We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.
On Analytically Tractable Multidimensional Diffusions via Doob h-Transforms, with Resetting and Applications to Wiener and Ornstein--Uhlenbeck Processes
Ciesielski's isomorphism between the space of alpha-Hölder continuous functions and the space of bounded sequences is used to give an alternative proof of the large deviation principle for Wiener processes with values in Hilbert space.
We study sample path deviations of the Wiener process from three different representations of its bridge: anticipative version, integral representation and space-time transform. Although these representations of the Wiener bridge are equal in law, their sample path behavior is quite different. Our results nicely demonstrate this fact. We calculate and compare the expected absolute, quadratic and conditional quadratic path deviations of the different representations of the Wiener bridge from the original Wiener process. It is further shown that the presented qualitative behavior of sample path …
We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by com…
A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are consid…
Analytic and Monte Carlo Approximations to the Distribution of the First Passage Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions
In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borel, and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $ν_α^n$, where $ν_α =Y_α^{-1}…
Motivated by the recent work of Benjamini, Haggstrom, Peres, and Steif (2003) on dynamical random walks, we: Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein-Uhlenbeck process in classical Wiener space; derive sharp tail-asymptotics for the probabilities of large deviations of the said dynamical walk; and characterize (by way of an integral test) the minimal envelop(es) for the growth-rate of the dynamical Gaussian walk. This development also implies the tail capacity-estimates of Mountford (1992) for large deviations in classical Wiener …
We study Karhunen-Loeve expansions of the process $(X_t^{(α)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(α)} = -\fracα{T-t} X_t^{(α)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(α)}=0,$ where $α>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ is a standard Wiener process. This process is called an $α$-Wiener bridge or a scaled Brownian bridge, and in the special case of $α=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(α)}$. As applications, we calculate the Laplace transform and the distribution function of the…
We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\frac{1}{N}$ times the su…
For a>0,let W^a(t) be the a-neighbourhood of standard Brownian motion in R^d starting at 0 and observed until time t.It is well-known that E|W^a(t)|~kappa_a t (t->infty) for d >= 3,with kappa_a the Newtonian capacity of the ball with radius a. We prove that lim_{t->infty} 1/t^{(d-2)/d}log P(|W^a(t)|<=bt) = -I^{kappa_a}(b) in (-infty,0) for all 0<b<kappa_a and derive a variational representation for the rate function I^{kappa_a}.We show that the optimal strategy to realise the above moderate deviation is for W^a(t) to look like a Swiss cheese: W^a(t) has random holes whose sizes are of order 1 …
We study Donsker-Watanabe's delta functions associated with strongly hypoelliptic diffusion processes indexed by a small parameter. They are finite Borel measures on the Wiener space and admit a rough path lift. Our main result is a large deviation principle of Schilder type for the lifted measures on the geometric rough path space as the scale parameter tends to zero. As a corollary, we obtain a large deviation principle conjectured by Takanobu and Watanabe, which is a generalization of a large deviation principle of Freidlin-Wentzell type for pinned diffusion processes.
We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
We consider a sequence $X^n=(X^n_t)_{t\ge 0},n\ge 1$ of semimartingales. Each $X^n$ is a weak solution to an Itô equation with respect to a Wiener process and a Poissonian martingale measure and is in general non-Markovian process. For this sequence, we prove the large deviation principle in the Skorokhod space $D=D_{[0,\infty)}$. We use a new approach based on of exponential tightness. This allows us to establish the large deviation principle under weaker assumptions than before.
We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…
We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associate…
In this article we establish a large deviation principle for the family {ν_ε:ε\in (0,1)} of distributions of the scaled stochastic processes {P_{-\log\sqrtε}Z_t}_{t\leq 1}, where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-Itô chaos decomposition of the terminal value Z_{1}. The result is established by developing a continuity theorem for large deviations, together with two essential tools, the hypercontractivity of the Ornstein-Uhlenb…
We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{ï} [KMSS94]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, A…
This paper explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell's definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infi…
We consider a family of continuous processes $\{X^\varepsilon\}_{\varepsilon>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C([0,1]^d:\mathbb{R})$, and have the Wiener chaos expansion \[ X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). \] We provide sufficient conditions for the large deviations principle of $\{X^\varepsilon\}_{\varepsilon>0}$ to hold in $C([0,1]^d:\mathbb{R})$, thereby refreshing a problem left open by Pérez-Abreu (1993) in the Brownian motion case. The proof is based on the …
We use a new method via $p$-Wasserstein bounds to prove Cramér-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0\leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(\log n+x^2)/\sqrt{n}$ for $P(W>x)/(1-Φ(x))\to 1$, where $Φ$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinat…
In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total ma…
We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space $D[0,1]$. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error type processes.
In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$α$ Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by $\sqrtα$. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as $α$ goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term o…
We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^ε,Y^ε),ε\to 0$ defined as $$ X^ε_t=\frac{1}{ε^κ}\int_0^t H(ξ^ε_s,Y^ε_s)ds, dY^ε_t=F(ξ^ε_t,Y^ε_t)dt+ Dε^{1/2-κ}G(ξ^ε_t,Y^ε_t)dW_t,$$ where $W_t$ is Wiener process and $ξ^ε_t$ is fast ergodic diffusion. We show that, under $κ<{1/2}$ or less and Veretennikov-Khasminskii type condition for fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.
We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $εD_{v}D_{u}$ where $0<ε\ll…
The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d \ W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d \ W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^…
