Let T (x, ε) denote the first hitting time of disc radius ε centered at x for Brownian motion on two dimensional torus 2 .We prove that sup x∈T ε)/| log ε| → 2/π as 0. The same applies to any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary.As a consequence, we conjecture, due Aldous (1989), number steps it takes simple random walk cover all points lattice Z n is asymptotic 4n (log n) /π.Determining these asymptotics an essential step toward...

Limit theorems are proved for the range of $d$-dimensional random walks in domain attraction a stable process index $\beta$. The $R_n$ is number distinct sites $\mathbb{Z}^d$ visited by walk before time $n$. Our results depend on value ratio $\beta/d$. most interesting obtained $2/3 < \beta/d \leq 1$. law large numbers then holds $R_n$, that is, sequence $R_n/E(R_n)$ converges toward some constant and we prove convergence distribution $(\operatorname{var} R_n)^{-1/2}(R_n - E(R_n))$...

We envision a network of $N$ paths in the plane determined by independent, two-dimensional Brownian motions $W_i(t), t \geq 0, i = 1, 2, \cdots, N$. Our problem is to study set "confluences" $z$ $\mathbb{R}^2$ where all meet and also $M_0$ $N$-tuples times $\mathbf{t} (t_1, t_N)$ at which confluences occur: $M_0 \{\mathbf{t}: W_1(t_1) \cdots W_N(t_N)\}$. The random analyzed constructing convenient stochastic process $X$, we call "confluent motion", for X^{-1}(0)$ using theory occupation...

Let $X$ be a strongly symmetric recurrent Markov process with state space $S$ and let $L_t^x$ denote the local time of at $X \in S$. For fixed element 0 in S, $$ \tau(t) := \inf \{s: L^0_s > t \}. The 0-potential density, $u_{\{0\}}(x, y)$, killed $T_0 = \{s:X_s =0\}$ is positive definite. $\eta \{\eta_x; x S \}$ mean-zero Gaussian covariance E_\eta (\eta_x \eta_y ) u_{\{0\}}(x, y). main result this paper following generalization classical second Ray–Knight theorem: for any $b R$ $t 0$,...

Necessary and sufficient conditions are obtained for the almost sure joint continuity of local time a strongly symmetric standard Markov process $X$. also global boundedness unboundedness continuity, in neighborhood point state space. The given terms 1-potential density proofs rely on an isomorphism theorem Dynkin which relates times processes related to $X$ mean zero Gaussian with covariance equal By showing equivalence sample path properties times, known necessary various carried over...

and conjectured that the limit exists equals 1/Tr a.s. The importance of determining value this is clarified in (1.3) below, where appears power laws governing local time walk. Erd6s-Taylor conjecture was quoted book by R~v~sz [19, w but to best our knowledge, bounds (1.1) were not improved prior present paper. As it turns out, an important step towards solution formulation Perkins Taylor [17] analogous problem on maximal occupation measure planar Brownian motion (run for unit time) can

We show how to use local times analyze the self-intersections of random fields. In particular, we compute Hausdorff dimension $r$-multiple for Brownian motion in plane, sheets and Levy's multiparameter motion.

This paper derives the asymptotic expansions of a wide class Gaussian function space integrals under assumption that minimum points action are isolated. Degenerate as well nondegenerate allowed. also limit theorems for related probability measures which correspond roughly to law large numbers and central theorem. In degenerate case, limits non-Gaussian.

For a new class of Gaussian function space integrals depending upon $n \in \{1, 2,\cdots\}$, the exponential rate growth or decay as \rightarrow \infty$ is determined. The result applied to calculation specific free energy in model statistical mechanics. physical discussion self-contained. paper ends by proving upper bounds on certain probabilities. These will be used sequel this paper, which asymptotic expansions and limit theorems proved for considered here.

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Let $\mathscr{T}(x,r)$ denote the total occupation measure of ball radius $r$ centered at $x$ for Brownian motion in $\mathbb{R}^3$. We prove that $\sup_{|x|\leq1}\mathscr{T}(x,r)/(r^{2}|\log r|)\rightarrow16/\pi^2$ a.s. as $r\rightarrow0$, thus solving a problem posed by Taylor 1974. Furthermore, any $a \in(0,16/\pi^2)$, Hausdorff dimension set “thick points” which $\lim\sup_{r \to 0}\mathscr{T}(x,r)/(r^2|\log r|) = a$ is almost surely $2-a\pi^2 /8$; this correct scaling to obtain...

Let $\{L^x_t, (t, x) \in R^+ \times R\}$ be the local time of a real-valued symmetric stable process order $1 < \beta \leq 2$ and let $\{\pi(n)\}$ sequence partitions $\lbrack 0, a\rbrack$. Results are obtained for $\lim_{n\rightarrow\infty} \sum_{x_i\in\pi(n)} |L^{x_i}_t - L^{x_{i-1}}_t|^{2/(\beta-1)}$ both almost surely in $L^r$ all $r > 0$. also similar expression but where supremum sum is taken over a\rbrack$ function other than power applied to increments times. The proofs use lemma...

We use a Tanaka-like formula to explain Varadhan's renormalization of the formally infinite measure Brownian self intersections given by $\int^T_0 \int^T_0 \delta(W_t - W_s) ds dt.$

Let $\mathcal{T}_n(x)$ denote the time of first visit a point $x$ on lattice torus $\mathbb {Z}_n^2=\mathbb{Z}^2/n\mathbb{Z}^2$ by simple random walk. The size set $α$, $n$-late points $\mathcal{L}_n(α)=\{x\in \mathbb {Z}_n^2:\mathcal{T}_n(x)\geq α\frac{4}π(n\log n)^2\}$ is approximately $n^{2(1-α)}$, for $α\in (0,1)$ [$\mathcal{L}_n(α)$ empty if $α&gt;1$ and $n$ large enough]. These sets have interesting clustering fractal properties: we show that $β\in (0,1)$, disc radius $n^β$ centered at...

We describe simple conditions on the transition density functions of two independent Markov processes $X$ and $Y$ which guarantee existence a continuous version for intersection local time, formally given by $\alpha (z, H) = \int_H\int \delta_z (Y_t - X_s) ds dt$. In analogous case self-intersections $\alpha$ can be discontinuous at $z 0$. develop Tanaka-like formula use this to show that singular part (z,\lbrack 0, T\rbrack^2)$ as \rightarrow 0$ is $2\int^T_0 U(X_t z, X_t) dt, a.s.$, where...

We study moderate deviations for the renormalized self-intersection local time of planar random walks.We also prove laws iterated logarithm such times