Incremental Input-to-State Stability and Equilibrium Tracking for Stochastic Contracting Dynamics

I N C R E M E N T A L I N P U T - T O - S T A T E S T A B I L I T Y A N D E Q U I L I B R I U M T R A C K I N G F O R S T O C H A S T I C C O N T R AC T I N G D Y N A M I C S T E C H N I C A L R E P O RT Y u Kawano Graduate School of Advanced Science and Engineering, Hiroshima Univ ersity , Higashi-Hiroshima, 739-8527, JP , ykawano[at]hiroshima-u.ac.jp ∗ Simone Betteti Robust and Intelligent Autonomous Systems Lab, The Italian Institute of Artificial Intelligence for Industry (AI4I), T orino, 10129, IT , simone.betteti[at]ai4i.it Alexander Davydo v Department of Mechanical Engineering and Ken K ennedy Institute, Rice Univ ersity , Houston, TX, 77005, US, davydov[at]rice.edu Francesco Bullo Center for Control, Dynamical Systems and Computation, Univ ersity of California at Santa Barbara, Santa Barbara, CA, 93106 US, bullo[at]ucsb.edu † A B S T R AC T In this paper , we study the contractivity of nonlinear stochastic dif ferential equations (SDEs) driv en by deterministic inputs and Brownian motions. Giv en a weighted ℓ 2 -norm for the state space, we show that an SDE is incrementally noise- and input-to-state stable if its vector field is uniformly contracting in the state and uniformly Lipschitz in the input. This result is applied to error estimation for time-v arying equilibrium tracking in the presence of noise af fecting both the system dynamics and the input signals. W e consider both Ornstein–Uhlenbeck processes modeling unbounded noise and Jacobi dif fusion processes modeling bounded noise. Finally , we turn our attention to the associated Fokker–Planck equation of an SDE. For this context, we pro ve incremental input-to-state stability with respect to an arbitrary p -W asserstein metric when the drift vector field is uniformly contracting in the state and uniformly Lipschitz in the input with respect to an arbitrary norm. Keyw ords Stochastic processes · Equilibrium tracking · Input-to-state stability · Contraction theory 1 Introduction Equilibrium tracking is the ability of an input-driven dynamical system to track an instantaneous input-dependent equilibrium point as the input varies with time. This property is critically important for dynamical systems solving optimization problems, where the equilibrium point corresponds to the optimizer of a time-v arying objective function. Recent work [ Davydo v et al. , 2025 ] has established equilibrium tracking guarantees for deterministic contracting dynamical systems. The key result is that deterministic contracting systems can maintain bounded tracking errors that depend e xplicitly on the rate of change of the input signal. These results pro vide a rigorous foundation for understanding how optimization algorithms perform in dynamic en vironments. In this work, we extend equilibrium tracking theory to stochastic differential equations, characterizing the effect of noise both in the system dynamics and in the input signal. While deterministic equilibrium tracking has been successfully applied to robotic motion planning and ener gy management systems, these and other applications ine vitably ∗ YK is supported in part by JST FOREST Program Grant Number JPMJFR222E. † FB is supported in part by AFOSR grant F A9550-22-1-0059. FB w ould lik e to thank Anand Gokhale for insightful con versations. Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T face stochastic disturbances arising from environmental fluctuations, measurement noise, and model uncertainties. Understanding how noise de grades tracking performance is essential for designing robust optimization algorithms and verifying safe beha vior in practical settings. Our analysis for SDEs quantifies this degradation through e xplicit bounds on the expected tracking error as function of both the intensity of stochastic perturbations and the geometric properties of the equilibrium manifold. Literatur e r eview: Contraction theory has recently established itself as a computationally-friendly modular rob ust stability notion for dynamical systems and algorithms; e.g., see [ Lohmiller and Slotine , 1998 , Bullo , 2026 ]. An important class of contracting dynamics is giv en by continuous-time solvers of con vex optimization problems: gradient descent [ Kachurovskii , 1960 ], primal-dual dynamics [ Qu and Li , 2019 ], distributed optimization and Nash seeking dynamics [ Gokhale et al. , 2023 ], and safe monotone flo ws [ Allibhoy and Cortés , 2025 ]. Asymptotic and exponential stability of dynamical systems solving con ve x optimization problems (not necessarily contracting) has long been studied, e.g., see [ W ang and Elia , 2011 , Cherukuri et al. , 2017 , Cortés and Niederländer , 2019 , Dhingra et al. , 2019 ] among many others. A main reference on stochastic contracting dynamical systems is [ Pham et al. , 2009 ]; Theorem 2 and Corollary 1 therein establish noise-to-state-stability (NSS) for systems without input; see also the revie w in [ Tsukamoto et al. , 2021 , Section 2.2.2]. Extensions include: [ T abareau et al. , 2010 ] on how synchronization protects from noise, [ Boffi and Slotine , 2020 , Theorem 1] on continuous-time distrib uted stochastic gradient, [ Ngoc , 2021 ] on exponential contractivity in mean square, and [ Ngoc and Hieu , 2026 ] on general decay rates. In a distinct line of in vestigation, Ahmad and Raha [ 2010 ] define the stochastic logarithmic norm and Aminzare [ 2022 ] the stochastic logarithmic Lipschitz constant to study Itô SDEs with multiplicativ e noise; see also [ Aminzare and Sriv astava , 2022 ]. Another related set of results concerns the contracti vity of the Fokker -Planck equation with respect to W asserstein metrics. When the drift is the gradient descent of a strongly con ve x function, the Gibbs distribution is a fixed point of the Fokker-Planck equation and that the measure dynamics is contracting with respect to the W asserstein metric, e.g., see [ V illani , 2009 , Chapter 2] and [ Bolley et al. , 2012 , Introduction]. Recent related works on contracti vity in the W asserstein metric include [ Bouvrie and Slotine , 2019 , Conger et al. , 2025 ]; specifically , Bouvrie and Slotine [ 2019 ] in vestigates the ef fect of distinct Bro wnian motions in W asserstein space. T ime-varying con vex optimization problems hav e been extensi vely studied; see the authoritati ve revie ws [ Simonetto et al. , 2020 , Dall’Anese et al. , 2020 ]. The work [ Davydo v et al. , 2025 ] offers a comprehensi ve treatment of equilibrium tracking and continuous-time solvers for time-varying con ve x optimization, demonstrating applications to the design and analysis of safety filters and control barrier functions. Theoretical advances in equilibrium tracking include: [ Marvi et al. , 2024 ] on the design of control barrier proximal dynamics (CBPD), a contracting dynamics with a particularly fa vorable computational structure, and [ Marvi et al. , 2025a ] on the ef fects of discretization and quantization of CBPD in safety filters. Practical applications include: robot collision av oidance [ Fazlyab et al. , 2018 , Da vydov et al. , 2025 ], and battery management problems with electro-thermal constraints [ Marvi et al. , 