A counterexample in parabolic potential theory Pawel Kr¨oger∗ Mathematisches Institut, Bismarckstr. 1 1/2 91054 Erlangen, Germany

0. Introduction In Section 1 of this note we will construct an example of a subset of R × Rn such that the parabolic capacity with respect to the heat equation is zero although its orthogonal projection onto {0} × Rn is the whole space. Such examples were already given by R. Kaufman and J.-M. Wu in [5] and [6]. However, our probabilistic approach seems to be more transparent since it does not depend on explicit formulas for Green functions. In Section 2, we modify our basic construction in order to show that the ”natural” analogue of a Theorem by J. Deny for capacities with respect to the Laplace equation fails to be true (see [1] and [8]). ∂ − 4 on R × Rn . The heat kernel g on R × Rn is We consider the heat operator ∂t given by g(t, x) ≡ (4πt)−n/2 exp(−|x|2 /4t) for every t > 0, x ∈ Rn and g(t, x) ≡ 0 if t ≤ 0. A Borel set M is polar with respect to the heat equation if there exists a positive R Borel measure µ such that the thermal potential Rn+1 g(t − s, x − y) dµ(s, y) is finite on a dense subset of R × Rn and identically equal to +∞ on M (cf. [2], Sect. 1.XVII.8). In probabilistic terms, M is polar if the hitting probability of M at a strictly positive ∂ time for a space-time Brownian motion process with differential generator ∂t + 4 starting n at an arbitrary point of R × R in negative time direction is equal to zero (cf. [2], Sect. 2.IX.5).

1. Construction of a polar set with respect to the heat equation We will first consider the case of one space dimension, i.e. n = 1. Qj Suppose that we are given a sequence {ri }∞ i=1 ri . i=1 of positive integers. We set lj ≡ + ∞ We aim to define in an inductive manner a sequence {Ki }i=0 of closed sets in R × R. Set K0 ≡ {0} × R. Assume now that K1 , ..., Kj−1 have been chosen. We set Kj ≡

1 1 −3/2 lj , x) | (t, x) ∈ Kj−1 and k ≤ lj x ≤ k + for an integer k} 10 2 1 ∪ {(t, x) | (t, x) ∈ Kj−1 and k − ≤ lj x ≤ k for an integer k}. 2 {(t +

We intend to choose the numbers ri sufficiently large. We consider a space-time Brownian motion process starting at (1, 0) in negative time direction. It is easy to see that for rj sufficiently large the events ”The Brownian particle hits Kj \ Kj−1 ” and ”The Brownian particle hits Kj ∩ Kj−1 ” are ”almost independent”. This motivates us to compare the number of intersections of a Brownian path with Kj with the number of individuals in the j th generation of suitable branching processes. First, we introduce a branching process of Galton-Watson type (cf. [4], Section 2.1). We consider a homogeneous Markov chain ∗

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which is given by successive integer valued random variables Z0 , Z1 , Z2 , ... The variable Zj can be interpreted as the number of individuals in the j th generation. We set Z0 ≡ 1. We suppose that the probability distribution of Z1 is given by P (Z1 = 0) = 1/4, P (Z1 = 1) = 1/2, and P (Z1 = 2) = 1/4. The characteristic property of a Galton-Watson process is the property that the variable Zj for j > 1 is defined as the sum of Zj−1 independent random variables with the same probability distribution as Z1 . In particular, the expectation value E(Zj ) of Zj is equal to 1 for every j. Steffensen’s Theorem states that E(Zj ) = E(Z1 ) ≤ 1 for every j yields that the extinction probability of the Galton-Watson process is equal (1) to 1, i.e. limj→∞ P (Zj = 0) = 1 (see [4], Section 2.6). Moreover, since limj→∞ P (Zj + (k) (1) (k) ... + Zj = 0) = 1 for any k-tuple Zj , ..., Zj of independent Galton-Watson processes with the same probability distribution as the above process, we can conclude that the extinction probability of any finite population with respect to the above Galton-Watson process is equal to 1. Now we consider a modified branching process (this process is not a process of GaltonWatson type). We set again Z˜0 ≡ 1. Let Z˜j for j > 1 be the sum of Z˜j−1 independent (1)

˜ (Z

)

