Brownian Motion and Thermal Capacity Yimin Xiao Michigan State University (Based on joint paper with Davar Khoshnevisan) December 14, 2012 Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 1 / 22
Outline Intersection of the Brownian images and thermal ca- pacity Hausdorff dimension of W ( E ) ∩ F Further research and open problems Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 2 / 22
1. Intersection of the Brownian images and thermal capacity Let W := { W ( t ) } t ≥ 0 denote standard d -dimensional Brow- nian motion where d ≥ 1, and let E and F be compact subsets of ( 0 , ∞ ) and R d , respectively. The following problems are of interest: When is P ( W ( E ) ∩ F � = Ø) > 0 ? 1 What is dim H ( W ( E ) ∩ F )? 2 Note that { W ( E ) ∩ F � = Ø } = { ( t , W ( t )) ∈ E × F for some t > 0 } . Problem 1 is an interesting problem in probabilistic poten- tial theory. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 3 / 22
Conditions for P ( W ( E ) ∩ F � = Ø) > 0 Necessary and sufficient condition in terms of “thermal ca- pacity” for P ( W ( E ) ∩ F � = Ø) > 0 were proved by Waston (1978) and Doob (1984). Waston and Taylor (1985) provided a simple-to-use condi- tion: � if dim H ( E × F ; ̺ ) > d , > 0 , P ( W ( E ) ∩ F � = Ø) if dim H ( E × F ; ̺ ) < d . = 0 , In the above, dim H ( E × F ; ̺ ) is the Hausdorff dimension of E × F using the metric | t − s | 1 / 2 , � x − y � � � ̺ (( s , x ) ; ( t , y )) := max . Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 4 / 22
As a by-product of our main result, we obtain a slightly improved version of the result of Waston (1978) and Doob (1984). Theorem 1.1 Suppose F ⊂ R d ( d ≥ 1) is compact and has Lebesgue measure 0. Then P { W ( E ) ∩ F � = ∅} > 0 ⇐ ⇒ ∃ µ ∈ P d ( E × F ) such that E 0 ( µ ) < ∞ , where P d ( E × F ) is the collection of all probability mea- sures µ on E × F such that µ ( { t } × F ) = 0 for all t > 0, and the energy E 0 ( µ ) will be defined below. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 5 / 22
2. Hausdorff dimension of dim H ( W ( E ) ∩ F ) Two common ways to compute the Hausdorff dimension of a set: Use a covering argument for obtaining an upper bound and a capacity argument for lower bound; The co-dimension argument. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 6 / 22
The co-dimension argument The “co-dimension argument” was initiated by S.J. Taylor (1966) for computing the Hausdorff dimension of the mul- evy process in R d . His method was tiple points of a stable L´ based on potential theory of L´ evy processes. Let Z α = { Z α ( t ) , t ∈ R + } be a (symmetric) stable L´ evy process in R d of index α ∈ ( 0 , 2 ] and let F ⊂ R d be a Borel set. Then P ( Z α (( 0 , ∞ )) ∩ F � = ∅ ) > 0 ⇐ ⇒ Cap d − α ( F ) > 0 , where Cap d − α is the Riesz-Bessel capacity of order d − α . Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 7 / 22
The co-dimension argument The above result and Frostman’s theorem lead to the stochas- tic co-dimension argument : If dim H F ≥ d − 2, then dim H F = sup { d − α : Z α (( 0 , ∞ )) ∩ F � = ∅ } � � = d − inf α > 0 : F is not polar for Z α . [The restriction dim H F ≥ d − 2 is caused by the fact that Z α (( 0 , ∞ )) ∩ F = Ø if dim H F < d − 2.] This method determines dim H F by intersecting F using a family of testing random sets. Hawkes (1971) applied the co-dimension method for com- puting the Hausdorff dimension of the inverse image X − 1 ( F ) of a stable L´ evy process. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 8 / 22
The co-dimension argument Families of testing random sets: ranges of symmetric stable L´ evy processes; fractal percolation sets [Peres (1996, 1999)]; ranges of additive L´ evy processes [Khoshnevisan and X. (2003, 2005), Khoshnevisan, Shieh and X. (2008)]. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 9 / 22
Hausdorff dimension of dim H ( W ( E ) ∩ F ) If F = R d , then dim H W ( E ) = min { d , 2dim H E } a.s. In general, dim H ( W ( E ) ∩ F ) is a (non-degenerate) random variable, an example was shown to us by Greg Lawler. Hence we compute � dim H ( W ( E ) ∩ F ) � L ∞ ( P ) , the L ∞ ( P ) - norm of dim H ( W ( E ) ∩ F ) . We distinguish two cases: | F | > 0 and | F | = 0, where | · | denotes the Lebesgue measure. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 10 / 22
