**Alfred Chukwuemeka Okoroafor**

Department of Mathematics, Abia State University, Uturu, Nigeria

We consider the packing measure properties of subsets of R

The exact packing dimension is determined for subset of R^{n},
n≥3 for which u(x, t) is unbounded for (x, t) ∈R^{n+1}.

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Alfred Chukwuemeka Okoroafor, 2008. The Exact Packing Dimension of a Set of Zero Heat Capacity. *Asian Journal of Mathematics & Statistics, 1: 90-99.*

**DOI:** 10.3923/ajms.2008.90.99

**URL:** https://scialert.net/abstract/?doi=ajms.2008.90.99

Taylor and Watson (1985) used a Hausdorff measure classification to determine which subsets of R^{n+1} are of zero heat capacity with respect to the heat equation:

(1) |

where, R^{n} is the n-dimensional Euclidean space.

Watson (1978) deduced that for a set in R^{n} with zero Lebesgue measure, there is a fundamental solution of (1) on R^{n}x(0, ∞) which tends to infinity as any point of the given set is approached and that for such Borel subset E⊃R^{n} = (R^{n}x(t_{0})) of a characteristics hyperplane, the heat capacity relative to the strip R^{n}x(b, c) is precisely the n-dimensional Lebesgue measure of E on the hyperplane.

In this research we exploit the well known fact that the transition density of Brownian motion in R^{n} is just the heat kernel

(2) |

which satisfies the heat Eq. 1. We show, in this study that packing measure zero is equivalent to heat capacity zero on the hyperplane and we also determine the exact packing dimension of subsets x∈R^{n} for which u(x, t) is unbounded for all t.

**Preliminaries **We will use Euclidean norm sign to denote both the distance ||x-y|| between two point in R

We use c_{1} c_{2}... to denote finite positive constants whose precise values are unimportant.

Now let us recall the general setting. Consider a situation where heat is distributed from a heat source over n+1 dimensional Euclidean space R^{n+1}. This heat source gives rise to a function V : R^{n+1}→R which assigns a heat potential V(x, t) to each point of (x, t)∈R^{n+1} through a generating kernel u(x, t) so that if we consider a postive Borel measure π on R^{n+1} as the distribution of heat source, then the integral:

(3) |

is called a heat potential if it is finite on a dense subset of R^{n+1}

We now define the heat capacity of E⊃R^{n+1}

**Definition 1 **For a compact set k, the heat capacity C(k) is defined by:

C(k) = sup{π(k) : V(x, t)≤1 on R^{n+1},π is supported by k, π≥0}.

Then the capacity of an arbitrary Borel set E is:

C(E) = sup{C(k):k is compact, K⊆E}

The heat capacity of E is zero if and only if V(x, t) is unbounded for every π such that π(E)>0 whenever (x, t)∈E.

Equivalently a set E is called polar if and only if there is a potential V on R^{n+1} such that V(x, t) = ∞,whenever (x, t)∈E.

The set E of heat capacity zero are precisely the polar sets.

Specifically, in the present study, we are mainly interested in the packing dimension of the subset S⊂R^{n} for which E⊂R^{n+1} has zero heat capacity.

So, in what follows, will concern ourselves with the case where the distribution of heat sources generating the potential V(x, t) is invariant under translation in the direction of the t axis so that π can be writen as the direct product of a completely additive function m of Borel sets in n-dimensions and Lebsegue measure on the t axis.

Thus we write:

(4) |

Where:

is the well known potential kernel of Brownian motion with respect to Lebesgue measure and:

(5) |

where, c_{1} is a constant.

**The Packing Dimension **One of the several distinct techniques in investigating the size of subsets of zero Lebesque in R

Packing dimension is defined via the packing measure as follows:

Start with a class Φ of monotone functions h : (0, δ)→(0, 1) which is non-decreasing, right continuous and satisfies h(0+) = 0 and for which there is a constant:

(6) |

We obtain the packing measure by a two stage definition. First, we define a pre-measure:

(7) |

where, B_{ri}(x_{i}) denotes the open ball centred at x_{i}, radius r_{i} Eq. 7 is not an outer measure because it is not countably subadditive:

However it leads to an outer measure by defining:

(8) |

which can be thought of as a generalization of the Lebesgue measure using maximal packing of E by balls, so that if h(s) = s^{n} then h-p() on R^{n} is n-dimensional Lebesgue measure.

