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We study small-ball probabilities for the stochastic heat equation with multiplicative noise in the moderate-deviations regime. We prove the existence of a small-ball constant and related it to other known quantities in the literature. These small-ball estimates are known to imply Chung-type laws of the iterated logarithm (LIL) at typical spatial points; these points can be thought of as "points of flat growth". For this result in a similar context in SPDEs see, for example, the recent work of Chen \cite{Ch2023}. We establish the existence of a new family of exceptional spatial points where the Chung-type LIL fails., Comment: 37 pages

We study the radius $R_T$ of a self-repellent fractional Brownian motion $\left\{B^H_t\right\}_{0\le t\le T}$ taking values in $\mathbb{R}^d$. Our sharpest result is for $d=1$, where we find that with high probability, \begin{equation*} R_T \asymp T^\nu, \quad \text{with $\nu=\frac{2}{3}\left(1+H\right)$.} \end{equation*} For $d>1$, we provide upper and lower bounds for the exponent $\nu$, but these bounds do not match.

We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes $N$ Brownian motions up to time $T$, started at the origin and subject to exponential penalization based on the amount of time they spend close to each other, or close to themselves. The effective radius measures the typical distance from the origin. Our main result gives estimates for the effective radius where in two and three dimensions we impose the restriction that $T \leq N$. One of the highlights of our results is that in two dimensions, we find that the radius is proportional to $T^{3/4}$, up to logarithmic corrections., Comment: 34 pages

We provide asymptotic bounds on the survival probability of a moving polymer in an environment of Poisson traps. Our model for the polymer is the vector-valued solution of a stochastic heat equation driven by additive spacetime white noise; solutions take values in ${\mathbb R}^d, d \geq 1$. We give upper and lower bounds for the survival probability in the cases of hard and soft obstacles. Our bounds decay exponentially with rate proportional to $T^{d/(d+2)}$, the same exponent that occurs in the case of Brownian motion. The exponents also depend on the length $J$ of the polymer, but here our upper and lower bounds involve different powers of $J$. Secondly, our main theorems imply upper and lower bounds for the growth of the Wiener sausage around our string. The Wiener sausage is the union of balls of a given radius centered at points of our random string, with time less than or equal to a given value.

We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as $t\to\infty$, and we establish an upper bound of $\exp\{-\text{const}\cdot t^{1/3}\}$ for the rate of decay. We demonstrate also that the length of a valley grows at least as $\exp\{+\text{const}\cdot t^{1/3}\}$ as $t\to\infty$. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails., Comment: 22 pages

We consider self-repelling elastic manifolds with a domain $[-N,N]^d \cap \mathbb{Z}^d$, that take values in $\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the effective radius $R_N$ of the manifold is approximately $N^{4/3}$. This verifies the conjecture of Kantor, Kardar and Nelson [7]. Our results for the case where $d \geq 3$ and $D

We study elastic manifolds with self-repelling terms and estimate their effective radius. This class of manifolds is modelled by a self-repelling vector-valued Gaussian free field with Neumann boundary conditions over the domain $[-N,N]^d\cap \mathbb{Z}^d$, that takes values in $\mathbb{R}^d$. Our main result states that in two dimensions ($d=2$), the effective radius $R_N$ of the manifold is approximately $N$. This verifies the conjecture of Kantor, Kardar and Nelson [8] up to a logarithmic correction. Our results in $d\geq 3$ give a similar lower bound on $R_N$ and an upper of order $N^{d/2}$. This result implies that self-repelling elastic manifolds undergo a substantial stretching at any dimension., Comment: 32 pages

We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\sigma:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function and $\dot{W}$ denotes space-time white noise. If additionally $\sigma(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\, \sup_{x\in[-1,1]}\, \log u(t\,,x)$ and $t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$ must coincide. As a consequence of this fact, we prove that, when $\sigma$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate., Comment: 32 pages

