Imagine measuring something like temperature at different points in a room using a series of increasingly precise instruments.

As you refine your measurements, your understanding of the temperature distribution becomes more detailed and eventually transitions from a series of discrete readings to a continuous temperature field.

This paper applies a similar concept to particle systems, proposing that, under suitable rescaling, the random and discrete nature of these systems actually converges towards a continuous and predictable behavior, encapsulated by a mathematical model known as a stochastic differential equation. This work serves as a foundation to understand random features present in the original discrete system, even at the continuous limit.

Consider a random particle making its way across an environment with unknown heterogeneous transition rules. This scenario describes a random walk in a random environment (RWRE). Now, consider that, from time to time, ressampling occurs, i.e thie environment shifts abruptly and all transition rules are updated independently from all previous occurences.

If the ressampling time intervals are constant, then the independence of the walk along these increments of time allow us to consider, when all the randomness is taken into account, the random walk process as a homogeneous random walk. In contrast, if no ressampling occurs then the environment is fixed and behaves as the classical RWRE which behaves very differently than homogeneous random walk. Our interest here is to change the time interval between the ressampling times so as to interpolate between homogeneous random walk (constant ressampling) and RWRE (frozen dynamics). This can be obtained by allowing the time interval between ressamplings to diverge. Since no ressampling corresponds to a "frozen" environment the case of interest for us is the 'cooling' dynamics. This introduces a new model - a random walk in a cooling random environment (RWCRE).

In RWCRE, when cooling is effective, i.e. when the ressampling intervals diverge, regardless of the rate at which it diverges, certain behaviors emerge. The particle's average speed and direction, over a long period, become predictable - the 'Strong Law of Large Numbers.'

Moreover, the deviations from the average, adhere to a certain pattern known as the 'Large Deviation Principle' (LDP). Intriguingly, despite the irregular and random changes of the environment, the LDP remains unaffected. The pattern of fluctuations, however, does change, highlighting the subtle interplay between the dynamism of the environment and the observable patterns in probabilistic systems. This offers a new perspective on understanding the intricacies of complex, changing environments.

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. An overarching goal of our paper is to investigate how the behaviour of a random process with a rich correlation structure can be affected by resettings.

We establish the existence of solutions to a class of nonlinear stochastic differential equations of reaction–diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained is the scaling limit of a sequence of interacting particle systems and satisfies the martingale problem corresponding to the target differential equation.

To study billiards, we need a table and a ball. Let our table be the region of points in the plane (x,y) with \(x>0\) \(|y|< x^\gamma\) for some \(\gamma \in (0,1)\). Consider a particle (our billiards ball) which moves linearly inside and reflects at the boundary according to a Markovian Kernel and the incoming angle, i.e. the outgoing angle is a function of the incoming angle and a uniform random variable which is independent of everything else in the system. We want to understand the recurrence behaviour of the process.

At the extreme value \(\gamma = 0\), our region corresponds to a tube that we close at the left to confine the \(x\)-coordinate of the particle to the positive half line. In this model the displacement of \(x\) is homogeneous in \(x\) and the we know that the walk with zero drift is recurrent and have standard tools to obtain quantitative insight on the long term behaviour of the walk.. When \(\gamma = 1\) provided there is a positive probability of reflecting towards the unbounded component determined by the ray which follows the normal, then the particle is transient as it never collides again with the boundary.

The interest of this model lies in allowing \(\gamma\) to take values between \(0\) and \(1\). To probe recurrence we consider Lamperti (critical) reflection Kernels and examine the influence that the curvature of the region has on displacements of the walk. We find expressions for the critical value \(\gamma_c\) which divides the walk into recurrent or transient and in case of recurrent we determine the moments of passage times for the process as a function of \(\gamma\) and the Kernel.

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the “cooling sequence”, and is kept fixed in between those times. This model interpolates between that of a homogeneous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the corresponding (static) RWRE is given by a s-stable random variable with \(s \in (1, 2)\). In this regime, due to the two extreme cases (resampling every step and never resampling, respectively), a crossover from Gaussian to stable limits for sufficiently regular cooling sequence was previously conjectured. Our first result answers affirmatively this conjecture by making clear critical exponent, norming sequences and limiting laws associated with the crossover which demonstrates a change from Gaussian to s-stable limits, passing at criticality through a certain generalized tempered stable distribution which have not appeared as limits of random walks in dynamic random environments previously. We then explore the resulting RWCRE scaling limits for general cooling sequences. On the one hand, we offer sets of operative sufficient conditions that guarantee asymptotic emergence of either Gaussian, s-stable or generalized tempered distributions from a certain class. On the other hand, we give explicit examples and describe how to construct irregular cooling sequences for which the corresponding limit law is characterized by mixtures of the three above mentioned laws. To obtain these results, we need and derive a number of refined asymptotic results for the static RWRE with \(s \in (1, 2)\) which are of independent interest.

Boundaries play an important role in the behaviour of confined systems. For example, particles near the surface of a material might move differently than those in the interior. In this paper, We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries.

Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peigné and Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.

Achievements of our study of partially homogeneous random walks are listed below.

1. We extend from Maximally-homogeneous walks to partially homogenous walks, a classification of recurrence and a quantification of the moments of return times;
2. We offer a broader framework (Partially homogeneous random walks) which allows us to include variant cases in the literature such as the case of Maximally-homogeneous walks with excitable boundaries, see for instance the paper by Menshikov and Petritis;
3. We treat the case Lindley random walks on the quadrant, which appear in the study of reflected processes;
4. We present the method of time change and uses stabilization to provide a complementary explanation of the (not so intuitive) Fredholm alternative method employed in the paper by Menshikov and Petritis;
5. We offer visual elements to the explanation of the choice of the test functions which enable the classification of recurrence and the quantification of the tails of the return times.