Theorem 2.1 [Khoshnevisan and X. (2012)] If F ⊂ R d ( d ≥ 1) is compact and | F | > 0, then � dim H ( W ( E ) ∩ F ) � L ∞ ( P ) = min { d , 2dim H E } . (1) If dim H E > 1 2 and d = 1, then P {| W ( E ) ∩ F | > 0 } > 0. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 11 / 22
Proof of Theorem 2.1 older continuity of W ( t ) on bounded Thanks to the uniform H¨ sets, we have dim H ( W ( E ) ∩ F ) ≤ min { d , 2dim H E } , a.s. This implies the upper bound in (1). For proving the lower bound in (1), we construct a random measure on W ( E ) ∩ F and use the capacity argument. The last part is proved by showing that the constructed ran- dom measure on W ( E ) ∩ F has a density function almost surely. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 12 / 22
Proof of Theorem 2.1 older continuity of W ( t ) on bounded Thanks to the uniform H¨ sets, we have dim H ( W ( E ) ∩ F ) ≤ min { d , 2dim H E } , a.s. This implies the upper bound in (1). For proving the lower bound in (1), we construct a random measure on W ( E ) ∩ F and use the capacity argument. The last part is proved by showing that the constructed ran- dom measure on W ( E ) ∩ F has a density function almost surely. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 12 / 22
Proof of Theorem 2.1 older continuity of W ( t ) on bounded Thanks to the uniform H¨ sets, we have dim H ( W ( E ) ∩ F ) ≤ min { d , 2dim H E } , a.s. This implies the upper bound in (1). For proving the lower bound in (1), we construct a random measure on W ( E ) ∩ F and use the capacity argument. The last part is proved by showing that the constructed ran- dom measure on W ( E ) ∩ F has a density function almost surely. Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 12 / 22
Theorem 2.2 [Khoshnevisan and X. (2012)] If F ⊂ R d ( d ≥ 1) is compact and | F | = 0, then � � � dim H ( W ( E ) ∩ F ) � L ∞ ( P ) (2) � � = sup γ ≥ 0 : µ ∈P d ( E × F ) E γ ( µ ) < ∞ inf , where P d ( E × F ) is the collection of all probability mea- sures µ on E × F such that µ ( { t } × F ) = 0 for all t > 0, and e −� x − y � 2 / ( 2 | t − s | ) �� E γ ( µ ) := | t − s | d / 2 · � y − x � γ µ ( ds dx ) µ ( dt dy ) . (3) Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 13 / 22
Theorem 2.2 [Khoshnevisan and X. (2012)] If F ⊂ R d ( d ≥ 1) is compact and | F | = 0, then � � � dim H ( W ( E ) ∩ F ) � L ∞ ( P ) (2) � � = sup γ ≥ 0 : µ ∈P d ( E × F ) E γ ( µ ) < ∞ inf , where P d ( E × F ) is the collection of all probability mea- sures µ on E × F such that µ ( { t } × F ) = 0 for all t > 0, and e −� x − y � 2 / ( 2 | t − s | ) �� E γ ( µ ) := | t − s | d / 2 · � y − x � γ µ ( ds dx ) µ ( dt dy ) . (3) Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 13 / 22
Hitting probability of random fields We prove Theorem 2.2 by checking whether or not W ( E ) ∩ F intersects the (closure of the) range of an additive L´ evy stable process. Let X ( 1 ) , . . . , X ( N ) be N isotropic stable processes with com- mon stability index α ∈ ( 0 , 2 ] . We assume that the X ( j ) ’s are independent from one another, as well as from the pro- cess W , and all take their values in R d . We assume also that X ( 1 ) , . . . , X ( N ) have right-continuous sample paths with left-limits and � e i � ξ, X ( k ) ( 1 ) � � = e −� ξ � α / 2 , ∀ ξ ∈ R d . E Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 14 / 22
Hitting probability of random fields We prove Theorem 2.2 by checking whether or not W ( E ) ∩ F intersects the (closure of the) range of an additive L´ evy stable process. Let X ( 1 ) , . . . , X ( N ) be N isotropic stable processes with com- mon stability index α ∈ ( 0 , 2 ] . We assume that the X ( j ) ’s are independent from one another, as well as from the pro- cess W , and all take their values in R d . We assume also that X ( 1 ) , . . . , X ( N ) have right-continuous sample paths with left-limits and � e i � ξ, X ( k ) ( 1 ) � � = e −� ξ � α / 2 , ∀ ξ ∈ R d . E Yimin Xiao (Michigan State University) Brownian Motion and Thermal Capacity December 14, 2012 14 / 22
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