Thus, to measure the Borel subset of E⊆R^{n} we need :

so that if h(s) = s^{α}, α > 0, there is a unique value α for which the packing measure h-p (E) drops from infinity to zero

that is, if E ⊆ R^{n} is bounded, there is a number say β∈[0, n] such that :

(9) |

This in turn means that E is less occupied than if it were α-ε dimensional for ε>0.

We define the index Dim E as

(10) |

the packing dimension of E.

This index gives the notion of size to sets of Lebesgue measure zero and takes value n on each subset of R^{n} of positive n-dimensional Lebesgue measure and for all subsets of R^{n},we have:

Dim E≤n

We refer to Taylor (1985) for a convenient reference on the use of packing measure for the analysis of random sets arising from the sample paths of Stocastic processes.

Another useful tool to study the size of a Polar set is the notion of multifractal analysis of occupation measure of a Brownian motion. Let B_{r}(x) denote the ball in R^{n} of radius r centred at x and {X(t)}_{t≥0} denote Brownian motion in R^{n}, n≥3, then the occupation measure of X(t) on B_{r} (x) is (B_{r}(x)),

(11) |

is the portion of the time interval [0, 1] which the process spends in the Borel set E.

Taylor and Triort (1985) relates the lower density of at x, denoted by :

**Lemma 1 **Let be a Borel measure on R

where c_{1} is a constant and

is the lower h-density of at x

This is a natural random measure associated with the range of a Brownian motion.

Let φ : R^{n}→R be a measureable function. Let D be a real-valued function defined on R^{n}

Then the spectrum with respect to φ and D is defined by D{x∈R^{n} : φ (x) = α}

Multifractal analysis consists of studying the size of the level set

In what follows, we consider spectra involving logarithmic order of magnitude. For a random measure associated to a Brownian motion, let φ (x) be the lower density of at x i.e.:

Then we compute the logarithmic multifractal spectrum of , which is defined as follows

**Definition 2 **For occupation measure associated to a Brownian motion and for every α≥0,

E_{α}= {x : φ (x) = α}

The mapping is the logarithimic spectrum of where Dim E denotes the packing dimension of E.

This provides information about the underlying geometric structure of the supports of , which are sets of Lebesgue measure zero.

Our first result is analogous to the one due to Watson (1978) for Lebesgue measure and states:

**Theorem 1**

Let E be a Borel set in R^{n}, n≥3 and suppose that s^{n} - p(E) = 0,

then C(E) = 0

**Proof: **Let (x

The potential V(x_{0}, t_{0}) can be written as the sum of integrals over B_{σ}(π_{0})xR and its complement.

Thus

Since this holds for all x_{0}, σ, t_{0} it follows that V(x, t) = ∞ if

A stansard application of lemma 1 then shows that s^{n} – p(E) = 0 and hence C(E) = 0

**Corollary 1 **If E⊂R

temperature V on R^{n}x(0, ∞) such that

Thus E = (Ex{0}) has zero capacity, in R^{n+1}

**Corollary 2 **Let E⊂R

**Brownian Motion and the Distribution of Heat Source **Let R

It is well known that for n≥2 the Brownian motion process hits a point x∈R^{n} with probability zero which implies that the Lebesgue measure of R_{T} is zero for n≥2.

This has an equally inportant interpretation that Brownian path X(t) avoids a set E with probability one i.e., P(R_{T}∩E = Φ ) = 1, if C(E)=0 (Takeuchi, 1964), so that on the set E⊆ R^{n}, V(x, t) fails to be bounded for all t.

In what follows we consider the occupation measure:

(E) = |{t∈[0, 1]: X(t)∈E}| as the distribution of heat source so that for any x∉<x_{t} : 0≤t≤1> and r, (B_{r}(x)) = 0

Note that for 0<t<1, (B_{r}(x))≤ T_{1}(r)+ T_{2}(r),

Where:

(12) |

is the total time spent by X(t) in B_{r}(0) and T_{2}(r) is the corresponding sojourn time for an independent copy of the dual X^{1} of X obtained by time reversal.

In order to obtain further information about the size of E such that E ∩ R_{T} = Φ , it is of interest to find a gauge function h such that

This allows us to consider the set

.

since for any and r small enough (B_{r} (x)) = 0

Dembo *et al*. (2000a, b) studied the random set

They proved the following

**Theorem 1 (Dembo et al., 2000a) **Let X(t), a Brownian motion in R

a.s.

whereand

**Theorem: 2 (Dembo et al., 2000b) **Let X(t) be a Brownian motion in R

a.s.

