We establish a Tanaka-like formula relating the local times of r and r + 1 fold self-intersections of a Brownian path in the plane.

Let G={G(x),x[set membership, variant]R1} be a mean zero Gaussian process with stationary increments and set [sigma]2(x-y)=E(G(x)-G(y))2. Let f be a symmetric function with Ef2([eta])[infinity], where [eta]=N(0,1). When [sigma]2(s) is concave or when [sigma]2(s)=sr, 1r=3/2, where...

We introduce the concept of capacitary modulus for a set , which is a function h that provides simple estimates for the capacity of [Lambda] with respect to an arbitrary kernel f, estimates which depend only on the L2 inner product (h,f). We show that for a large class of Lévy processes, which...

We study the occupation measure of various sets for a symmetric transient random walk in Zd with finite variances. Let denote the occupation time of the set A up to time n. It is shown that tends to a finite limit as n--[infinity]. The limit is expressed in terms of the largest eigenvalue of a...

We study the object formally defined as where Xt denotes the symmetric stable processes of index 0[beta]=2 in Rd. When , this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm...

Let G={G(x),x=0} be a mean zero Gaussian process with stationary increments and set [sigma]2(x-y)=E(G(x)-G(y))2. Let f be a function with Ef2([eta])<[infinity], where [eta]=N(0,1). When [sigma]2 is regularly varying at zero and is locally integrable for some integer j0>=1, and satisfies some additional regularity conditions, in L2. Here Hj is the jth Hermite polynomial. Also :(G')j:(I[a,b]) is a jth order Wick...</[infinity],>

We show how to renormalize the intersection local time of fractional Brownian motion of index [beta] in the plane, when½< [beta] <¾. When[beta] = ½, i.e., planar Brownian, such a renormalization is due to Varadhan.

We define renormalized intersection local times for random interlacements of Lévy processes in Rd and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.