(k)

random variables Xj , ..., Xj j−1 with the probability distribution P (Xj = 0) = 1/4 − (k) (k) 2δj , P (Xj = 1) = 1/2 + δj , and P (Xj = 2) = 1/4 + δj for every k, the positive number δj will be chosen later. We claim that the extinction probability of the modified process is again equal to 1 if the δj are sufficiently small. Suppose that we are given δ1 , ..., δjk such that P (Z˜jk = 0) ≥ 1 − 2−k . Obviously, Z˜jk ≤ 2jk . We choose an integer l such that the extinction probability after l generations of an initial population of 2jk objects with respect to the above Galton-Watson process is greater than 1/2. On the other hand, the extinction probability after a finite number of generations of a given finite population with respect to the modified process depends continuously on the corresponding δj . Hence, we can choose positive numbers δjk , ..., δjk +l such that P (Z˜jk +l = 0) ≥ 1 − 2−(k+1) . We have obtained a sequence of positive numbers δj such that limj→∞ P (Z˜j = 0) = 1. This establishes the claim. We aim to construct in an inductive manner sequences {rj }∞ j=1 of natural numbers ∞ and {j }j=1 of positive real numbers such that the following properties are satisfied: 1.) The probability that the Brownian motion process starting at (1, 0) hits the set Kj at least c times is less than P (Z˜j ≥ c) for every c > 0. 2.) Kj is contained in the i -neighbourhood Ui of Ki for every i < j. 3.) The probability that the Brownian motion process starting at (1, 0) does not hit Ui is greater or equal than P (Z˜i = 0). Suppose that we are given r1 , ..., rj−1 and 1 , ..., j−1 . Let Tj−1 be an arbitrary subset of the orthogonal projection of Kj−1 on the time axis and let τ be a mapping from Tj−1 to 1 −3/2 lj τ (t)). the set of all subsets of {0; 1}. We set Tj ≡ ∪t∈Tj−1 (t + 10 Our construction is based on the fact that the Markov property and the invariance of the Browning motion process starting at (t0 , x0 ) with respect to the scaling t − t0 7→ α2 (t − t0 ) and x − x0 7→ α(x − x0 ) yield that lim P (Xt ∈ Kj for every t ∈ Tj | Xt ∈ Kj−1 for every t ∈ Tj−1 )

rj →∞

2

=

2−c

(∗)

where c denotes the number of elements of Tj . On the other hand, limrj →∞ Kj = Kj−1 . Thus, the properties 1.) and 2.) are satisfied for rj sufficiently large. Finally, we mention that the limit set K of the sequence {Kj }∞ j=1 is contained in Ui for every i. If we take into account that limi→∞ P (Z˜i = 0) = 1, we can conclude that K is polar with respect to the heat equation. On the other hand, the orthogonal projection of K onto {0} × R is obviously equal to {0} × R. The case n > 1 can be reduced to the case n = 1 if we consider the set K × Rn−1 .

2. Modifications of the basic construction First, we will modify our construction in such a way that the parabolic projection of the ˆ from (1, 0) on {0} × Rn , i.e. the set of the points (0, x) for all x such that modified set K ˆ is not empty, contains the intersection of the parabolic arc {(1 − α2 , αx) | α > 0} with K an open subset of {0} × Rn . Again, we will restrict ourselves to the case n = 1. ˜ 0 ≡ R × [0, 1] and We set K ˜j ≡ K

k + 1/4 1 −3/2 k + 1/4 −1/2 lj , + (1 + lj )(x − )) 10 lj lj ˜ j−1 and k ≤ lj x ≤ k + 1 for an integer k} | (t, x) ∈ K 2 1 ˜ j−1 and k − ≤ lj x ≤ k for an integer k}. ∪ {(t, x) | (t, x) ∈ K 2 {(t +

