On a family of coupled diffusions that can never change their initial order

We introduce a real-valued family of interacting diffusions where their paths can meet but cannot cross each other in a way that would alter their initial order. Any given interacting pair is a solution to coupled stochastic differential equations with time-dependent coefficients satisfying certain regularity conditions with respect to each other. These coefficients explicitly determine whether these processes bounce away from each other or stick to one another if/when their paths collide. When all interacting diffusions in the system follow a martingale behaviour, and if all these paths ultimately come into collision, we show that the system reaches a random steady-state with zero fluctuation thereafter. We prove that in a special case when certain paths abide to a deterministic trend, the system reduces down to the topology of captive diffusions. We also show that square-root diffusions form a subclass of the proposed family of processes. Applications include order-driven interacting particle systems in physics, adhesive microbial dynamics in biology and risk-bounded quadratic optimization solutions in control theory.

We introduce a real-valued family of interacting diffusions where their paths can meet but cannot cross each other in a way that would alter their initial order. Any given interacting pair is a solution to coupled stochastic differential equations with time-dependent coefficients satisfying certain regularity conditions with respect to each other. These coefficients explicitly determine whether these processes bounce away from each other or stick to one another if/when their paths collide. When all interacting diffusions in the system follow a martingale behaviour, and if all these paths ultimately come into collision, we show that the system reaches a random steady-state with zero fluctuation thereafter. We prove that in a special case when certain paths abide to a deterministic trend, the system reduces down to the topology of captive diffusions. We also show that square-root diffusions form a subclass of the proposed family of processes. Applications include order-driven interacting particle systems in physics, adhesive microbial dynamics in biology and risk-bounded quadratic optimization solutions in control theory.
Keywords: coupled processes, captive dynamics, stochastic domains, interacting systems (Some figures may appear in colour only in the online journal) * Author to whom any correspondence should be addressed.
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Introduction
This paper introduces a family of stochastic processes that maintain their initial order throughout their lifetime. In other words, by virtue of order-preservation, these random processes display a fundamental feature: they are able to collide with but never trespass each other's randomly evolving paths, hence act as pairwise stochastic boundaries to one another. Our framework can help address a wide range of questions in physical systems that include two random objects of equivalent speed and/or velocity (moving simultaneously in the same direction), which would either bounce away from each other, or stick to one another, at time of collision, depending on their underlying parameters (e.g. mass, magnetism, shape, spin). Some possible applications may surface in the study of approximating elastic collisions in particle physics (e.g. behaviour of atoms in thermal agitation under black-body radiation), or modelling pathogenic mechanisms of microbial infections in mathematical biology, or pricing financial instruments whose maximum-minimum payoffs are capped by each other's evolution.
For the majority of this work, order will simply refer to any two points on a totally-ordered set that can be compared with each other via the binary relation '⩽' (later in the paper, we shall relax this to a partial-order on the convex cone of positive semi-definite matrices). In this sense, for points in time t ∈ [0, T] = T with T < ∞, the attribute of order-preservation in a t ⩽ b t is satisfied when a t is bound to remain smaller or equal to b t for the duration of its lifetime, unable to break free from its initial order with respect to b t until T. For what follows, we shall choose the totally-ordered set to be R, where the points refer to the values of R- . . , n} where n ⩾ 2 and n ∈ N + . More specifically, we shall guarantee that where P(.) is a probability measure. Accordingly, every pair {(X )} t∈T in the system manifests either bouncy (reflective) or sticky (absorbing) behaviour if/when the two paths collide.
The study of random processes with reflective or absorbing dynamics within restricted domains has attracted significant attention in the literature due to their wide applicability in physics, engineering, economics and biology-see [1][2][3][4][5][6][7][8][9][10][11] amongst many more, including random walkers with Kardar-Parisi-Zhang fluctuations driven by the underlying geometry (see [12]). Essentially, the topological boundaries control the space of all possible paths that a stochastic entity may follow-e.g. a particle or a microbe. In most cases however, the domains that restrict the space of all possible paths are pre-determined from the outset, and hence, the architecture of these domains are usually not randomly-formed. As such, the boundaries that restrict the evolution of a stochastic phenomenon typically arise from the exogenouslyassigned deterministic topologies that represent the relevant underlying space, which may certainly be a desired framework depending on the type of application. On the other hand, one of the blooming areas in natural sciences, in which stochastic boundaries arise randomly and endogenously, is random matrix theory, where non-colliding eigenvalue dynamics tend to appear in the specific form of Dyson's Brownian-motion, which can be formed via harmonic Doob transforms in Weyl chambers [13][14][15][16][17]. In our proposed framework, we diverge from the construction of random matrix theory, and allow a high level of flexibility in defining systems where stochastic diffusions act as stochastic boundaries to each other-particles can collide and in turn encapsulate a considerably large family of dynamics involving absorption and reflection properties, paving way to the investigation of random paths that evolve within self-induced topologies displaying highly versatile stochastic geometries that satisfy order-preservation. Therefore, in relation with the existing literature, we may still speak of topological boundaries that control the space of all possible paths that a stochastic entity may follow, but this time the boundaries are naturally and endogenously moulded in time as a result of the randomness of the agents involved. Given the mathematical generality of our approach, we can simulate systems where each particle may behave very differently, while still respecting their order within the system to sustain a degree of unity. To the best of our knowledge, there is no literature that studies this aspect as presented in this paper. In addition, we prove that captive diffusions of [18], that evolve within deterministic boundaries, form a special subclass of the processes we introduce in this paper when certain pairs of diffusion coefficients are set to zero (this will be clear to readers later on).
We note that our framework can be applied in random particle systems that interact with each other based on their order (see [18][19][20][21]), risk-controlled quadratic optimization problems (see, [22]), and sticky microbial dynamics (see, [23][24][25]), amongst others. We leave a more detailed account of these applications for future and focus on establishing the mathematical groundwork in this paper. This work is organised as follows: In section 2, we present our main results together with simulations for demonstration. In section 2.1, we generate Hermitianvalued processes that respect their initial Loewner-order, and produce order-preserving efficient frontiers in quadratic optimization. Section 3 concludes. Section 5 is the appendix, where we offer a generalization of our framework.