2024 , 2025b ]. Contributions: In Section 3 , we establish an incremental noise- and input-to-state stability (NISS) property for SDEs. Our treatment generalizes known earlier results on (i) the incremental ISS property of contracting dynamics with input [ Bullo , 2026 , Chapter 3], (ii) NSS of contracting dynamics [ Pham et al. , 2009 ], and (iii) the mean-square response of Ornstein-Uhlenbeck processes. Given a weighted ℓ 2 norm on the state space, we consider vector fields that are uniformly contracting with respect to the state and uniformly Lipschitz with respect to the input signal. For the mean-square distance between trajectories, we provide a bound valid for all times and we simplify it in the limit of large times. Additionally , we consider both the case of two stochastic trajectories and the case of one stochastic and one deterministic trajectory . Section 4 contains the main result of this paper . This section extends equilibrium tracking theory from deterministic to stochastic settings, establishing rigorous bounds for tracking errors when both system dynamics and input signals are subject to noise. W e consider noisy contracting dynamics driven by two distinct noise processes: Ornstein-Uhlenbeck (OU) processes for unbounded noise and Jacobi dif fusion (JD) processes for bounded noise scenarios. For each noise model, the analysis cov ers three increasingly more complex scenarios: (1) deterministic inputs tracking deterministic equilibrium curves, (2) stochastic inputs tracking deterministic curves, and (3) stochastic inputs tracking stochastic curves. This hierarchy provides a complete characterization of equilibrium tracking under uncertainty . Our theorems establish explicit, computable upper bounds on the expected squared tracking error in terms of fundamental system parameters, including the contraction rate, the input Lipschitz constant, the multiple rele v ant noise intensities, and the rate of change of the reference trajectory . These bounds rev eal how tracking performance degrades with increasing noise and input variation. Finally , Section 5 considers the Fokk er-Planck equation in W asserstein metric space. W e extend the classic results on contractivity of the F okker-Planck equation when the drift is the gradient descent of a strongly conv ex function. Giv en 2 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T an arbitrary norm in the state space, we prove incremental input-to-state stability with respect to arbitrary p -W asserstein metrics for drift vector fields that are uniformly contracting in the state and uniformly Lipschitz in the input signal. P aper organization W e re view preliminary concepts about contracti vity , equilibrium tracking and stochastic differential equations in Section 2 . Section 3 presents our results on incremental noise- and input-to-state stability . Section 4 discusses stochastic equilibrium tracking. Section 5 discusses incremental input-to-state stability in W asserstein space. W e provide some concluding remarks in Section 6 and proofs of main theorems in the Appendices. 2 Preliminaries 2.1 Notation The field of real numbers is denoted by R . The n -component vector and n × m -matrix whose all components are 0 are denoted by 0 n and 0 n × m , respectiv ely . The n × n identity matrix is denoted by I n . The Hadamard product (i.e. element-wise product) of two vectors x, y ∈ R n is denoted by x ⊙ y . F or vectors x, y ∈ R n , x < y means the element-wise inequality , i.e., x i < y i for all i = 1 , . . . , n . For symmetric matrices P , Q ∈ R n × n , P ≻ Q means that P − Q is positiv e definite. For P ≻ 0 n × n , ∥ · ∥ 2 ,P 1 / 2 denote the P -weighted ℓ 2 -norm, i.e., ∥ x ∥ 2 ,P 1 / 2 := √ x ⊤ P x for x ∈ R n . When P = I n , the ℓ 2 -norm is denoted by ∥ · ∥ 2 . Also, the induced matrix ℓ 2 -norm is denoted by ∥ · ∥ 2 . The one-sided Lipschitz constant of a mapping F : R n → R n with respect to norm ∥ · ∥ 2 ,P 1 / 2 , denoted by osLip 2 ,P 1 / 2 ( F ) is the minimum constant b ∈ R satisfying ( F ( y ) − F ( x )) ⊤ P ( y − x ) ≤ b ∥ y − x ∥ 2 2 ,P 1 / 2 for all ( x, y ) ∈ R n × R n . The Lipschitz constant of a map F : U → R n from normed space ( ∥ · ∥ U , U ) to normed space ( ∥ · ∥ 2 ,P 1 / 2 , R n ) , denoted by Lip U → 2 ,P 1 / 2 ( F ) is the minimum constant ℓ ∈ R satisfying ∥ F ( v ) − F ( u ) ∥ 2 ,P 1 / 2 ≤ ℓ ∥ v − u ∥ U for all ( u, v ) ∈ U × U . 2.2 Input-to-State Stability of Deterministic Systems and Equilibrium T racking In this subsection, we recall [ Bullo , 2026 , Corollary 3.17] for incremental input-to-state stability (ISS) and equilibrium tracking [ Davydo v et al. , 2025 , Theorem 2] in the deterministic case. For the sake of self-containedness, we recall these existing result. Consider an ordinary differential equation (ODE): ˙ x ( t ) = F ( x ( t ) , u ( t )) . (1) Proposition 1 (Incremental input-to-state stability) . Given an input-dependent vector field F : R n × U → R n and deterministic measurable inputs u x , u y : R → U , wher e U ⊂ R m is compact, consider a pair of systems: ˙ x ( t ) = F ( x ( t ) , u x ( t )) , ˙ y ( t ) = F ( y ( t ) , u y ( t )) . Assume that ther e exists a matrix P = P ⊤ ≻ 0 n × n such that A1. Contraction in x : with r espect to the state x , the map F is str ongly infinitesimally contracting with rate c > 0 with r espect to norm ∥ · ∥ , uniformly in u ∈ U ; A2. Lipschitz in u : with r espect to the input u , the map F fr om normed space ( R n , ∥ · ∥ ) to normed space ( U , ∥ · ∥ U ) is Lipschitz continuous with constant ℓ > 0 , uniformly in x ∈ R n ; Then, for any finite t ≥ 0 , the ODE ( 1 ) is incr ementally ISS in the sense that for any trajectories x ( t ) and y ( t ) fr om initial conditions x (0) ∈ R n and y (0) ∈ R n subject to deterministic measurable inputs u x and u y , we have ∥ x ( t ) − y ( t ) ∥ ≤ ∥ x (0) − y (0) ∥ e − ct + ℓ Z t 0 e − c ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ U dτ . (2) W e no w denote u by θ to emphasize its role as a parameter and consider θ -dependent equilibrium x ⋆ ( θ ) that is a solution to F ( x ⋆ ( θ ) , θ ) = 0 n . Under uniform contractivity of F in x , for each θ ∈ U , θ -dependent equilibrium x ⋆ ( θ ) exists uniquely . Moreover , under Lipschitzness in θ , we can estimate a tracking error of x ( t ) − x ⋆ ( θ ( t )) as follows. Proposition 2 (Equilibrium T racking) . Given F : R n × U → R n , wher e U ⊂ R m is compact, we impose the same assumptions as Pr oposition 1 . Let x ⋆ ( θ ( t )) be a time-varying equilibrium curve, i.e ., F ( x ⋆ ( θ ( t )) , θ ( t )) ≡ 0 n . Then, 3 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T for any finite t ≥ 0 , for any initial condition x (0) ∈ R n , and continuously differ entiable θ : R → U , the solution to the ODE ( 1 ) satisfies ∥ x ( t ) − x ⋆ ( θ ( t )) ∥ ≤ ∥ x (0) − x ⋆ ( θ (0)) ∥ e − ct + ℓ c Z t 0 e − c ( t − τ ) ∥ ˙ θ ( τ ) ∥ U dτ , (3) and lim sup t →∞ ∥ x ( t ) − x ⋆ ( θ ( t )) ∥ ≤ ℓ c 2 lim sup t →∞ ∥ ˙ θ ( τ ) ∥ U . (4) 2.3 Stochastic Differential Equations In this paper , our goal is to generalize Propositions 1 and 2 for the ODE ( 1 ) to the stochastic dif ferential equation (SDE) in the so-called Itô sense: dx t = F ( x t , u ( t )) dt + Σ( x t , u ( t )) d B t , (5) where Σ : R n × U → R n × r