this paper we examine properties of B<\/span> + f<\/span> when f<\/span>D<\/span>. We start by establishing a general 0-1 law, which in particular implies that for any fixed f<\/span>, the Hausdorff dimension of the image and the graph of B<\/span> + f<\/span> are constants a.s. (This 0-1 law applies to any L\u00e9vy process.) Then we show that if the function f<\/span> is H\u00f6lder(1\/2), then B<\/span> + f<\/span> is intersection equivalent<\/span> to B<\/span>. Moreover, B<\/span> + f<\/span> has double points a.s. in dimensions d<\/span> \u2264 3, while in d<\/span> \u2265 4 it does not. We also give examples of functions which are H\u00f6lder with exponent less than 1\/2, that yield double points in dimensions greater than 4. Finally, we show that for d<\/span> \u2265 2, the Hausdorff dimension of the image of B<\/span> + f<\/span> is a.s. at least the maximum of 2 and the dimension of the image of f<\/span>.<\/p>\n","protected":false},"excerpt":{"rendered":"

By the Cameron\u2013Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. 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