Main Results
We choose a filtered probability space (Ω, F, {F t } t⩽∞ , P), where all filtrations are rightcontinuous and complete with F ∞ = F. We let Ω = C(T × R) be the space of continuous paths and X Accordingly, if we have n = 2, we write X t = X In addition, we introduce the following notation that will be useful throughout this paper: to specify that the ith coordinate of X t takes the value y ∈ R. If i = 1 or i = n, then (2) should be understood accordingly, where we have We are now in the position to introduce the main object of this work.
given that X We note that definition 2.1 stands fairly abstract, and to keep the setup general, we deliberately avoid specifying sufficiency conditions on the SDEs above (e.g. local Lipschitz continuity and linear growth) for the existence and uniqueness of their solutions, and instead define our processes as solutions when they exist. There is a well-established literature on existence and uniqueness of SDE solutions (see, [26,27]), which is not the focus of our study. The next result is what gives the name of the main object of our work and what in turn proves (1).

Proposition 2.2.
Let {X t } t∈T be an order-preserving coupled system as in definition 2.1. Then, for all t ∈ T and every i = 1, . . . , n − 1.

Proof. Fix any i and
at that t ∈ T, given that µ (i) and µ (i+1) are locally bounded functions (since they are continuous). Hence, from Property 3., in definition 2.1, {X given that {X (4) and (5) and Property 2., in definition 2.1, we have for any t ∈ T. Since this holds at any t ∈ T where X for all t ∈ T, P-a.s. Same steps can be taken for every chosen pair i and i + 1 from i = 1, . . . , n − 1, and the result follows.
The statement above essentially implies that an initially-assigned order cannot be violated thereafter, hence, the naming of these stochastic processes. This order is maintained due to the interacting (time-dependent) coefficients that satisfy the aformentioned regularity conditions with respect to each other. Many models can be constructed that belong to this family-we provide an example below.
The following is an order-preserving coupled system: for all t ∈ T, and the parameters satisfy: κ ∈ [0, ∞), 0 < |α| < ∞ and 0 < |β| < ∞. We shall provide an extended version of this example in the appendix together with a demonstrative simulation. Now, for parsimony let {W (1) t } t∈T and {W (2) t } t∈T in (6) be mutually independent and p(x, t) be the probability density function for X t ∈ R 2 in (6). Then the Fokker-Planck equation (i.e. the Kolmogorov forward equation) is given by with given initial condition. At a time of collision when X x, the Fokker-Planck equation reduces down to the following: Using proposition 2.2, when we have n > 2, we can construct systems whereby {X The following is an order-preserving coupled system: where each {W t } t∈T and {f (3) t } t∈T are deterministic R-valued continuous functions such that for all t ∈ T, and where 0 < |α| < ∞, 0 < |ξ| < ∞ and 0 < |β| < ∞.
Note that in figure 1 {X  (7) be mutually independent and p(x, t) be the probability density function for X t ∈ R 3 in (7). We shall now provide the Kolmogorov backward equation given by (2) p(x, t) with given terminal condition. Note that (8) is a specific case of the Feynman-Kac formula. Depending on µ (i) and µ (i+1) , order-preserving coupled processes can exhibit both bouncing (reflecting) and binding (absorbing) behaviour with respect to each other over non-overlapping time frames. Since the statement below follows from proposition 2.2, we shall omit its proof.
} t∈T absorb each other after they collide.
Corollary 2.5 tells us that the system shows reflecting vs absorbing properties explicitly through the drift terms. For instance, a fully reflective structure of µ (i) and µ (i+1) can model elastic collisions of atoms with random trajectories. Example 2.6. Using corollary 2.5, we can also model systems where the coordinates of {X t } t∈T branch out from each other only after a given timet ∈ T, until which time their trajectories have been the same. More precisely, if we keep the following: = x 0 , then the system would demonstrate branching dynamics. As an example, in figure 2, where we set n = 2, for every i ∈ I as a canonical representation using Brownian motions.
