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Let $u(s,t)$ be a continuous potential density of a symmetric L\'evy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $\lambda \wedge T_{0}$, where $\lambda$ is an independent exponential time. Let \[ f(t)=\int_{T} u(t,v)\,d\mu(v), \] where $\mu$ is a finite positive measure on $T$. Let $X_{\alpha}=\{X_{\alpha}(t),t\in T \}$ be an $\alpha-$permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{t\to 0}u(t,t)=0$, \[ \limsup_{t\downarrow 0}\frac{X_{\alpha}(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and \[ \limsup_{t\downarrow 0}\frac{X_{\alpha}(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,\mu}\le |\mu|$ is a constant that depends on both $u$ and $\mu$, which is given explicitly, and is different in the different examples., Comment: arXiv admin note: text overlap with arXiv:2302.10262

Let $Y$ be a symmetric Borel right process with locally compact state space $T\subseteq R^{1}$ and potential densities $u(x,y)$ with respect to some $\sigma$-finite measure on $T$. Let $g$ and $f$ be finite excessive functions for $ Y$. Set $$ u_{g, f}(x,y)= u(x,y)+g(x)f(y),\qquad x,y\in T.$$ In this paper we take $Y$ to be a symmetric L\'evy process, or a diffusion, that is killed at the end of an independent exponential time or the first time it hits 0. Under general smoothness conditions on $g$, $f$, $u$ and points $d\in T$, laws of the iterated logarithm are found for $X_{k/2} =\{X_{k/2}(t), t\in T \}$, a $k/2-$permanental process with kernel $ \{u_{g, f}(x,y),x,y\in T \}$, of the following form: For all integers $k\geq 1$, $$\limsup_{x \to 0}\frac{| X_{k/2}( d+x)- X_{k/2} (d)|}{ \left( 2 \sigma^{2}\left(x\right)\log\log 1/x\right)^{1/2}}= \left( 2 X _{k/2} (d)\right)^{1/2}, \qquad a.s. ,$$ where, $$\sigma^2(x)=u(d+x,d+x)+u(x,x)-2u(d+x,x).$$ Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the iterated logarithm are found for the local times of certain Markov processes with potential densities that have the form of $ \{u_{g, f}(x,y),x,y\in T \}$ or are slight modifications of it.

Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ \theta$ be the first exit time of the unit disk $D_{1}$. Let \[\mu_{ \theta } ( x,\epsilon) =\frac{1}{\pi\epsilon^{ 2} }\int_{0}^{ \theta }1_{\{ B( x,\epsilon)\}}( W_{t})\,dt,\] and set $\mu^{ \ast}_{ \theta } (\epsilon)=\sup_{x\in D_{1}}\mu_{ \theta } ( x,\epsilon)$. We show that \[\sqrt{\mu^{\ast}_{\theta} (\epsilon)}-\sqrt{2/\pi} \left(\log \epsilon^{-1}- \frac{1}{2}\log\log \epsilon^{-1}\right)\] is tight.