is a state- and input-dependent matrix, and B t is an r -dimensional Brownian motion. It is customary to introduce the following standard assumption for the Lipschitzness and linear gro wth. Assumption 3 (Lipschitzness and linear gro wth) . In the SDE ( 5 ) , the drift vector field F and the dispersion matrix Σ satisfy the global Lipschitz and linear gr owth conditions: ∥ F ( x, u ) − F ( y , u ) ∥ 2 + ∥ Σ( x, u ) − Σ( y , u ) ∥ F ≤ L ∥ x − y ∥ 2 , (6a) ∥ F ( x, u ) ∥ 2 2 + ∥ Σ( x, u ) ∥ 2 F ≤ L (1 + ∥ x ∥ 2 2 ) (6b) for each x, y ∈ R n and every u ∈ U , wher e L is a positive constant, and ∥ X ∥ 2 F = trace( X ⊤ X ) is the F r obenius norm of the matrix X . ◁ Assumption 3 of Lipschitz continuity and linear growth for the drift vector field F and the dispersion matrix Σ guarantees the unique existence of a strong solution to ( 5 ) , where we refer to [ Karatzas and Shrev e , 2014 , Section 5.2] for the definition of a strong solution. Proposition 4 (Existence, uniqueness, and joint measurability of strong solutions) . F or an SDE ( 5 ) with globally Lipschitz and linearly gr owing drift and dispersion (Assumption 3 ), ther e exists the unique str ong solution { x t } t ≥ 0 with continuous sample paths and with finite second moment on finite horizons such that E x 0 [ ∥ x t ∥ 2 ] ≤ (1 + ∥ x 0 ∥ 2 )e (1+ L ) t 2 , ∀ t ≥ 0 , (7a) E x 0 [ ∥ x t ∥ 2 2 ] ≤ (1 + ∥ x 0 ∥ 2 2 )e (1+ L ) t , ∀ t ≥ 0 , (7b) wher e E x 0 [ · ] denotes the conditional expectation given x 0 . Mor eover , sup τ ∈ [0 ,t ] E [ ∥ x τ ∥ 2 ] < + ∞ for each finite t > 0 if the initial condition x 0 is squar e integr able, that is, E [ ∥ x 0 ∥ 2 ] < ∞ . ◁ W e ne xt introduce two central tools in the analysis of SDEs: the infinitesimal generator [ Øksendal , 2013 , Theorem 7.3.3] and Dynkin’ s formula [ Øksendal , 2013 , Theorem 7.4.1]. Definition 5 (Infinitesimal generator) . F or φ : R n → R of class C 2 , the infinitesimal generator L of the SDE ( 5 ) acting on φ is given by L φ ( x, u ) = ∂ φ ( x ) ∂ x F ( x, u ) + 1 2 trace  Σ( x, u ) ⊤ Hess( φ ( x ))Σ( x, u )  . (8) Proposition 6 (Dynkin’ s formula) . Consider an SDE ( 5 ) with globally Lipschitz and linearly growing drift and dispersion (Assumption 3 ) and with squar e-integr able initial condition x 0 , that is, E [ ∥ x 0 ∥ 2 ] < ∞ . Let φ : R n → R be of class C 2 , L be the infinitesimal gener ator of the SDE ( 5 ) acting on φ , and t > 0 be a finite time. Then, the unique str ong solution { x t } t ≥ 0 to the SDE satisfies E x 0 [ φ ( x t )] = φ ( x 0 ) + E x 0  Z t 0 ( L φ )( x τ , u ( τ )) dτ  , wher e E x 0 [ · ] denotes the conditional expectation given x 0 . ◁ 4 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T 2.4 Ornstein-Uhlenbeck Pr ocess An example of SDEs satisfying Assumption 3 of Lipschitz continuity and linear growth for the drift vector field F and the dispersion matrix Σ is an Ornstein-Uhlenbeck (OU) process: dx t = − cx t dt + σ √ n d B t , (9) where c, σ > 0 . The mean and expected squared norm of x t are E [ x t ] = E [ x 0 ]e − ct , E [ ∥ x t ∥ 2 2 ] = e − 2 ct ∥ x 0 ∥ 2 + σ 2 2 c (1 − e − 2 ct ) . (10) 2.5 Jacobi Diffusion Pr ocess Even if Assumption 3 of Lipschitz continuity and linear growth for the drift v ector field F and the dispersion matrix Σ does not hold, we can still work with weak solutions. Consider the following SDE: dx t = − c ( x t − θ ) dt + σ diag ( x t ⊙ ( a − x t )) 1 2 d B t , (11) where c > 0 and θ , a > 0 n . Each component of the SDE ( 11 ) is called a Jacobi diffusion (JD), and its stationary distribution is a β -distribution [ Coskun and K orn , 2021 ]. The JD ( 11 ) does not satisfy the Lipschitz assumption ( 6a ) , but does the linear growth assumption ( 6b ) . Thus, the JD ( 11 ) admits a weak solution [ Karatzas and Shrev e , 2014 , Problem 3.15], which satisfies ( 7 ) . Moreov er, sup τ ∈ [0 ,t ] E [ ∥ x τ ∥ 2 ] < + ∞ for each finite t > 0 if the initial condition x 0 is square integrable, that is, E [ ∥ x 0 ∥ 2 ] < ∞ . Also, by Feller’ s test for explosions [ Karatzas and Shrev e , 2014 , Theorem 5.29], the weak solution stay in the the interval ( 0 m , a ) := (0 , a 1 ) × · · · × (0 , a m ) almost surely if σ 2 ξ 2 c a ≤ θ ≤ 1 − σ 2 ξ 2 c ! a, σ 2 ξ < c (12) Therefore, Dynkin’ s formula can be applied on ( 0 m , a ) . From ( 12 ) , a JD process is helpful when modeling a bounded noise. 3 Incremental Noise- and Input-to-State Stability W e present the first main result of this paper for contracting SDEs. Specifically , we establish an incremental noise- and input-to-state stability (NISS) property for SDEs as a generalization of Proposition 1 for ISS of ODEs. Theorem 7 (Incremental noise- and input-to-state stability) . Given an input-dependent vector field F : R n × U → R n , a state- and input-dependent matrix Σ : R n × U → R n × r , and deterministic measurable inputs u x , u y : R → U , wher e U ⊂ R m is compact, consider the r ealizations with independent noises of dx t = F ( x t , u x ( t )) dt + Σ( x t , u x ( t )) d B x t , dy t = F ( y t , u y ( t )) dt + Σ( y t , u y ( t )) d B y t . W e impose Assumption 3 of Lipschitz continuity and linear gr owth for the drift vector field F and the dispersion matrix Σ . W e additionally assume that ther e exists a matrix P = P ⊤ ≻ 0 n × n such that A1. Contr action in x : there e xists c > 0 such that osLip 2 ,P 1 / 2 ( F ) ≤ − c , uniformly in u ∈ U ; A2. Lipsc hitz in u : there e xists ℓ > 0 such that Lip U → 2 ,P 1 / 2 ( F ) ≤ ℓ , uniformly in x ∈ R n ; A3. Bounded disper sion: the dispersion is uniformly bounded in the P -weighted F r obenius norm, namely σ 2 x := sup ( x,u ) ∈ R n ×U ∥ Σ( x, u ) ∥ 2 F ,P 1 / 2 = sup ( x,u ) ∈ R n ×U trace  Σ( x, u ) ⊤ P Σ( x, u )  < + ∞ . (13) Then, for each α ∈ (0 , 1) and any finite t ≥ 0 , 5 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T 1. the SDE ( 5 ) is incr ementally NISS in the sense that, for any two r ealizations x t and y t fr om random squar e inte grable initial conditions x 0 and y 0 and subject to deterministic measurable inputs u x and u y and differ ent r ealizations of the Br ownian motion, we have E  ∥ x t − y t ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − y 0 ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x c (1 − e − 2 cαt ) + 1 1 − α ℓ 2 2 c Z t 0 e − 2 cα ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ 2 U dτ , (14) and, without assuming the measurability of u x or u y , lim sup t →∞ E  ∥ x t − y t ∥ 2 2 ,P 1 / 2  ≤ 1 α σ 2 x c + 1 α (1 − α ) ℓ 2 4 c 2 lim sup t →∞ ∥ u x ( t ) − u y ( t ) ∥ 2 U ; (15) 2. for any solution y ( t ) to the ODE ( 1 ) and any r ealization x t of the SDE ( 5 ) fr om random square inte grable initial condition x 0 under deterministic measurable inputs u y and u x , we have E  ∥ x t − y ( t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − y (0) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c (1 − e − 2 cαt ) + 1 1 − α ℓ 2 2 c Z t 0 e − 2 cα ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ 2 U dτ , (16) and, without assuming the measurability of u x or u y , lim sup t →∞ E  ∥ x t − y ( t ) ∥ 2 2 ,P 1 / 2  ≤ 1 α σ 2 x 2 c + 1 α (1 − α ) ℓ 2 4 c 2 lim sup t →∞ ∥ u x ( t ) − u y ( t ) ∥ 2 U . Pr oof. (Proof of item 1 ) we consider the combined SDE:  dx t dy t  =  F ( x t , u x ( t )) F ( y t , u y ( t ))  dt +  Σ( x t , u x ( t )) 0 n × r 0 n × r Σ( y t , u y ( t ))  d B t , (17) where B t is a 2 r dimensional Bro wnian motion, i.e., we consider independent noise. For the combined SDE ( 17 ) , we consider a class C 2 function φ ( x, y ) := ∥ x − y ∥ 2 2 ,P 1 / 2 and compute the Hessian matrix: Hess  ∥ x − y ∥ 2 2 ,P 1 / 2  = Hess  x y  ⊤  P − P − P P   x y  ! = 2  P − P − P P  , and thus, from the bounded dispersion assumption (Assumption A3 ) trace  Σ( x, u x ) 0 n × r 0 n × r Σ( y , u y )  ⊤  P − P − P P   Σ( x, u x ) 0 n × r 0 n × r Σ( y , u y )  ! = trace(Σ( x, u x ) ⊤ P Σ( x, u x )) + trace(Σ( y , u y ) ⊤ P Σ( y , u y )) ≤ 2 sup ( x,u x ) ∈ R n ×U trace(Σ( x, u x ) ⊤ P Σ( x, u x )) = 2 σ 2 x . (18) Giv en the contractivity (with respect to x ) and Lipschitzness (with respect to u ) assumptions (Assumptions A1 and A2 ) on F and from ( 8 ) and ( 18 ), the infinitesimal generator on the function φ ( x, y ) = ∥ x − y ∥ 2 2 ,P 1 / 2 satisfies L∥ x t − y t ∥ 2 2 ,P 1 / 2 ≤ − 2 c ∥ x t − y t ∥ 2 2 ,P 1 / 2 + 2 ℓ ∥ x t − y t ∥ 2 ,P 1 / 2 ∥ u x ( t ) − u y ( t ) ∥ U + 2 σ 2 x a.s. (19) Next, we apply Dynkin’ s formula in Proposition 6 . For an y finite t ≥ s ≥ 0 , taking the conditional expectation gi ven x s and y s leads to E x s ,y s  ∥ x t − y t ∥ 2 2 ,P 1 / 2  − ∥ x s − y s ∥ 2 2 ,P 1 / 2 = E x s ,y s  Z t s L∥ x τ − y τ ∥ 2 2 ,P 1 / 2 dτ  ≤ − 2 c E x s ,y s  Z t s ∥ x τ − y τ ∥ 2 2 ,P 1 / 2 dτ  + 2 ℓ E x s ,y s  Z t s ∥ x τ − y τ ∥ 2 ,P 1 / 2 ∥ u x ( τ ) − u y ( τ ) ∥ U dτ  + Z t s 2 σ 2 x dτ a.s. , (20) 6 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T where the inequality follows from ( 19 ). From ( 7a ) and the triangular inequality , it follo ws that Z t s E x s ,y s [ ∥ x τ − y τ ∥ 2 ] dτ ≤ Z t s ( E x s [ ∥ x τ ∥ 2 ] + E y s [ ∥ y τ ∥ 2 ]) dτ ≤ (2 + ∥ x 0 ∥ 2 + ∥ y 0 ∥ 2 ) Z t s e (1+ L ) τ 2 dτ a.s. Thus, R t s E x s ,y s [ ∥ x τ − y τ ∥ 2 ,P 1 / 2 ] dτ is finite almost surely . Similarly , from ( 7b ) and ∥ x − y ∥ 2 2 ≤ 2( ∥ x ∥ 2 2 + ∥ y ∥ 2 2 ) , R t s E x s ,y s [ ∥ x τ − y τ ∥ 2 2 ,P 1 / 2 ] dτ is finite almost surely . Therefore, from T onelli’ s theorem for measurable non-negati ve functions [ Brezis , 2011 , Theorems 4.4-4.5], we can exchange the order of taking the time-integration and expectation in ( 20 ). Namely , we have E x s ,y s  ∥ x t − y t ∥ 2 2 ,P 1 / 2  − ∥ x s − y s ∥ 2 2 ,P 1 / 2 ≤ − 2 c Z t s E x s ,y s  ∥ x τ − y τ ∥ 2 2 ,P 1 / 2  dτ + 2 ℓ Z t s ∥ u x ( τ ) − u y ( τ ) ∥ U E x s ,y s  ∥ x τ − y τ ∥ 2 ,P 1 / 2  dτ + Z t s 2 σ 2 x dτ a.s. (21) Since √ · is a concav e function, Jensen’ s inequality [ Brezis , 2011 , Proposition 4.9] yields E x s ,y s [ ∥ x τ − y τ ∥ 2 ,P 1 / 2 ] ≤ q E x s ,y s  ∥ x τ − y τ ∥ 2 2 ,P 1 / 2  . (22) Denoting g ( t ) := E x s ,y s  ∥ x t − y t ∥ 2 2 ,P 1 / 2  , ( 21 ) and ( 22 ) lead to g ( t ) − g ( s ) ≤ Z t s  − 2 cg ( τ ) + 2 σ 2 x + 2 ℓ ∥ u x ( τ ) − u y ( τ ) ∥ U p g ( τ )  dτ a.s. , and consequently , g ( t ) − g ( s ) t − s ≤ 1 t − s Z t s  − 2 cg ( τ ) + 2 σ 2 x + 2 ℓ ∥ u x ( τ ) − u y ( τ ) ∥ U p g ( τ )  dτ a.s. for any t > s . Since g ( t ) is continuous, its Dini deri vati ve [ Bullo , 2026 , Chapter 2.4.2] is D + g ( s ) := lim t → s + g ( t ) − g ( s ) t − s ≤ − 2 cg ( s ) + 2 σ 2 x + 2 ℓ ∥ u x ( s ) − u y ( s ) ∥ U p g ( s ) a.s. Using non-negati vity of g ( s ) = ∥ x s − y s ∥ 2 2 ,P 1 / 2 , compute by using Y oung’ s inequality: 2 ℓ ∥ u x ( s ) − u y ( s ) ∥ U p g ( s ) ≤ 2 c (1 − α ) g ( s ) + ℓ 2 2 c (1 − α ) ∥ u x ( s ) − u y ( s ) ∥ 2 U a.s. , (23) where α ∈ (0 , 1) , and consequently , D + g ( s ) ≤ − 2 cαg ( s ) + 2 σ 2 x + ℓ 2 2 c (1 − α ) ∥ u x ( s ) − u y ( s ) ∥ 2 U a.s. Since this holds for arbitrary finite s ≥ 0 , the comparison principle [ Khalil , 2002 , Lemma 3.4] yields ∥ x t − y t ∥ 2 2 ,P 1 / 2 ≤ ∥ x 0 − y 0 ∥ 2 2 ,P 1 / 2 e − 2 cαt + σ 2 x cα (1 − e − 2 cαt ) + ℓ 2 2 c (1 − α ) Z t 0 e − 2 cα ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ 2 U dτ a.s. , where recall the compactness of U and the measurability of both u x and u y , guaranteeing the boundedness of the integral. T aking the expectation concludes ( 14 ). T o show ( 15 ), we compute an upper bound on the right-hand side of ( 14 ), yielding E  ∥ x t − y t ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − y 0 ∥ 2 2 ,P 1 / 2  e − 2 cαt + σ 2 x cα (1 − e − 2 cαt ) + ℓ 2 4 c 2 α (1 − α ) (1 − e − 2 cαt ) sup τ ∈ [0 ,t ] ∥ u x ( τ ) − u y ( τ ) ∥ 2 U . 7 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T This holds for any finite t ≥ 0 . The compactness of U implies that lim sup t →∞ ∥ u x ( τ ) − u y ( τ ) ∥ 2 U is alw ays finite e ven without assuming the measurability of u x or u y . Also, from the square inte grability assumption of the random initial conditions x 0 and y 0 , the right-hand side is finite for any t ≥ 0 . Therefore, we can take the limit superiors of both sides, leading to ( 15 ). (Proof of item 2 ) Instead of ( 17 ), consider  dx t dy t  =  F ( x t , u x ( t )) F ( y t , u y ( t ))  dt +  Σ( x t , u x ) 0 n × r 0 n × r 0 n × r  d B t . Also, instead of ( 18 ), compute trace  Σ( x, u x ) 0 n × r 0 n × r 0 n × r  ⊤  P − P − P P   Σ( x, u x ) 0 n × r 0 n × r 0 n × r  ! = trace(Σ( x, u x ) ⊤ P Σ( x, u x )) ≤ sup ( x,u x ) ∈ R n ×U trace(Σ( x, u x ) ⊤ P Σ( x, u x )) = σ 2 x . Thus, instead of ( 19 ), we hav e L∥ x t − z t ∥ 2 2 ,P 1 / 2 ≤ − 2 c ∥ x t − z t ∥ 2 2 ,P 1 / 2 + 2 ℓ ∥ x t − z t ∥ 2 ,P 1 / 2 ∥ u x ( t ) − u y ( t ) ∥ U + σ 2 x a.s. The rest of the proof is similar to that of item 1 . Remark 1 (Comparisons) . i. In the deterministic case when σ x = 0 , Theor em 7 r educes to Proposition 1 for ISS (note that Theor em 7 uses the squar ed norm to evaluate the second moment). The additional parameter α ∈ (0 , 1) arises from the application of Y oung’s inequality ( 23 ) for estimating an upper bound under the pr esence of both noise and deterministic inputs. The term 1 α σ 2 x c (1 − e − 2 cαt ) in ( 14 ) r epresents the effect of noise and is similar to the e xpected squared norm of an OU pr ocess ( 10 ) . In the denominator , we have c instead of 2 c because we consider two r ealizations x t and y t of SDE ( 5 ) . In fact, in ( 16 ) considering a r ealization x t of SDE ( 5 ) , the term repr esenting the effect of noise is 1 α σ 2 x 2 c (1 − e − 2 cαt ) . ii. In the stochastic case without deterministic input u , or mor e precisely when u x ( t ) ≡ u y ( t ) , Theorem 7 r ecovers [ Pham et al. , 2009 , Theor em 2] for noise-to-state stability in the constant metric case. In summary , Theor em 7 can be viewed as a generalization of Pr oposition 1 and [ Pham et al. , 2009 , Theor em 2]. ◁ 4 Stochastic Equilibrium T racking In this section, we generalize Proposition 2 for equilibrium tracking to the stochastic setting. As discussed in the Introduction, equilibrium tracking has been applied in sev eral deterministic applications. In practice, howe ver , both systems and signals are subject to stochastic fluctuations and noise. T o address this, we deriv e bounds for equilibrium tracking performance in the presence of stochastic disturbances. As stochastic disturbances, we consider two dif ferent noises driv en by A) OU process; and B) JD process, where JD process is useful to represent bounded noise. 