By construction, every curve R → R : t 7→ x(t) with x(0) ∈ [0, 1] and | ∂x | ≤ 5/2 for ∂t ˜ every t ∈ [0, 1] has a non-empty intersection with Kj . We suppose that ri ≥ 2 for every ˜ j ⊂ [0, 1/4] for every j. Since the restriction of every parabolic arc i. This implies that K | ≤ 5/2, we can {(1 − α2 , αx)|α > 0} for x ∈ [0, 1] to [0, 1/4] × R satisfies the condition | ∂x ∂t ˜ ˜ conclude that the parabolic projections of every Kj and of the corresponding limit set K from (1, 0) on {0} × R contain {0} × [0, 1]. A similar continuity argument as at the end of the previous section shows that the ˜ is polar with respect to the heat equation if the ri are sufficiently large (take into set K account that the analogue of (*) is again true). Now we consider the set ˜ and k a positive integer }. ˆ ≡ {(1 − s , x ) | (1 − s, x) ∈ K K k2 k Since polarity is invariant under the scaling t − t0 7→ α2 (t − t0 ) and x − x0 7→ α(x − x0 ), ˆ is again polar for the heat equation. On the other hand, the we can conclude that K intersection of the parabolic arc {(1 − α2 , αx) | α > 0} for an arbitrary x ∈ [0, 1] with ˆ contains points which are arbitrarily close to (1, 0). This shows that the ”natural” K analogue of Theorem 1 from [1] fails to be true (cf. [8]). ˇ ⊂ R × Rn such that K ˇ is polar with Finally, we claim that it is possible to find a set K Pn ∂ ∂2 respect to L ≡ ∂t − i,j=1 aij (x) ∂xi ∂xj for arbitrary bounded H¨older continuous functions aij on Rn with (aij (x))i,j uniformly positive definite for every x ∈ Rn and such that the ˇ on {0} × Rn coincides with {0} × Rn . The proof of the claim orthogonal projection of K 3

is based on the fact that the fundamental solutions of the parabolic equations Lu = 0 are uniformly continuous with respect to all variables if the H¨older constants of the coefficients, the bounds on the moduli of the coefficients, and the ellipticity constant are ˇ in the uniformly bounded (see [3], Chapter 1 and [7], Theorem 3.2.1). We construct K n−1 ˇ ≡ K ×R same way as in Section 1, i.e. we will set K where K denotes the set defined ∞ in Section 1 and where {rj }j=1 is an appropriate sequence of natural numbers which will be chosen later. Let Lk be the set of all operators L of the above form with coefficients aij such that (1/k)1 ≤ (aij )i,j ≤ k1 and |aij (x) − aij (y)| ≤ k|x − y|1/k for every x, y with P 2 ∂ − ni,j=1 aij (α(x − x0 ) + x0 ) ∂x∂i ∂xj ∈ Lk for every α ≤ 1 |x − y| ≤ 1. We notice that ∂t and every x0 ∈ Rn . By the Markov property, the equality (*) is still true if we consider probabilities with respect to diffusion processes with differential generators from Lk . We can conlude in the same way as in Section 1 that the set K is polar with respect to every diffusion with a differential generator from Lk if the sequence {rj }∞ j=1 satisfies the condi(k) (k) ∞ tion rj ≥ rj for every j ≥ j0 for an appropriate sequence {rj }j=1 and an arbitrary j0 . We finally obtain a set which is polar for every L ∈ ∪∞ k=1 Lk if we consider the diagonal (j) ∞ sequence {rj }j=1 . Acknowledgement. The author is very grateful to M. Brzezina for drawing his attention to the above problem. The author wishes to thank N. A. Watson for informing him on Kaufman’s and Wu’s work. References [1] Deny, J. Un th´eor`eme sur les ensembles effil´es. Ann. Univ. Grenoble Sect. Sci. Math. Phys. 23, 139–142 (1948). [2] Doob, J. L. Classical potential theory and its probabilistic counter part. Springer: New York 1984 [3] Friedman, A. Partial differential equations of parabolic type. Prentice Hall: Englewood Cliffs, N.J. 1964 [4] Harris, Th. E. The theory of branching processes. Springer: Berlin 1963 [5] Kaufman, R.; Wu, J.-M. Singularity of parabolic measures. Comp. Math. 40, 243–250 (1980). [6] Kaufman, R. and Wu, J.-M. Parabolic potential theory. J. Diff. Eq. 43,204–234 (1982). [7] Stroock, D. W.; Varadhan, S. R. S. Multidimensional diffusion processes. Springer: Berlin 1979 [8] Watson, N. A. Thinness and boundary behaviour of potentials for the heat equation. Mathematika 32, 90–95 (1985).

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