With a little modification, their methods extend, only with obvious changes to the following

**Theorem 3**

For X(t) a Brownian motion in R^{n}, n≥3

a.s

We focus on constructing a measure function h for which

To this end we state the following

**Theorem 4**

If X(t) is a Brownian motion in R^{n}, nâ‰¥3, T_{1} (r) and T_{2} (r) are independent copies of

Suppose

Then, with probability 1:

To prove the above theorem, we will need a few preliminary facts, in particular, the following lemmas

**Lemma 2 (Taylor, 1972) **For a Brownian motion X(t) in n-space, there is a λ

Whenever λ>λ_{0} = λ_{0} (ε, n)

**Lemma 3 (Perkins and Taylor, 1987)**

If

With |x| = ρ and Γ_{1} = Br_{1} (0) is a ball with centre of the origin and radius r_{1}

Then for a standard Brownian motion process in n-space, n≥3:

**Lemma 4 **For a Brownian path X(t) in R

Define

Then P(D_{k}) = 0 a.s for α >1

**Proof **From the obvious relationship

{ω : T(r, ω)≥s}⊆{w: X(s, ω)<r}

We have

We replace this event by a larger event.

X(t) does not leave Bα_{k+1} (0) before the time t>λ h(α_{k})

Let ,

where,

C_{k} = { X(t) does not leave Bα_{k+1} (0) Before time t > λ λ h(α_{k})}

By the choice of a_{k,} we take ρ large enough so that p(B_{k}\A_{k})≤1-ε = C_{1}.

By Lemma 3 P(C_{k}\B_{k}∩A_{k})≤C_{2}

so that

But, , by scaling

Thus by Lemma 2

we have:

By Borel Cantelli Lemma

P(D_{k}) = 0

We now prove Theorem 4

For a fixed

Define

Then

so that , where

for a constant c

But by Lemma 4

P(D_{k}) = 0 so that P(E_{k}) = 0

and thus E_{k} happens finitely often a.s for each λ and hence

since λ is any fixed number.

By the blumental zero-one law,we have

and so that and hence

a.s

**Theorem 5**

Suppose X(t) is Brownian motion in R^{n}, n≥3 and

Then for any compact set ECR^{n}

**Proof **We first show that Dim E<2 implies R

This follows from the fact that if x∉R_{T} then (B_{r}(x)) = 0 and by theorem 3 that

Dim E < 2 implies

R_{1}∩E = Φ a.s.

It remains to show that if Dim E > 2 then R_{T}∩E ≠ Φ a.s.

Pruit and Taylor (1996) have proved that if λ > 2

then Dim R_{1}≥λ a.s.

This implies that whenever Dim(E)>2, then R_{1}∩E ≠ Φ a.s.

We have characterized the geometric structure of the set of points x∈R^{n} for which the solution u(x, t) of the heat equation in R^{n} is unbounded for all t, by giving the size of such set of points (more precisely, the packing dimension of such set of points). Such points have, until very recently, been considered of little interest since the set of such points is of Lebesque measure zero. We have shown that the set has packing dimension 2 and thus is big. Moreover, we have shown that a set of zero heat capacity has zero packing measure in n-dimensional space.

- Dembo, A., Y. Peres, J. Rosen and O. Zeitouni, 2000. Thick points for spatial Brownian motion: Multifractal analysis of occupation measure. Ann. Probability, 8: 1-35.

Direct Link - Dembo, A., Y. Peres, J. Rosen and O. Zeitouni, 2000. Thin points for Brownian motion. Ann. Inst. Henri Poincare Probability Statistics, 36: 749-774.

Direct Link - Perkins, E.A. and S.J. Taylor, 1987. Uniform measure results for the image of subsets under Brownian motion. Probability Theory Relat. Fields, 76: 257-289.

CrossRefDirect Link - Pruit, W.E. and S.T. Taylor, 1996. Packing and covering indices for a general Levy process. Ann. Probability, 24: 971-986.

Direct Link - Takeuchi, J., 1964. On the sample paths of the symmetric stable processes in spaces. J. Math. Soc. Jpn., 16: 109-127.

CrossRefDirect Link - Taylor, S.J., 1972. Exact asymptotic estimates of Brownian path variation. Duke Math. J., 39: 219-241.

CrossRefDirect Link - Taylor, S.J., 1985. The use of packing measure in the analysis of random sets. Proceedings of the International Conference held in Nagoya, LNM., 1203, July 2-6, 1985, Springer Verlag, Berlin, pp: 214-222.

Direct Link - Taylor, S.J. and C. Triort, 1985. Packing measure and its evaluation for a Brownian path. Trans. Am. Math. Soc., 288: 679-699.

CrossRefDirect Link