4.1 Driven by Or nstein-Uhlenbeck Processes W e consider equilibrium tracking for noisy contracting dynamics: dx t = F ( x t , u t ) dt + Σ( x t , u t ) d B x t , (24) where contraction rate is osLip 2 ,P 1 / 2 ( F ) ≤ − c < 0 uniformly in u ∈ U , Lipschitz constant is Lip U → 2 ,P 1 / 2 ( F ) ≤ ℓ uniformly in x ∈ R n , and dispersion is uniformly bounded on σ 2 x with respect to the P -weighted Frobenius norm, i.e., ∥ Σ( x, u ) ∥ 2 F ,P 1 / 2 ≤ σ 2 x . Our objective is to estimate tracking errors x t − x ⋆ ( v t ) for parameter-dependent equilibrium x ⋆ ( v t ) , i.e., F ( x ⋆ ( v t ) , v t ) = 0 n . There are sev eral possible scenarios: 1) deterministic input u t = θ ( t ) and deterministic equilibrium curve x ⋆ ( θ ( t )) , i.e., u t = v t = θ ( t ) ; 2) stochastic input u t = θ ( t ) + ξ t and deterministic equilibrium curve x ⋆ ( θ ( t )) , i.e., v t = θ ( t ) ; 3) stochastic input u t = θ ( t ) + ξ t and stochastic equilibrium curve x ⋆ ( θ ( t ) + ξ t ) , i.e., u t = v t = θ ( t ) + ξ t . In this subsection, we assume that ξ t is driv en by an OU process. Namely , u t = θ ( t ) + ξ t , dξ t = − cξ t dt + σ ξ √ m d B ξ t . (25) W e summarize the three main results of this subsection, where roles of parameter α and constant h OU , named the Itô drift correction constant associated with the OU generator , are explained in Remark 2 below: 8 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T 1. (Theorem 8 ) deterministic input u t = θ ( t ) , tracking a deterministic curve ∥ x t − x ⋆ ( θ ( t )) ∥ : lim sup t →∞ E  ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2  ≤ 1 α σ 2 x 2 c + 1 4 α (1 − α ) ℓ 2 c 4 lim sup t →∞ ∥ ˙ θ ( t ) ∥ 2 U . (26) 2. (Theorem 9 ) stochastic input u t = θ ( t ) + ξ t , tracking a deterministic curve ∥ x t − x ⋆ ( θ ( t )) ∥ : lim sup t →∞ E [ ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 ] ≤ 1 α σ 2 x c + 1 α (1 − α ) ℓ 2 c 4 lim sup t →∞ ∥ ˙ θ ( t ) ∥ 2 2 + 1 α ℓ 2 c 2 σ 2 ξ c . (27) 3. (Theorem 10 ) stochastic input u t = θ ( t ) + ξ t , tracking a stochastic curve ∥ x t − x ⋆ ( u t ) ∥ : lim sup t →∞ E [ ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P ] ≤ 1 α σ 2 x 2 c + 1 α (1 − α ) ℓ 2 c 4 lim sup t →∞ ∥ ˙ θ ( t ) ∥ 2 2 + 1 α (1 − α ) (2 − α ) ℓ 2 c 2 σ 2 ξ 2 c + h 2 OU 2 σ 4 ξ 4 c 2 ! , (28) where h OU := 1 m sup u ∈ R m           trace  Hess( x ⋆ 1 ( u ))  . . . trace  Hess( x ⋆ n ( u ))            2 ,P 1 / 2 . (29) Remark 2 (Comparisons) . i. Theor em 8 can be re gar ded as a generalization of Pr oposition 2 , extending de- terministic equilibrium trac king to the stochastic setting. As noted pre viously for Theorem 7 on NISS, in the stochastic case, we use the squar ed norm to evaluate the second moment. The additional parameter α ∈ (0 , 1) arises fr om applying Y oung’ s inequality to estimate an upper bound in the pr esence of both noise and deterministic input. Compar ed with ( 4 ) for the deterministic equilibrium trac king, the tracking error bound ( 33 ) contains additional term 1 α σ 2 x 2 c caused by noise. As afor ementioned, this term corresponds to the expected squar ed norm of the OU pr ocess ( 10 ) . ii. Theor ems 9 and 10 can be r e garded as two differ ent g eneralizations of Theor em 8 . In both cases, the additional term ℓ 2 c 2 σ 2 ξ c arises fr om the stochastic input ξ t . The coefficient ℓ 2 c 2 is standar d in ISS estimates. In addition, by means of a second-moment analysis, we obtain the bounds for the case in which ∥ · ∥ U is the ℓ 2 -norm. iii. In Theorem 10 , there is a further additional constant h OU . This is called the Itô drift correction constant associated with the OU gener ator because it corr esponds to the Itô drift corr ection associated with the OU generator that appear s when computing the SDE satisfied by stochastic equilibrium x ⋆ ( u t ) : dx ⋆ ( u t ) = F ( x ⋆ ( u t ) , u t ) dt + v ( θ ( t ) , ξ t , u t ) dt + Λ( u t ) d B ξ t , (30) wher e v ( θ ( t ) , ξ t , u t ) = ∂ x ⋆ k ∂ u ( u t )( ˙ θ ( t ) − cξ t ) + 1 2 σ 2 ξ m    trace  Hess( x ⋆ 1 ( u t ))  . . . trace  Hess( x ⋆ n ( u t ))     (31a) Λ( u t ) = σ ξ √ m ∂ x ⋆ ∂ u ( u t ) . (31b) This SDE is obtained by the Itô formula to x ⋆ ( u t ) . F or mor e details, see Appendix B . ◁ 4.1.1 Deterministic Input, T racking a Deterministic Curve W e first focus on deterministic parameter -dependent equilibrium x ⋆ ( θ ) , i.e., F ( x ⋆ ( θ ) , θ ) = 0 n and then estimate the tracking error x t − x ⋆ ( θ ( t )) for the solution to the following SDE: dx t = F ( x t , θ ( t )) dt + Σ( x t , θ ( t )) d B x t . (32) Theorem 8 (Stochastic Equilibrium T racking: Deterministic Input, T racking a Deterministic Curve) . Given an input- dependent vector field F : R n × U → R n , and matrix Σ : R n × R m → R n × r , wher e U ⊂ R m is compact, we impose the same assumptions as Theor em 7 (i.e., i) Assumption 3 of Lipschitz continuity and linear gr owth for the drift vector 9 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T field F and the disper sion matrix Σ ; and ii) the existence of a matrix P = P ⊤ ≻ 0 n × n satisfying Assumptions A1 t o A3 for contractivity of F , Lipschitzness of F , and the boundedness of the dispersion of Σ ). Let x ⋆ ( θ ( t )) be a time-varying equilibrium curve, i.e ., F ( x ⋆ ( θ ( t )) , θ ( t )) ≡ 0 n . Then, for eac h α ∈ (0 , 1) and any finite t ≥ 0 , for any r ealization x t of the SDE ( 32 ) fr om random squar e inte grable initial condition x 0 under continuously differ entiable deterministic parameter θ : R → U , we have E  ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( θ (0)) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c (1 − e − 2 cαt ) + 1 1 − α ℓ 2 2 c 3 Z t 0 e − 2 cα ( t − τ ) ∥ ˙ θ ( t ) ∥ 2 U dτ , (33) and ( 26 ) . Pr oof. Consider an auxiliary dynamics of the SDE ( 32 ): dx t = F ( x t , θ ( t )) dt + v ( t ) dt + Σ( x t , θ ( t )) d B x t . (34) When v ( t ) ≡ 0 n , this is nothing but ( 32 ) . When v ( t ) ≡ ˙ x ⋆ ( θ ( t )) and noise free (i.e., d B x t is identical to zero), we hav e x t = x ⋆ ( θ ( t )) . Namely , x ⋆ ( θ ( t )) is a solution to ( 34 ). Applying item 2 of Theorem 7 to the auxiliary SDE ( 34 ) as v ( t ) as the input, we ha ve E  ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( θ (0)) ∥ 2 2 ,P 1 / 2  e − 2 cαt + σ 2 x 2 cα (1 − e − 2 cαt ) + 1 2 c (1 − α ) Z t 0 e − 2 cα ( t − τ ) ∥ ˙ x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 dτ , From ∥ ˙ x ⋆ ( θ ( τ )) ∥ 2 ,P 1 / 2 ≤ ℓ c ∥ ˙ θ ( τ ) ∥ U , we hav e ( 33 ). Finally , ( 26 ) is obtained by taking the limit superior of ( 33 ). 4.1.2 Stochastic Input, T racking a Deterministic Curve Theorem 9 (Stochastic Equilibrium T racking with OU process: Stochastic Input, T racking a Deterministic Curve) . Given F : R n × U → R n and Σ : R n × R m → R n × r , wher e U ⊂ R m is compact, we impose the same assumptions as Theor em 7 , where ∥ · ∥ U = ∥ · ∥ 2 . In addition, we assume that A4. Normalization of P : ∥ P ∥ 2 = 1 . Let x ⋆ ( θ ( t )) be a time-varying equilibrium curve. Then, for each α ∈ (0 , 1) and any finite t ≥ 0 , for any r ealiza- tion ( x t , ξ t ) of the SDEs ( 24 ) and ( 25 ) fr om random squar e inte grable initial condition ( x 0 , ξ 0 ) under continuously differ entiable deterministic parameter θ : R → U , we have E [ ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 ] ≤ E [ ∥ x 0 − x ⋆ ( θ (0)) ∥ 2 2 ,P 1 / 2 ]e − cαt + 1 α σ 2 x c (1 − e − cαt ) + 1 1 − α ℓ 2 c 3 Z t 0 e − cα ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + ℓ 2 c 2 E [ ∥ ξ 0 ∥ 2 2 ]e − cαt + 1 α ℓ 2 c 2 σ 2 ξ c (1 − e − cαt ) , (35) and ( 27 ) . Pr oof. The proof is in Appendix A . 