On diffusions that cannot escape from a convex set

Korzeniowski, Andrzej

doi:10.1016/0167-7152(89)90127-2

Cited by 13 publications

(16 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[1]. We are strongly motivated by the past mathematical research, whose roots can be traced back to Knight's "taboo processes", [17][18][19].…”

mentioning

confidence: 99%

Killing (absorption) versus survival in random motion

Garbaczewski

2017

Phys. Rev. E

View full text Add to dashboard Cite

We address diffusion processes in a bounded domain, while focusing on somewhat unexplored affinities between the presence of absorbing and/or inaccessible boundaries. For the Brownian motion (Lévy-stable cases are briefly mentioned) model-independent features are established, of the dynamical law that underlies the short time behavior of these random paths, whose overall lifetime is predefined to be long. As a by-product, the limiting regime of a permanent trapping in a domain is obtained. We demonstrate that the adopted conditioning method, involving the so-called Bernstein transition function, works properly also in an unbounded domain, for stochastic processes with killing (Feynman-Kac kernels play the role of transition densities), provided the spectrum of the related semigroup operator is discrete. The method is shown to be useful in the case, when the spectrum of the generator goes down to zero and no isolated minimal (ground state) eigenvalue is in existence, like e.g. in the problem of the long-term survival on a half-line with a sink at origin. I. MOTIVATION.Diffusion processes in a bounded domain (likewise the jump-type Lévy processes) serve as important model systems in the description of varied spatio-temporal phenomena of random origin in Nature. When arbitrary domain shapes are considered, one deals with highly sophisticated problems on their own, an object of extensive investigations in the mathematical literature.A standard physical inventory, in case of absorbing boundary conditions which are our concern in the present paper, refers mostly to the statistics of exits, e.g. first and mean first exit times, probability of survival and its asymptotic decay, thence various aspects of the lifetime of the pertinent stochastic process in a bounded domain, [1]-[6], see also [7][8][9].Typically one interprets the survival probability as the probability that not a single particle may hit the domain boundary before a given time T . The long-time survival is definitely not a property of the free Brownian motion in a domain with absorbing boundaries, where the survival probability is known to decay to zero exponentially with T → ∞, [4,5]. Therefore, the physical conditions that ultimately give rise to a long-living random system, like e.g. those considered in [4,5], see also [6], must result in a specific remodeling (conditioning, deformation, emergent or "engineered" drift) of the "plain" Brownian motion.For simple geometries (interval, disk and the sphere) the exponential decay of the single-particle survival probability has been identified to scale the stationary (most of the time) gas density profile, while that profile and the decay rates stem directly from spectral solutions of the related eigenvalue problem for the Laplacian with the Dirichlet boundary data, respectively on the interval, disk and sphere, see e.g. [1,4,5]. In fact, the square of the lowest eigenfunction, upon normalization, has been found to play the role of the pertinent gas profile density, while the associated lowest eigenvalue of th...

show abstract

“…[1]. We are strongly motivated by the past mathematical research, whose roots can be traced back to Knight's "taboo processes", [17][18][19].…”

mentioning

confidence: 99%

Killing (absorption) versus survival in random motion

Garbaczewski

2017

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…Over the years, many different conditioning constraints have been considered, including the Brownian excursion [26,27], the Brownian meander [28], the taboo processes [29][30][31][32][33][34], or nonintersecting Brownian bridges [35]. Let us also mention the conditioning in the presence of killing rates [18,[36][37][38][39][40][41][42][43] or when the killing occurs only via an absorbing boundary condition [44][45][46][47].…”

Section: Reminder On the Conditioning Of Stochastic Processes With Re...mentioning

confidence: 99%

Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes

Mazzolo

Monthus

2023

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $ \Sigma(t)=A(t)+B(t)$. Their asymptotic statistics for large time $t \to + \infty$ involves two very different cases : (i) when the diffusion process $X(t)$ is transient, the two local times $[A(t);B(t)]$ remain finite random variables $[A^*(\infty),B^*(\infty)]$ and we analyze their limiting joint distribution ; (ii) when the diffusion process $X(t)$ is recurrent, we describe the large deviations properties of the two intensive local times $a = \frac{A(t)}{t}$ and $b = \frac{B(t)}{t}$ and of their intensive sum $\sigma = \frac{\Sigma(t)}{t}=a+b$. These properties are then used to construct various conditioned processes $[X^*(t),A^*(t),B^*(t)]$ satisfying certain constraints involving the two local times, thereby generalizing our previous work [arXiv:2205.15818] concerning the conditioning with respect to a single local time $A(t)$. In particular for the infinite time horizon $T \to +\infty$, we consider the conditioning towards the finite asymptotic values $[A^*(\infty),B^*(\infty)]$ or $\Sigma^*(\infty) $, as well as the conditioning towards the intensive values $[a^*,b^*] $ or $\sigma^*$, that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local times in the regime of large deviations. This general construction is then applied to the simplest case where the unconditioned diffusion is the Brownian motion of uniform drift $\mu$.

show abstract

“…With such a drift, the boundary now corresponds to an entrance boundary [1], which means that the boundary cannot be reached from the interior of the state space (here the interval ] − ∞, a[). Originally introduced in the mathematical literature, and since then widely studied in this field [10,109,110] for both semi-infinite and finite domains, the taboo process and its later generalizations have recently found applications in physics [111] where they are relevant for studying confined polymers [112]. For a physicist-oriented survey, we refer to the recent article [9].…”

Section: Application To the Brownian Motion With Drift µ For Finite H...mentioning

confidence: 99%

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Monthus¹,

Mazzolo²

2022

Preprint

View full text Add to dashboard Cite

When the unconditioned process is a diffusion living on the half-line x ∈] − ∞, a[ in the presence of an absorbing boundary condition at position x = a, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite T < +∞, the conditioning consists in imposing the probability P * (y, T ) to be surviving at time T and at the position y ∈ ] − ∞, a[, as well as the probability γ * (Ta) to have been absorbed at the previous time Ta ∈ [0, T ]. When the time horizon is infinite T = +∞, the conditioning consists in imposing the probability γ * (Ta) to have been absorbed at the time Ta ∈ [0, +∞[, whose normalization [1−S * (∞)] determines the conditioned probability S * (∞) ∈ [0, 1] of forever-survival. This case of infinite horizon T = +∞ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passagetime properties at position a. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift µ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

show abstract

On diffusions that cannot escape from a convex set

Cited by 13 publications

References 1 publication

Killing (absorption) versus survival in random motion

Killing (absorption) versus survival in random motion

Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Contact Info

Product

Resources

About