Let $\eta=\{\eta(t);t\in [0,1]\}$ be a mean zero continuous Gaussian process with covariance $U=\{U(s,t),s,t\in [ 0,1]\},$ with $U(0,0)>0$. Let $\{\eta_{i};i=1,\ldots, k\}$ be independent copies of $\eta$ and set $ Y_{k}(t)=\sum_{i=1}^{k} \eta^2_{i}(t), t\in [ 0,1].$ The stochastic process $Y_{k } =\{Y_{k }(t),t\in [ 0,1] \}$ is referred to as a chi--square process of order $k $ with kernel $U$. Let $\phi(t)$ be a positive function on $[0,\delta]$ for some $\delta>0$. If \[\limsup_{t\to 0}\frac{ \eta(t)-\eta(0)}{ \phi(t) }=1 \qquad a.s., \] then for all integers $k\ge 1$, \[ \limsup_{t\to 0} \frac{Y_{k }(t)-Y_{k }(0)} { \phi (t)} = 2 Y^{1/2}_{k}(0) \qquad a.s.\] Set \[ \sigma^2(u,v)=E(\eta(u)-\eta(v))^2\quad\text{and}\quad \widetilde\sigma^2(x)=\sup_{|u-v|\le x}\sigma^2(u,v).\] Assume that $\inf_{t\in [0,1]}U(t,t)>0$ and, \[ \lim_{x\to0}\widetilde\sigma^2(x)\log 1/x =0. \] Let $\varphi(t)$ be a positive function on $[0,1]$. Then if \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\in\Delta}}\frac{ \eta(u)-\eta(v)}{ \varphi(|u-v|) }=1 \qquad a.s.\] for all intervals $\Delta\subset [0,1]$, it follows that for all intervals $\Delta\subset [0,1]$ and all integers $k\ge 1$, \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\in\Delta}} \frac{Y_{k }(u)-Y_{k }(v) }{ \varphi (|u-v|)} = 2 \sup_{u\in\Delta}Y_{k }^{1/2}(u), \hspace{.2 in}a.s.\], Comment: 11 pages. arXiv admin note: text overlap with arXiv:2006.14457

Let $ \overline B=\{ \overline B_{t},t\in R^{1} \}$ be Brownian motion killed after an independent exponential time with mean $2/\lambda^{2}$. The process $\overline B$ has potential densities, \[ u(x,y) ={e^{-\lambda |y-x|}\over \lambda},\qquad x,y\in R^{ 1}, \] which is also the covariance of an Ornstein--Uhlenbeck process. Let $f$ be an excessive function for $\overline B$. Then, \[ {e^{-\lambda |y-x|}\over \lambda}+f(y),\qquad x,y\in R^{ 1}, \] is the kernel of an $\alpha$-permanental process $ X_{\alpha}=\{ X_{\alpha}(t), t\in R^{ 1}\}$ for all $\alpha>0$. It is shown that for all $k\ge 1$ and intervals $\Delta \subseteq [0,1] $, \[ \limsup_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\in\Delta}} \frac{|X_{k/2} (u)-X_{k/2} (v)|}{ 2 ( |u-v| \log 1/|u-v|)^{1/2}}= \sqrt 2 \sup_{t\in\Delta}X_{k/2}^{1/2}(t)\qquad a.s.\] The local modulus of continuity of $X_{k/2}$ for all $k\ge 1$ is also obtained. Local and uniform moduli of continuity are also obtained for chi--square processes which are defined by, \[ Y_{k/2}(t)=\sum_{i=1}^{k}\frac{\eta^2_{i}(t)}{2},\qquad t\in [0,1], \] where $\eta=\{\eta(t);t\in [0,1]\}$ is a mean zero Gaussian process and $\{\eta_{i};i=1,\ldots, k\}$ are independent copies of $\eta.$, Comment: There was an error in (3.23) in calculating derivatives

Permanental sequences with non-symmetric kernels that are generalization of the potentials of a Markov chain with state space $\{0,1/2, \ldots, 1/n,\ldots\}$ that was introduced by Kolmogorov, are studied. Depending on a parameter in the kernels we obtain an exact rate of divergence of the sequence at $0$, an exact local modulus of continuity of the sequence at $0$, or a precise bounded discontinuity for the sequence at $0$.