4.1.3 Stochastic Input, T racking a Stochastic Curve Theorem 10 (Stochastic Equilibrium Tracking with OU process: Stochastic Input, T racking a Stochastic Curve) . Given F : R n × U → R n , and Σ : R n × R m → R n × r , wher e U ⊂ R m is compact, we impose the same assumptions as Theor em 8 . Let x ⋆ ( u t ) be a stochastic equilibrium curve, wher e u t is generated by ( 25 ) . In addition, we assume that A5. T wice continuous differ entiability of F : F : R n × U → R n is of class C 2 ; A6. Boundedness and Lipshitzness of the Hessian of x ⋆ i ( v ) : Hess( x ⋆ i ( v )) is bounded and Lipschitz on R m for all i = 1 , . . . , m . 10 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T Then, for each α ∈ (0 , 1) and any finite t ≥ 0 , for any weak solution ( ξ t , x t ) of SDE ( 24 ) and ( 25 ) fr om random squar e inte grable initial conditions ( x 0 , ξ 0 ) under continuously differ entiable deterministic parameter θ : R → U , we have E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( u 0 ) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c (1 − e − 2 cαt ) + 2 1 − α ℓ 2 c 3 Z t 0 e − 2 cα ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + 1 (1 − α ) 2 ℓ 2 2 c 2 E  ∥ ξ 0 ∥ 2 2  ( e − 2 cαt − e − 2 ct ) + 1 α ℓ 2 c 2 σ 2 ξ 2 c + 1 1 − α h 2 OU 2 σ 4 ξ 4 c 2 ! (1 − e − 2 cαt ) + 1 1 − α ℓ 2 c 2 σ 2 ξ 2 c  1 α − 1 α (1 − α ) e − 2 cαt + 1 1 − α e − 2 ct  , (36) and ( 28 ) . Pr oof. The proof is in Appendix B . 4.2 Driven by J acobi Diffusion Processes T o deal with a case where stochastic input u t is bounded, in this subsection we consider noise driv en by a multi variate JD process: du t = − c ( u t − θ ( t )) dt + σ u diag( u t ⊙ ( a − u t )) 1 2 d B u t . (37) W e summarize the two main results of this subsection for noise driven by a JD process, which are parallel to those obtained in the OU case. In each case, we consider the noisy contracting dynamics ( 24 ) characterized by contraction rate c , Lipschitz constant ℓ , and dispersion bound σ 2 x . The parameter α emerges from the application of Y oung’ s inequality to bound the combined ef fect of stochastic disturbances and deterministic inputs, as in the OU case. Also, the role of constant h JD , named the Itô drift correction constant associated with the JD generator , is simitar to that of h OU for the OU process. 1. (Theorem 11 ) stochastic input u t , tracking a deterministic curve ∥ x t − x ⋆ ( θ ( t )) ∥ : lim sup t →∞ E [ ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 ] ≤ 1 α σ 2 x c + 1 α (1 − α ) ℓ 2 c 4 lim sup t →∞ ∥ ˙ θ ( t ) ∥ 2 2 + 1 α ℓ 2 c 2 ∥ a ∥ 2 2 4 σ 2 u c . (38) 2. (Theorem 12 ) stochastic input u t , tracking a stochastic curve ∥ x t − x ⋆ ( u t ) ∥ : lim sup t →∞ E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ 1 α σ 2 x 2 c + 1 2 α (1 − α ) ℓ 2 c 4 lim sup t →∞ ∥ ˙ θ ( t ) ∥ 2 2 + 1 α (1 − α )  (4 − 3 α ) ℓ 2 c 2 ∥ a ∥ 2 2 4 σ 2 u 2 c + h 2 JD 2 σ 4 u 4 c 2  , (39) where h JD := sup θ ∈ ( 0 m ,a )      m X i =1 u i ( a i − u i ) ∂ 2 x ⋆ ( u ) ∂ u 2 i      2 ,P 1 / 2 . (40) Although the JD process has a dif ferent structure form the OU process, the basic structure of the error bounds is similar . In the JD case, the SDE satisfied by stochastic equilibrium x ⋆ ( u t ) is dx ⋆ ( u t ) = F ( x ⋆ ( u t ) , u t ) dt + v ( θ ( t ) , u t ) dt + Λ( u t ) d B ξ t , where v ( θ ( t ) , u t ) = ∂ x ⋆ ∂ θ ( u t )( − c ( u t − θ ( t ))) + 1 2 σ 2 u m X i =1 u t,i ( a i − u t,i ) ∂ 2 x ⋆ ( u t ) ∂ u 2 t,i (41a) Λ( u t ) = σ u ∂ x ⋆ ∂ u ( u t ) diag ( u t ⊙ ( a − u t )) 1 2 . (41b) Thus, h JD is called the Itô drift correction constant associated with the JD generator . 11 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T 4.2.1 Stochastic Input, T racking a Deterministic Curve Theorem 11 (Stochastic Equilibrium T racking with respect to Jacobi Dif fusion Equilibrium Curve: Stochastic Input, T racking a Deterministic Curve) . Given F : R n × U → R n , and Σ : R n × R m → R n × r , wher e U ⊂ R m is compact, we impose the same assumptions as Theor em 8 . In addition, we assume that A7. F eller condition: for any t ≥ 0 , σ 2 u 2 c a ≤ θ ( t ) ≤  1 − σ 2 u 2 c  a. Let x ⋆ ( θ ( t )) be a time-varying equilibrium curve . Then, for each α ∈ (0 , 1) and any finite t ≥ 0 , for any r ealization ( x t , u t ) of the SDE ( 24 ) with ( 37 ) fr om r andom squar e integr able initial condition ( x 0 , u 0 ) under continuously differ entiable deterministic parameter θ : R → U , we have E [ ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 ] ≤ E [ ∥ x 0 − x ⋆ ( θ (0)) ∥ 2 2 ,P 1 / 2 ]e − cαt + 1 α σ 2 x c (1 − e − cαt ) + 1 1 − α ℓ 2 c 3 Z t 0 e − cα ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + ℓ 2 c 2 E [ ∥ ξ 0 ∥ 2 2 ]e − cαt + 1 α ℓ 2 c 2 ∥ a ∥ 2 2 4 σ 2 u c (1 − e − cαt ) , (42) and ( 38 ) . Pr oof. The proof is in Appendix C . 4.2.2 Stochastic Input, T racking a Stochastic Curve Theorem 12 (Stochastic Equilibrium T racking with respect to Jacobi Dif fusion Equilibrium Curve: Stochastic Input, T racking a Stochastic Curve) . Given F : R n × U → R n , and Σ : R n × R m → R n × r , wher e U ⊂ R m is compact, we impose the same assumptions as Theor em 11 and Assumption A5 of Theor em 10 . Let x ⋆ ( u t ) be a stochastic equilibrium curve, where u t is generated by ( 37 ) . Then, for each α ∈ (0 , 1) and any finite t ≥ 0 , for any weak solution ( x t , u t ) of the SDE ( 24 ) with ( 37 ) fr om random squar e inte grable initial condition ( x 0 , u 0 ) under continuously diff er entiable deterministic parameter θ : R → U , we have E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( u 0 ) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c (1 − e − 2 cαt ) + 1 1 − α ℓ 2 c 2 Z t 0 e − 2 cα ( t − τ ) Z τ 0 e − c ( τ − r ) ∥ ˙ θ ( r ) ∥ 2 2 dr dτ + 1 1 − α ℓ 2 c E  ∥ u 0 − θ (0) ∥ 2 2  Z t 0 e − 2 cα ( t − τ ) e − cτ dτ + 1 α  ℓ 2 c 2 3 ∥ a ∥ 2 2 4 σ 2 u 2 c + 1 1 − α h 2 JD 2 σ 4 u 4 c 2  (1 − e − 2 cαt ) + 1 1 − α ℓ 2 c ∥ a ∥ 2 2 4 σ 2 u c Z t 0 e − 2 cα ( t − τ ) (1 − e − cτ ) dτ . (43) and, if α ≥ 1 / 2 , ( 39 ) . Pr oof. The proof is in Appendix D . 