STRICT FINE MAXIMA

We provide a simple probabilistic proof of a result of J. Kr(cid:19)al and I. Netuka: If f is a measurable real-valued function on R d ( d (cid:21) 2 ) then the set of points at which f has a strict (cid:12)ne local maximum value is polar. Let X = ( X t ) t (cid:21) 0 be standard Brownian motion in R d ( d (cid:21) 2), and recall that the (cid:12)ne topology on R d is the coarsest topology with respect to which all superharmonic functions are continuous. Here is an alternative description of the (cid:12)ne topology: A set V (cid:26) R d is a (cid:12)ne neighborhood of x 2 V if and only if there is a Borel set U such that x 2 U (cid:26) V and x = 2 ( R d n U ) r . Here B r := f x 2 R d : P x [ T B = 0] = 1 g is the set of regular points for B , P x is the law of X started at X 0 = x , and T B := inf f t > 0 : X t 2 B g is the hitting time of B . It is well known that the set of points in R d at which a function f : R d ! R has a strict local maximum value is at most countable. Of course, \local" here refers to the usual topology on R d . The situation changes if the (cid:12)ne topology is substituted for the usual topology.

Let X = (X t ) t≥0 be standard Brownian motion in R d (d ≥ 2), and recall that the fine topology on R d is the coarsest topology with respect to which all superharmonic functions are continuous. Here is an alternative description of the fine topology: A set V ⊂ R d is a fine neighborhood of x ∈ V if and only if there is a Borel set U such that x ∈ U ⊂ V and x / ∈ (R d \ U ) r . Here B r := {x ∈ R d : P x [T B = 0] = 1} is the set of regular points for B, P x is the law of X started at X 0 = x, and T B := inf{t > 0 : X t ∈ B} is the hitting time of B. It is well known that the set of points in R d at which a function f : R d → R has a strict local maximum value is at most countable. Of course, "local" here refers to the usual topology on R d . The situation changes if the fine topology is substituted for the usual topology. For example, suppose d ≥ 3 and let M ⊂ R d be a subspace of dimension d − 2. Then M is uncountable, but the indicator function 1 M admits a strict local maximum value (with respect to the fine topology!) at each point of M . To see this simply notice that if x ∈ M then {x} ∪ M c is a fine neighborhood of x, because M is polar. Our purpose in this note is to give a new proof of a result of J. Král and I. Netuka [7], to the effect that the preceding example is the worst case scenario. We say that a Borel measurable function f : R d → R has a strict fine local maximum value at x ∈ R d provided there is a fine then, because singletons are polar for X, f has a strict fine local maximum value at x if and

Remarks.
(a) Our proof is quite different from the one given by Král and Netuka, and perhaps more accessible to a probabilistic audience. The proof in [7] is based on a deep result of A. Ancona [1], which states that every non-polar set contains a compact set K with K r = K. We use the strong Markov property and the invariance of Brownian motion under time reversal.
(b) Our method is quite robust and applies to any (transient) symmetric strong Markov process for which singletons are polar. The method can even be applied (with somewhat modified conclusions) to non-symmetric processes for which semipolar sets (and singletons) are polar. We leave these extensions to the interested reader.
(c) Král and Netuka show that even if f is not Borel measurable the set M (f ) is inner polar, in that each of its Borel subsets is polar.
Proof of the Theorem. We assume d ≥ 3; the recurrent case d = 2 can be handled similarly, by killing X at the first exit time from a large enough ball. We refer the reader to [7] for a proof that the function ϕ( Suppose, contrary to the assertion of the Theorem, that M (f ) is non-polar. Then, by Choquet's capacitability theorem there exists a compact non-polar set K contained in M (f ). Because X is transient, the hitting probability x → P x [T K < ∞] is the Green potential of a measure π-the equilibrium measure of K. This measure is carried by K, charges no polar set, and has total mass equal to the Newtonian capacity of K. Let Y = (Y t ) t∈R be the stationary Brownian motion in R d . This is a strong Markov process with the same transition probabilities as X, defined for all times t ∈ R, and with onedimensional distributions given by Lebesgue measure on R d . Associated with the equilibrium measure π is a homogeneous random measure κ of Y characterized by for all Borel functions g : (Here E denotes expectation with respect to Y .) In fact, κ is the dual predictable projection of the random measure placing a unit mass at the last exit time from K. The random measure κ is diffuse: κ{t} = 0 for all t ∈ R, almost surely. See [8], [2], [6], [5; §5.1], and [4; §5]. Let Z t (ω) be the indicator of the event that the function s → f (Y s (ω)) has a strict local maximum at time t ∈ R. Classically, the set {t ∈ R : Z t (ω) = 1} is countable for each sample point ω. Thus if φ is a strictly positive Borel function on R with finite Lebesgue integral, then Let ΠZ denote the optional projection of Z with respect to the filtration F t := σ{Y s : s ≤ t}, t ∈ R; that is, ΠZ is the unique (F t ) optional process such that E(Z T |F T ) = (ΠZ) T a.s. for every (F t ) stopping time T . Observe that where ω + t = (Y t+s (ω); s > 0), H(x, ω + t ) is the indicator of the event that T A f (x) (ω + t ) > 0, and Z − t (ω) is the indicator of the event that there is an (ω) > 0 such that f (Y s (ω)) < f(Y t (ω)) for all − (ω) < s < 0. If T is an (F t ) stopping time then Z − T is F T measurable, so by the strong Markov property of Y we have It follows that Because the random measure κ is (F t ) optional, See, for example, Théorème VI.2.66 in [3]. The time reversed processŶ : t → Y −t has the same law as Y . Therefore the optional projectionΠ(ΠZ) of ΠZ with respect to the reverse filtrationF t := σ{Y s : s ≥ t}, t ∈ R, is given by because ϕ is {0, 1}-valued by Blumenthal's 0-1 law. Now κ is also adapted to (F t ), so (9) E R φ(t)(ΠZ) t κ(dt) = E R φ(t)(Π(ΠZ)) t κ(dt).