Let $U=\{U_{j,k},j,k\in \overline {\mathbb N}\}$ be the potential of a transient symmetric Borel right process $X$ with state space $\overline {\mathbb N}$. For any excessive function $f=\{f_{k,k\in \overline {\mathbb N}}\}$ for $X$ , $\widetilde U=\{\widetilde U_{j,k},j,k\in\overline {\mathbb N}\}$, where \begin{equation} \widetilde U_{j,k}= U_{j,k} +f_{ k},\qquad j,k\in\overline {\mathbb N},\label{a.1} \end{equation} is the kernel of an $\alpha$-permanental sequence $\widetilde X_{\alpha}=(\widetilde X_{\alpha, 1} ,\ldots)$ for all $\alpha>0$. The symmetric potential $U$ is also the covariance of a mean zero Gaussian sequence $\eta=\{\eta_{j},j\in \overline {\mathbb N}\}$. Conditions are given on the potentials $U$ and excessive functions $f$ under which, \begin{equation} \limsup_{j\to \infty}\frac{ \eta_{j}}{( 2\,\phi_{j})^{1/2} }=1 \quad a.s. \quad \implies \quad \limsup_{n\to \infty}\frac{\widetilde X_{\alpha, j}}{\phi_{j} }=1\quad a.s.,\label{a.2} \end{equation} for all $\alpha>0$, and sequences $\phi=\{\phi_{j}\}$ such that $f_{j}=o(\phi_{j})$. The function $\phi$ is determined by $U$. Many examples are given in which $U$ is the potential of symmetric birth and death processes with and without emigration, first and higher order Gaussian autoregressive sequences and L\'evy processes on $\mathbf Z$.

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.

Kernels of $\alpha$-permanental processes of the form \[ v(x,y)=u(x,y)+f(y),\qquad x,y\in S, \] in which $u(x,y)$ is symmetric, and $f$ is an excessive function for the Borel right process with potential densities $u(x,y)$, are considered. Conditions are given that determine whether $\{v(x,y);x,y\in S\}$ is symmetrizable or asymptotically symmetrizable.

Let $C^*_{\epsilon,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $\epsilon$. We prove that $$\sqrt{C^{*}_{\epsilon,S^2} } -\sqrt{\frac{2A_{S^2}}{\pi}}(\log \epsilon^{-1}-\frac14\log\log \epsilon^{-1})$$ is tight, where $A_{S^2}=4\pi$ denotes the Riemannian area of $S^2$., Comment: Third version deals only with the sphere, because the reduction from general manifold to the sphere in the second version contains a mistake. V5 corrects an error in the statement of Lemma 9.1, and its use in the first and second moment estimates (replacing the former erroneous estimates (4.56) and (4.87)). V6 corrects minor typos, and added details to the statement and proof of Lemma 9.2

Let $X_{\alpha}=\{X_{\alpha}(t),t\in T\}$, $\alpha>0$, be an $\alpha$-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha}$ is a subgaussian process with respect to the metric $\sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2}$. This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha$-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient L\'evy processes that are not necessarily symmetric, or with kernels of the form $ \hat u(x,y)= u(x,y)+f(y)$, where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

For the critical Galton--Watson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if $C_L$ denotes the cover time of the binary tree of depth $L$ by simple walk, then $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ is tight. The latter improves results of Aldous (1991), Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for the two-dimensional sphere.

We explore some properties of a recent representation of permanental vectors which expresses them as sums of independent vectors with components that are independent gamma random variables.

An $\al$-permanental process $\{X_{ t},t\in T \}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T \}$, with the property that for all $t_{1},\ldots,t_{n}\in T $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ is the Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $ K( t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i}, t_{j})\}_{i,j=1}^{n}$ and $S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n} $. $ (X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector. Under the condition that $K$ is the potential density of a transient Markov process, $(X_{t_{1}},\ldots,X_{t_{n}})$ is represented as a random mixture of $n$-dimensional random variables with components that are independent gamma random variables. This representation leads to a Sudakov type inequality for the sup-norm of $(X_{t_{1}},\ldots,X_{t_{n}})$ that is used to obtain sufficient conditions for a large class of permanental processes to be unbounded almost surely. These results are used to obtain conditions for permanental processes associated with certain L\'evy processes to be unbounded. Because $K$ is the potential density of a transient Markov process, for all $t_{1},\ldots,t_{n}\in T $, $A( t_{1},\ldots,t_{n}):= (K( t_{1},\ldots,t_{n}))^{-1}$ are $M$-matrices. The results in this paper are obtained by working with these $M$-matrices.

We construct Markov loop measures without assuming the existence of densities for transition probabilities.