5 Incremental input-to-state stability in W asserstein space In this section, as a different ISS property of SDEs, we study ISS of probability densities in W asserstein spaces. Similarly to the pre vious sections, we assume Lipschitz continuity and linear gro wth for the drift vector field and the dispersion matrix as well as the contractivity and Lipschitz properties of the drift vector field. Ho wev er, we do not impose boundedness of dispersion 12 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T W e consider a simpler SDE than ( 5 ): dx t = F ( x t , u ( t )) dt + ˆ Σ( t ) d B t , (44) where ˆ Σ : R → R n × r is time-dependent matrix. The time evolution of the probability density µ ( t, x ) of the process x t gov erned by the SDE ( 44 ), is gi ven by the F okker -Planck equation (or called the K olmogorov forward equation): ∂ µ ( t, x ) ∂ t = − n X i =1 ∂ ( µ ( t, x ) F i ( x, u ( t ))) ∂ x i + 1 2 trace  ˆ Σ( t ) ⊤ Hess x ( µ ( t, x )) ˆ Σ( t )  . (45) As a distance of two probability measures, we employ the W asserstein distance. Definition 13 (W asserstein metric) . Given an arbitrary norm ∥ · ∥ and p ∈ [1 , ∞ ] , the p -W asserstein distance between the pr obability measur es µ x and µ y with finite p -moments on R n is defined by W p ( µ x , µ y ) := inf π ∈ Π( µ x , µ y )  E π [ ∥ x − y ∥ p ]  1 p , wher e Π( µ x , µ y ) is the set of joint probability measur es with mar ginals µ x and µ y , that is µ x ( x ) = R R n π ( x, y ) dy and µ y ( y ) = R R n π ( x, y ) dx . ◁ As the main result of this section, we study incremental input-to-state stability in a W asserstein distance as follows. Theorem 14 (Incremental input-to-state stability in W asserstein distance) . Given an input-dependent vector field F : R n × U → R n , time-dependent matrix ˆ Σ : R → R n × r , and deterministic measurable inputs u x , u y : R → U , wher e U ⊂ R m is compact, consider the r ealizations driven by the same Br ownian motion: dx t = F ( x t , u x ( t )) dt + ˆ Σ( t ) d B t , (46a) dy t = F ( y t , u y ( t )) dt + ˆ Σ( t ) d B t , (46b) and the corr esponding F okker -Planck equations ( 45 ) . W e impose Assumption 3 of Lipsc hitz continuity and linear gr owth for the drift vector field F and the dispersion matrix ˆ Σ . W e additionally assume that ther e exists a norm ∥ · ∥ such that A1. Contraction in x : with r espect to the state x , the map F is str ongly infinitesimally contracting with rate c > 0 with r espect to norm ∥ · ∥ , uniformly in u ∈ U ; A2. Lipschitz in u : with r espect to the input u , the map F fr om normed space ( R n , ∥ · ∥ ) to normed space ( U , ∥ · ∥ U ) is Lipschitz continuous with constant ℓ > 0 , uniformly in x ∈ R n . Then, for each p ∈ [1 , ∞ ] and any finite t ≥ 0 , any pair of solutions to the F okk er-Planc k equations µ x t ( x ) = µ x ( t, x ) and µ y t ( y ) = µ y ( t, y ) with the initial distributions µ x (0 , x ) = µ x 0 ( x ) and µ y (0 , y ) = µ y 0 ( y ) with finite p -moments satisfies W p ( µ x t , µ y t ) ≤ e − ct W p ( µ x 0 , µ y 0 ) + ℓ Z t 0 e − c ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ U d τ . (47) Pr oof. Let x t and y t denote the solution to the SDE ( 46 ) with x 0 drawn from µ x 0 and y 0 drawn from µ y 0 , respecti vely and further assume x t and y t are driv en by the same realization of Brownian motion. Since x t and y t are driv en by the same Bro wnian motion, the process x t − y t is subject to no process noise. From the contracti vity (with respect to z ) and Lipschitzness (wit respect to u ) assumptions (Assumptions A1 and A2 ) on F , we obtain ∥ x t − y t ∥ ≤ e − ct ∥ x 0 − y 0 ∥ + ℓ Z t 0 e − c ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ U d τ a.s. (48) Let π 0 denote a joint probability distribution with mar ginals µ x 0 and µ y 0 , and let π t be the corresponding joint distribution for the joint solution ( x t , y t ) . W e take the expectation E π 0 [ · ] of ( 48 ) . Select X = ∥ x t − y t ∥ , Y = e − ct ∥ x 0 − y 0 ∥ , and b = ℓ R t 0 e − c ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ U d τ . Since E π t 0 [ ∥ Y ∥ p ] 1 /p is finite for any p ∈ [1 , ∞ ] (where E [ ∥ · ∥ p ] 1 /p is read as ess sup ∥ · ∥ when p = ∞ ), applying Lemma 18 to ( 48 ) yields W p ( µ x , µ y ) ≤ E π t  ∥ x t − y t ∥ p  1 /p = E π 0  ∥ x t − y t ∥ p  1 /p ≤ e − ct E π 0  ∥ x 0 − y 0 ∥ p  1 /p + ℓ Z t 0 e − c ( t − τ ) ∥ u x ( τ ) − u y ( τ ) ∥ U d τ . T aking the infimum with respect to π 0 ∈ Π( µ x 0 , µ y 0 ) leads to ( 47 ) for any p ∈ [1 , ∞ ] . 13 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T Let ( X, d ) be a complete, separable metric space. For any finite p ≥ 1 , the space ( P p ( X ) , W p ) is complete [ Bolley , 2008 ]. Moreov er , when p = + ∞ , also ( P b ∞ ( X ) , W ∞ ) is known to be complete [ Giv ens and Shortt , 1984 ], with P b ∞ ( X ) being the space of probability measures with bounded support. Thus, if u x ( t ) and u y ( t ) are identical and constant, and ˆ Σ is constant, then an equilibrium distrib ution of the Fokker -Planck equation is globally exponentially stable for any finite p ≥ 1 . Corollary 15 (Global Exponential Stability of Equilibrium Distribution) . Given F : R n × U → R n and constant ˆ Σ ∈ R n × r , consider the pair of SDEs ( 46 ) , wher e u x ( t ) ≡ u y ( t ) ≡ ¯ u for some constant ¯ u ∈ U . If the same assumptions as Theor em 14 hold, then 1. for each p ∈ [1 , ∞ ] and any finite t ≥ 0 , any pair of solutions to the F okker -Planck equations µ x t ( x ) = µ x ( t, x ) and µ y t ( y ) = µ y ( t, y ) with the initial distributions µ x (0 , x ) = µ x 0 ( x ) and µ y (0 , y ) = µ y 0 ( y ) with finite p - moments satisfies W p ( µ x t , µ y t ) ≤ e − ct W p ( µ x 0 , µ y 0 ); 2. for any p ∈ [1 , ∞ ) , an equilibrium distribution µ ∗ for the F okker -Planck equation ( 45 ) is unique, and globally exponentially stable; 3. mor eover , if F ( x, ¯ u ) = −∇ f ( x ) for a continuously dif fer entiable c -str ongly con vex function f : R n → R n , and ˆ Σ = σ I n , then the equilibrium distrib ution to which the F okker -Planck equation con ver ges is the Gibbs distribution with ener gy f and temperatur e σ 2 / 2 : µ ∗ ( x ) ∝ e − 2 f ( x ) /σ 2 . Pr oof. Item 1 is a special case of Theorem 14 . Item 2 follows from Banach contraction theorem [ Bullo , 2026 , Theorem 1.6]. W e show item 3 , i.e., n X i =1 ∂ ( µ ∗ ( x ) F i ( x, ¯ u )) ∂ x i = σ 2 2 n X i =1 ∂ 2 µ ∗ ( x ) ∂ x 2 i , where F i ( x, ¯ u ) = − ∂ f ( x ) ∂ x i and µ ∗ ( x ) ∝ e − 2 f ( x ) /σ 2 . Since the constant scaling coef ficient can be canceled, it suffices to show − ∂ ∂ x i  ∂ f ( x ) ∂ x i e − 2 f ( x ) /σ 2  = σ 2 2 ∂ 2 e − 2 f ( x ) /σ 2 ∂ x 2 i . This can readily be shown by taking one round of deri vati ves in the right hand side. 6 Conclusion In this paper , we hav e developed contraction theory for SDEs driv en by deterministic inputs and stochastic noise. Giv en a weighted ℓ 2 -norm for the state space, we have sho wn that the standard ISS conditions for deterministic control systems imply NISS for the corresponding SDEs, providing a natural generalization of conv entional noise-to-state (NSS) and ISS analysis. W e have further applied our NISS analysis to estimate error bounds for stochastic equilibrium tracking under different scenarios: 1) deterministic input with a deterministic equilibrium curve, 2) stochastic input with a deterministic equilibrium curve, and 3) stochastic input with a stochastic equilibrium curve, considering two types of stochastic processes, OU process and JD process, to represent unbounded and bounded noise, respectiv ely . Finally , we ha ve studied contracti vity of SDEs with respect to a W asserstein metric and sho wn that the standard ISS conditions for deterministic control systems also imply ISS with respect to an arbitrary p -W asserstein metric. A Proof of Theor em 9 Before estimating the tracking error x t − x ⋆ ( θ ( t )) , we estimate the contractivity rate and Lipschitz constant of the cascade interconnection: ˙ ξ ( t ) = − cξ ( t ) , (49a) ˙ x ( t ) = F ( x ( t ) , u ( t )) , u ( t ) = θ ( t ) + ξ ( t ) . (49b) 14 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T Lemma 16. Given