From this point onwards, we shall let {X t } t∈T stand for a Markovian order-preserving coupled system as in remark 2.7. We are interested in transformations of {X t } t∈T that retain the system to be order-preserving as in definition 2.1-this allows one to form increasingly more sophisticated systems starting from simpler models by applying monotonic function compositions in succession. For the statement below, we let C 2 b (R) ⊂ C(T × R) be the subspace of continuous locally bounded measurable functions that are also twice-differentiable with continuous locally bounded derivatives. Proposition 2.8. Let h ∈ C 2 b (R) be a strictly increasing monotonic function. Then, h(X (i) t ) for every t ∈ T and every i ∈ I form an order-preserving coupled system.
for every t ∈ T. We need to check if all the properties in definition 2.1 are satisfied. First of all, since h ∈ C 2 b (R), using Itô's lemma, we have for all t ∈ T and for every i ∈ I, where we are able to writeμ (i) andσ (i) in (11) in terms of Y, since h is a strictly increasing monotonic function, which means it has an inverse, and since h is applied to every i ∈ I; hence, we can findμ (i) andσ (i) in term of Y that produces (10). In addition, having h ∈ C 2 b (R), bothμ (i) andσ (i) are measurable continuous maps. Moreover, since h is a strictly increasing monotonic function, y and ∂h/∂x > 0 for any x, which implies we get for every i = 1, . . . , n − 1 using (10), and we also havê for every i = 1, . . . , n − 1 that follows directly from (10), which completes the proof. Example 2.9. Let h(x) = x 3 for any x ∈ R and consider the system given in (6) in example 2.3. Then, we reach s as an order-preserving coupled system. Remark 2.10. Using proposition 2.2, we can deduce that the stochastic integration below is order-preserving that can be mapped into a non-negative random variable as follows: The following result provides the postcollision behaviour when the coupled processes are strict martingales-if/when two paths meet, the paths necessarily stick to each other thereafter. For the following statements, we adopt the convention inf ∅ = ∞. Proposition 2.11. Let each coordinate of {X t } t∈T be a (P, {F X t })-martingale and let the random variable τ (i,i+1) be given by as the first collision-time of {X and the pair reaches a steady-state and becomes constant such that Proof. Since {X -martingales, we must have the following condition on the drift terms: for all t ∈ T, which means the system is governed by From Property 3., in definition 2.1, we further have and using corollary 2.5 and (12), X i+1) , T], which proves (13). In addition, this means and hence, from (16), writing we have (18) and (19) imply that dX , T], which proves (14).
In theoretical physics, martingale property has been interpreted as a stochastic analogue for the conservation law of energy (see [28][29][30]). In our framework, proposition 2.11 tells us that martingales can be interpreted as systems that (sequentially) reach a collection of random steady-states as the paths meet each other-i.e. if the particles collide with at least one other particle after some finite time. Surely, we can have more than two paths colliding with each other to attain their random steady-state. The proposition below gives the case of three paths meeting at the same level, which can be generalised to any number of meeting paths via following a similar logic. Proposition 2.12. Let each coordinate of {X t } t∈T be a (P, {F X t })-martingale and let the random variables τ (i,i+1) and τ (i+1,i+2) be given by (20) as the first collision-times of {X and the triplet reaches a steady-state and becomes constant such that The proof is similar to that of proposition 2.11, where we must now have for all t ∈ T, since every coordinate is a (P, {F X t })-martingale. Hence, From Property 3., in definition 2.1, we have which from (23) and (24) imply that X (21). This further implies Therefore, using (23), we have which, by using (25), completes the proof.