These notes originated in a series of lectures I gave in Marseille in May, 2013. I was invited to give an introduction to the isomorphism theorems, originating with Dynkin, which connect Markov local times and Gaussian processes. This is an area I had worked on some time ago, and even written a book about, but had then moved on to other things. However, isomorphism theorems have become of interest once again, both because of new applications to the study of cover times of graphs and Gaussian fields, and because of new isomorphism theorems for non-symmetric Markov processes and their connection with loop soups and Poisson processes. Thus I felt the time was ripe for a new introduction to this topic. In these notes I have tried to preserve the informal atmosphere of the lectures, and often simply refer the reader to my book and other sources for details.

The loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an in nitesimal variation of the generator a ecting the killing rates or the jumping rates.

We define renormalized intersection local times for random interlacements of L\'evy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials., Comment: SPA, to appear

Several stochastic processes related to transient L\'evy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $V$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(V,d)$. The processes include $n$-fold self-intersection local times of transient L\'evy processes and permanental chaoses, which are `loop soup $n$-fold self-intersection local times' constructed from the loop soup of the L\'evy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of $n$-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.

A permanental field, $\psi=\{\psi(\nu),\nu\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $\psi$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $\psi$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^{\nu},(\nu ,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm., Comment: Published in at http://dx.doi.org/10.1214/13-AOP893 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This is a survey of results about permanental processes, real valued positive processes which are a generalization of squares of Gaussian processes. In a certain sense the symmetric positive definite function that determines a Gaussian process is replaced by a function that is not necessarily symmetric nor positive definite, but that nevertheless determines a stochastic process. This is a new avenue of research with very many open problems., Comment: 31 pages

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times., Comment: Published in at http://dx.doi.org/10.1214/12-AOP744 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study the object formally defined as \gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<\beta\le 2$ in $R^{d}$. When $\beta\le\sigma<\displaystyle\min \Big\{{3\over 2}\beta, d\Big\}$, 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 for $\gamma$. This is applied to obtain results about stable processes in random potentials.

We give a stochastic calculus proof of the Central Limit Theorem \[ {\int (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\Longrightarrow}c(\int (L^{x}_{t})^{2} dx)^{1/2} \eta\] as $h\to 0$ for Brownian local time $L^{x}_{t}$. Here $\eta$ is an independent normal random variable with mean zero and variance one.

Let G={G(x), x\in R_+}, G(0)=0, be a mean zero Gaussian process with $E(G(x)-G(y))^2=\sigma ^2(x-y) $. Let $ \rho (x)= \frac12{d^{2}\over dx^2}\sigma^2(x)$, $x\ne 0 $. When $\rho^{k}$ is integrable at zero and satisfies some additional regularity conditions, \[ \lim_{h\downarrow 0} \int :(\frac{G(x+h)-G(x)}{h})^{k}:g(x) dx=:(G') ^{k}:(g){.3 in}a.s. \] for all $g\in B_{0}(R^{+})$, the set of bounded Lebesgue measurable functions on $R_+$ with compact support. Here $G'$ is a generalized derivative of $G$ and $:(\cd)^{k}:$ is the $k$--th order Wick power.

We study $\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $\gamma_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.

Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'{e}vy process with local time $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'{e}vy exponent $\psi(\la)$ is regularly varying at zero with index $1<\beta\leq 2$, and satisfies some additional regularity conditions, \begin{eqnarray*} && {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- E(\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\over t\sqrt{\psi^{-1}(1/t)}}\label{r5.0tweaksabs} && \stackrel{\mathcal{L}}{\Longrightarrow}(8c_{\psi,1 })^{1/2}(\int_{-\infty}^{\finfty} (L_{\beta,1}^{x})^{2} dx)^{1/2} \eta \end{eqnarray*} as $t\rar\infty$, where $L_{\bb,1}=\{L^{x}_{\beta, 1} ; x \in R^{1} \}$ denotes the local time, at time 1, of a symmetric stable process with index $\beta$, $\eta$ is a normal random variable with mean zero and variance one that is independent of $L_{\beta,1}$, and $c_{\psi,1}$ is a known constant that depends on $\psi$.