We consider a reaction-diffusion equation of the type \[ \partial_t\psi = \partial^2_x\psi + V(\psi) + \lambda\sigma(\psi)\dot{W} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice" initial value and periodic boundary, where $\mathbb{T}=[-1\,,1]$ and $\dot{W}$ denotes space-time white noise. The reaction term $V:\mathbb{R}\to\mathbb{R}$ belongs to a large family of functions that includes Fisher--KPP nonlinearities [$V(x)=x(1-x)$] as well as Allen-Cahn potentials [$V(x)=x(1-x)(1+x)$], the multiplicative nonlinearity $\sigma:\mathbb{R}\to\mathbb{R}$ is non random and Lipschitz continuous, and $\lambda>0$ is a non-random number that measures the strength of the effect of the noise $\dot{W}$. The principal finding of this paper is that: (i) When $\lambda$ is sufficiently large, the above equation has a unique invariant measure; and (ii) When $\lambda$ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures., Comment: 69 pages

In order to extend the study of uniqueness property of multi-dimensional systems of stochastic differential equations, in this paper, we look at the following three-dimensional system of equations, of which the two-dimensional case was well-studied before: $dX_t=Y_tdt\quad, dY_t=Z_tdt,\quad dZ_t=|X_t|^{\alpha}dB_t$. We proved that if $(X_0,Y_0,Z_0)\neq(0,0,0)$, and $\frac{3}{4}<\alpha<1$, then the system of equations has a unique solution in the strong sense.

We consider the following stochastic partial differential equation on $t \geq 0, x\in[0,J], J \geq 1$ where we consider $[0,J]$ to be the circle with end points identified: \begin{equation*} \partial_t{\mathbf u}(t,x) =\frac{1}{2}\,\partial_x^2 {\mathbf u}(t,x) + {\mathbf g}(t,x,\mathbf u) + {\mathbf \sigma}(t,x, {\mathbf u})\dot {\mathbf W}(t,x) , \end{equation*} and $\dot {\mathbf W }(t,x)$ is 2-parameter $d$-dimensional vector valued white noise and ${\mathbf \sigma}$ is function from ${\mathbb R}_+\times {\mathbb R} \times {\mathbb R}^d \rightarrow {\mathbb R}^d$ to space of symmetric $d\times d$ matrices which is Lipschitz in $\mathbf u$. We assume that $\sigma$ is uniformly elliptic and that $\mathbf g$ is uniformly bounded. Assuming that ${\mathbf u}(0,x) \equiv \mathbf 0$, we prove small-ball probabilities for the solution $\mathbf u$. We also prove a support theorem for solutions, when ${\mathbf u}(0,x)$ is not necessarily zero.

We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length $J$ taking values in $(0,\infty)$. Our main result states that the effective radius of the polymer is approximately $J^{5/3}$; evidently for large $J$ the polymer undergoes stretching. This contrasts with the equilibrium situation without the time variable, where many earlier results show that the effective radius is approximately $J$. For such a moving polymer taking values in $\mathbf{R}^2$, we offer a conjecture that the effective radius is approximately $J^{5/4}$., Comment: 37 pages

We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+\sigma(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar., Comment: 30 pages

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations., Comment: to appear in EJP

We study the asymptotic speed of a random front for solutions $u_t(x)$ to stochastic reaction-diffusion equations of the form \[ \partial_tu=\farc{1}{2}\partial_x^2u+f(u)+\sigma\sqrt{u(1-u)}\dot{W}(t,x),~t\ge 0,~x\in\Rm, \] arising in population genetics. Here, $f$ is a continuous function with $f(0)=f(1)=0$, and such that~$|f(u)|\le K|u(1-u)|^\gamma$ with~$\gamma\ge 1/2$, and $\dot{W}(t,x)$ is a space-time Gaussian white noise. We assume that the initial condition $u_0(x)$ satisfies $0\le u_0(x)\le 1$ for all $x\in\Rm$, $u_0(x)=1$ for~$xR_0$. We show that when $\sigma>0$, for each $t>0$ there exist~$R(u_t)<+\infty$ and~$L(u_t)<-\infty$ such that $u_t(x)=0$ for $x>R(u_t)$ and $u_t(x)=1$ for~$x0$ there exists a finite deterministic speed~$V(\sigma)\in\Rm$ so that~$R(u_t)/t\to V(\sigma)$ as $t\to+\infty$, almost surely. This is in dramatic contrast with the deterministic case $\sigma=0$ for nonlinearities of the type $f(u)=u^m(1-u)$ with $01/2$ there exists $c_f\in\Rm$, so that~$\sigma^2V(\sigma)\to c_f$ as~$\sigma\to+\infty$ and give a characterization of $c_f$. The last result complements a lower bound obtained by Conlon and Doering \cite{cd05} for the special case of $f(u)=u(1-u)$ where a duality argument is available.