Loewner-order preserving coupled processes
In a multivariate setting, the framework naturally lends itself to Hermitian-valued coupled stochastic processes that cannot change their initially-assigned Loewner-order. More precisely, we can produce matrix-valued Hermitian processes {H (i) t } t∈T for i ∈ I on the space of positive semi-definite Hermitian matrices H m×m + for some m ∈ N + that maintain their initially given order on the induced convex cone, i.e.
where ' ' is the Loewner-order; a partial-order on H m×m is a positive semi-definite matrix. In doing so, we can select a collection of m order-preserving coupled systems from definition 2.1 as follows: , . . . , Efficient-frontiers are models to quantify risk-return trade-offs in decision-making (see [31,32])) and the relation in (38) provides a particular way to order risk via definition (2.16). For demonstration, figure 4. displays an example where the efficient-frontiers satisfy the order given in (38). We believe such Loewner-order preserving matrix constructs are relevant in the fields of control theory, operations research and economics, where one can produce stochastic covariance processes that evolve between pairs of other stochastic covariance processes in the Loewner-sense, which in turn generate order-preserving efficient-frontier dynamics as in (38).

Conclusion
The main objective of this paper has been to establish a theoretical framework for representing random systems where paths of stochastic processes can collide but cannot cross each other, by virtue of preserving their initially given order over their lifetime. The main objective is geared towards building a robust but an adaptable mathematical foundation, rather than discussing in detail the practical applications which are introduced to spur new ideas but are left for future research. The proposed setup includes multiple stochastic paths as solutions to SDEs defined through their drift and diffusion coefficients interacting with each other. The particular nature of these cross-communicating coefficients that satisfy certain regularity conditions forms the basis of how these paths draw in space such stochastic trajectories that in turn manifest as stochastic boundaries to one another. This construction includes cases of potential collisions that would either cause particles to bounce away from each other, or to bind to one another, depending on the coupling force characterised through their drift components. We show that, if all interacting particles follow a martingale behaviour, the system ultimately reaches a random steady-state with zero fluctuation at the time of final collision, forcing all paths to necessarily remain attached to one another thereafter. We also conclude that the manifestation of the captive diffusion topology, as introduced in [18], boils down to being a specific case of our proposed framework. Furthermore, we prove that order-preserving coupled systems are invariant under monotonic transformations, which enables one to transition from simple models to more sophisticated, if not more realistic, representations of certain physical dynamics. One of the key discussions in the paper revolves around the framework's high degree of flexibility in allowing stochastic paths to endogenously sculpt the geometry of their own system, which brings a level of self-induced dynamic control to the system through order-preservation. Due to the versatility of our framework, supported by numerical simulations to inspire imagination, we envisage its use in many areas such as particle physics, microbiology and control theory as mentioned in the paper (e.g. the modelling, calibration and simulation of interacting particle systems, where collisions may exhibit either reflection or absorption dynamics against each other).

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.