The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability., Comment: 32 pages

We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=\Delta u(t,x)+u^{-\alpha}(t,x)+g(u(t,x))\dot{W}(t,x) \] where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $0<\alpha<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$. (ii) If $\alpha>3$ then with probability one, $u(t,x)\ne0$ for all $(t,x)$., Comment: 35 pages

We consider a stochastic flow $\phi_t(x,\omega)$ in $\mathbb{R}^n$ with initial point $\phi_0(x,\omega)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|\phi_t(x)\|)}{\|\phi_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin., Comment: 34 pages, 3 figures

As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha<1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha<1$ and $(x_0,y_0)\neq(0,0)$. We also show that blowup in finite time holds if $\alpha>1$ and $(x_0,y_0)\neq(0,0)$., Comment: 20 pages

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space-time white noise, or colored noise in spatial dimensions $k\geq 1$. Our approach builds on a delicate covering argument developed by M. Talagrand for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic pde's.

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities., Comment: 26 pages

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points., Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x) +\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is H\"{o}lder continuous of index $\gamma>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks., Comment: Published in at http://dx.doi.org/10.1214/13-AOP870 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $\pi$ is equal to the number of pairs $i \pi_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.

We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution.

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot W, $ and $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u}\dot W, $ where $\dot W= \dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically $ 2-\pi^2 |\log \epsilon^2|^{-2} $ up to a factor of order $ (\log|\log\epsilon|)|\log\epsilon|^{-3}$.

In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is established to describe the large time evolution, distinguishing between certain death or possible life of the population.

We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.

We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution., Comment: To appear in TAMS

We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean.

For a superprocess under a stochastic flow, we prove that it has a density with respect to the Lebesgue measure for d=1 and is singular for d>1. For d=1, a stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's L_p-theory for linear SPDE. A snake representation for this superprocess is established. As applications of this representation, we prove the compact support property for general d and singularity of the process when d>1.

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the Skorokhod integral or Wick product. We give a criterion for existence based on the dimension of the space variable, and the parameter p of the stable noise. Our arguments are different for p<1 and p>1.

A super-Brownian motion in two and three dimensions is constructed where "particles" give birth at a higher rate, if they approach the origin. Via a log-Laplace approach, the construction is based on Albeverio et al. (1995) who calculated the fundamental solutions of the heat equation with one-point potential in dimensions less than four.

We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.

We give a new example of a measure-valued process without a density, which arises from a stochastic partial differential equation with a multiplicative noise term. This process has some unusual properties. We work with the heat equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number, and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2). The exponent 2 is critical in the following sense. For exponents less than 2, the equation has function-valued solutions, and for exponents higher than 2, we do not expect solutions to exist. This model is closely related to the parabolic Anderson model; we expect solutions to be small, except for a collection of high peaks. This phenomenon is called intermittency, and is reflected in the singular nature of our process. Solutions exist as singular measures, under suitable assumptions on the initial conditions and for sufficiently small k. We investigate various properties of the solutions, such as dimension of the support and long-time behavior. As opposed to the super-Brownian motion, which satisfies a similar equation, our process does not have compact support, nor does it die out in finite time. We use such tools as scaling, self-duality and moment formulae.

We consider Funaki's model of a random string taking values in R^d. It is specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is two-parameter white noise, also taking values in R^d. We study hitting properties, double points, and recurrence. The main difficulty is that the process has the Markov property in time, but not in space. We find: (1) The string hits points if d<6. (2) For fixed t, there are points x,y such that u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y) iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12. (5) The string is recurrent iff d < 7.

Consider the stochastic partial differential equation u_t=u_{xx}+u^gamma dot{W}, where x in [0,J], dot{W}=dot{W}(t,x) is 2-parameter white noise, and we assume that the initial function u(0,x) is nonnegative and not identically 0. We impose Dirichlet boundary conditions on u. We say that u blows up in finite time, with positive probability, if there is a finite random time T such that P(\lim_{t->T}sup_x u(t,x)=infty)>0. It was known that if gamma<3/2, then with probability 1, u does not blow up in finite time. It was also known that there is a positive probability of finite time blow-up for gamma sufficiently large. In this paper, we show that if gamma>3/2, then there is a positive probability that u blows up in finite time.