a vector field F : R n × R m → R n , assume that ther e exists a matrix P = P ⊤ ≻ 0 n × n such that A1. Normalization of P : ∥ P ∥ 2 = 1 ; A2. Contraction in x : there e xists c > 0 such that osLip 2 ,P 1 / 2 ( F ) ≤ − c , uniformly in u ∈ U ; A3. Lipschitz in u : for ∥ · ∥ U = ∥ · ∥ 2 , ther e exists ℓ > 0 such that Lip U → 2 ,P 1 / 2 ( F ) ≤ ℓ , uniformly in x ∈ R n . Then, with r espect to the state ( ξ , x ) , the cascade inter connection ( 49 ) is str ongly infinitesimally contracting with rate c 2 > 0 in the weighted norm ∥ · ∥ 2 ,P 1 / 2 c ℓ , P c ℓ :=  ℓ c I m 0 m × n 0 n × m c ℓ P  uniformly in u ∈ R m . Pr oof. Define P ε :=  ε − 1 I m 0 m × n 0 n × m εP  , ε > 0 . Similarly to [ Bullo , 2026 , E2.40], we hav e osLip 2 ,P 1 / 2 ε  − c id F  ≤ max {− c, osLip x 2 ,P 1 / 2 ( F ) } + ε 2 Lip u 2 ( P 1 / 2 F ) ≤ − c + εℓ 2 ∥ P 1 / 2 ∥ 2 = − c + εℓ 2 . The statement holds by selecting ε = c ℓ . Pr oof of Theor em 9 . Consider an auxiliary dynamics of the SDE ( 24 ) and ( 25 ): u t = θ ( t ) + ξ t , dξ t = − cξ t dt + σ ξ √ m d B ξ t , (50a) dx t = F ( x t , u t ) dt + v ( t ) dt + Σ( x t , u t ) d B x t . (50b) When v ( t ) ≡ 0 n , this is nothing b ut ( 24 ) and ( 25 ) . When v ( t ) ≡ ˙ x ⋆ ( θ ( t )) and noise free (i.e., d B ξ t and d B x t are both identically equal to zero and ξ 0 = 0 m ), we hav e x t = x ⋆ ( θ ( t )) . Namely , x ⋆ ( θ ( t )) is a solution to ( 50 ). W e consider applying item 2 of Theorem 7 to the auxiliary SDE ( 50 ) . From Lemma 16 , with respect to the state ( ξ , x ) , the vector field  − cξ F ( x, θ + ξ ) + v  is strongly infinitesimally contracting with rate c 2 > 0 in the weighted norm ∥ · ∥ 2 ,P 1 / 2 c ℓ , uniformly in θ ∈ U and v ∈ R n , and with respect to the input v , the Lipschitz constant is 1 . Also, from the bounded dispersion assumption (Assumption A3 ), we hav e trace  σ ξ √ m 0 m × n 0 m × n Σ( x, u )  ⊤  ℓ c I m 0 m × n 0 n × m c ℓ P   σ ξ √ m 0 m × n 0 m × n Σ( x, u )  ! = ℓ cm σ 2 ξ trace ( I m ) + c ℓ trace(Σ( x, u ) ⊤ P Σ( x, u )) ≤ ℓ c σ 2 ξ + c ℓ σ 2 x . Applying item 2 of Theorem 7 to the auxiliary SDE ( 50 ), we hav e for ( 24 ) and ( 25 ) and x ⋆ ( θ ( t )) , E       ξ t x t  −  0 m x ⋆ ( θ ( t ))      2 2 ,P 1 / 2 c ℓ  ≤ E       ξ 0 x 0  −  0 m x ⋆ ( θ (0))      2 2 ,P 1 / 2 c ℓ  e − cαt + 1 α ℓ c σ 2 ξ c + c ℓ σ 2 x c ! (1 − e − cαt ) + 1 c (1 − α ) Z t 0 e − cα ( t − τ ) ∥ ˙ x ⋆ ( θ ( τ )) ∥ 2 2 , ( c ℓ P ) 1 / 2 dτ . (51) From ∥ ˙ x ⋆ ( θ ( τ )) ∥ 2 ,P 1 / 2 ≤ ℓ c ∥ ˙ θ ( τ ) ∥ 2 , we hav e ∥ ˙ x ⋆ ( θ ( τ )) ∥ 2 2 , ( c ℓ P ) 1 / 2 ≤ ℓ c ∥ ˙ θ ( τ ) ∥ 2 2 . (52) Also, from the definition of P c ℓ , we obtain c ℓ E [ ∥ x t − x ⋆ ( θ ( t )) ∥ 2 2 ,P 1 / 2 ] ≤ E       ξ t x t  −  0 m x ⋆ ( θ ( t ))      2 2 ,P 1 / 2 c ℓ  . (53) Combining ( 51 )–( 53 ) yields ( 35 ). Finally , ( 27 ) is obtained by taking the limit superior of ( 35 ). 15 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T B Proof of Theor em 10 Pr oof. Consider an auxiliary dynamics of the SDE ( 24 ): dx t = F ( x t , u t ) dt + Σ( x t , u t ) d B x t + v ( θ ( t ) , ξ t , u t ) dt + Λ( u t ) d B ξ t . (54) Note that v ( t ) dt in ( 34 ) is replaced by v ( θ ( t ) , ξ t , u t ) dt + Λ( u t ) d B v t . If v ( θ ( t ) , ξ t , u t ) and Λ( u t ) are identically equal to zero, this is nothing but ( 24 ). When d B x t is identical to zero, ( 54 ) becomes dx t = F ( x t , u t ) dt + v ( θ ( t ) , ξ t , u t ) dt + Λ( u t ) d B ξ t . (55) W e first show that x ⋆ ( u t ) satisfies this, i.e., ( 30 ) with ( 31 ). By the Itô formula, the k th component of x ⋆ ( u t ) satisfies dx ⋆ k ( u t ) = m X i =1 ∂ x ⋆ k ∂ u i ( u t ) du i,t + 1 2 m X i =1 m X j =1 ∂ 2 x ⋆ k ∂ u i ∂ u j ( u t ) du i,t du j,t , where d B ξ i,t d B ξ j,t = δ i,j dt and dtdt = d B ξ i,t dt = dtd B ξ i,t = 0 m . From ( 25 ), we hav e m X i =1 ∂ x ⋆ k ∂ u i ( u t ) du i,t = ∂ x ⋆ k ∂ u ( u t ) du t = ∂ x ⋆ k ∂ u ( u t )  ( ˙ θ ( t ) − cξ t ) dt + σ ξ √ m d B ξ t  , and m X i =1 m X j =1 ∂ 2 x ⋆ k ∂ u i ∂ u j ( u t ) du i,t du j,t = σ 2 ξ m trace  Hess( x ⋆ k ( u t ))  dt. Thus, x ⋆ ( u t ) is a solution to ( 55 ) for ( 31 ). W e consider applying item 2 of Theorem 7 to the auxiliary dynamics ( 54 ) . W e need to concern v ( θ ( t ) , ξ t , u t ) and Λ( u t ) . From boundedness and Lipschitzness of ∂ 2 x ⋆ i ∂ θ 2 , i = 1 , . . . , m (Assumption A6 ), they satisfy the Lipschitz continuity and linear growth assumptions. Next, from the bounded dispersion assumption (Assumption A3 in Theorem 7 ) and ( 31b ), we ha ve trace  Σ( x, u ) 0 n × m 0 n × r Λ( u )  ⊤  P − P − P P   Σ( x, u ) 0 n × m 0 n × r Λ( u )  = trace(Σ( x, u ) ⊤ P Σ( x, u )) + trace(Λ( u ) ⊤ P Λ( u )) ≤ σ 2 x + σ 2 ξ m trace  ∂ x ⋆ ∂ v ( v ) ⊤ P ∂ x ⋆ ∂ v ( v )  ≤ σ 2 x + ℓ 2 c 2 σ 2 ξ ∥ P ∥ 2 = σ 2 x + ℓ 2 c 2 σ 2 ξ . Thus, repeating a similar calculation as ( 19 ) for the auxiliary SDE ( 54 ), we hav e for ( 24 ) and ( 30 ), L∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2 ≤ − 2 c ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2 + σ 2 x + ℓ 2 c 2 σ 2 ξ + 2 ∥ x t − x ⋆ ( u t ) ∥ 2 ,P 1 / 2 ∥ v ( θ ( t ) , ξ t , u t ) ∥ 2 ,P 1 / 2 ≤ − 2 cα ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2 + σ 2 x + ℓ 2 c 2 σ 2 ξ + 1 2 c (1 − α ) ∥ v ( θ ( t ) , ξ t , u t ) ∥ 2 ,P 1 / 2 a.s. for each α ∈ (0 , 1) . Similarly to the proof of Theorem 7 , we ha ve E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( u 0 ) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c + ℓ 2 c 2 σ 2 ξ 2 c ! (1 − e − 2 cαt ) + 1 1 − α 1 2 c Z t 0 e − 2 cα ( t − τ ) E  ∥ v ( θ ( τ ) , ξ τ , u τ ) ∥ 2 2 ,P 1 / 2  dτ . (56) for each α ∈ (0 , 1) . 16 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T It remains to estimate an upper bound on E  ∥ v ( θ ( t ) , ξ t , u t ) ∥ 2 2 ,P 1 / 2  . From ( 29 ) and ( 31a ) with Y oung’ s inequality , we hav e ∥ v ( θ ( t ) , ξ t , u t ) ∥ 2 2 ,P 1 / 2 ≤ 2     ∂ x ⋆ k ∂ u ( u t )( ˙ θ ( t ) − cξ t )     2 2 ,P 1 / 2 + σ 4 ξ 2 m 2           trace  Hess( x ⋆ 1 ( u t ))  . . . trace  Hess( x ⋆ n ( u t ))            2 2 ,P 1 / 2 ≤ 2 ℓ 2 c 2 ∥ ˙ θ ( t ) − cξ t ∥ 2 2 + h 2 OU 2 σ 4 ξ . Compute with Y oung’ s inequality , ∥ ˙ θ ( t ) − cξ t ∥ 2 2 ≤ 2 ∥ ˙ θ ( t ) ∥ 2 2 + 2 c 2 ∥ ξ t ∥ 2 2 . For the OU process, we ha ve E  ∥ ξ t ∥ 2 2  = e − 2 ct E  ∥ ξ 0 ∥ 2 2  + σ 2 ξ 2 c (1 − e − 2 ct ) . In summary , we obtain E [ ∥ v ( θ ( t ) , ξ t , u t ) ∥ 2 2 ,P 1 / 2 ] ≤ 2 ℓ 2 c 2 E [ ∥ ˙ θ ( t ) − cξ t ∥ 2 2 ] + h 2 OU 2 σ 4 ξ . = 4 ℓ 2 e − 2 ct E  ∥ ξ 0 ∥ 2 2  + 2 ℓ 2 σ 2 ξ c (1 − e − 2 ct ) + 4 ℓ 2 c 2 ∥ ˙ θ ( t ) ∥ 2 2 + h 2 OU 2 σ 4 ξ . (57) Combining ( 56 ) and ( 57 ) yields E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( u 0 ) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α σ 2 x 2 c + ℓ 2 c 2 σ 2 ξ 2 c ! (1 − e − 2 cαt ) + 2 1 − α ℓ 2 c E  ∥ ξ 0 ∥ 2 2  Z t 0 e − 2 cα ( t − τ ) e − 2 cτ dτ + 1 1 − α ℓ 2 c σ 2 ξ c Z t 0 e − 2 cα ( t − τ ) (1 − e − 2 cτ ) dτ + 2 1 − α ℓ 2 c 3 Z t 0 e − 2 cα ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + 1 1 − α h 2 OU 4 σ 4 ξ c Z t 0 e − 2 cα ( t − τ ) dτ Computing the time integrations, we ha ve ( 36 ). Finally , ( 28 ) is obtained by taking the limit superior of ( 36 ). C Proof of Theor em 11 Pr oof. Consider an auxiliary dynamics of the SDE ( 24 ) with ( 37 ): du t = − c ( u t − θ ( t )) dt + σ u diag( u t ⊙ ( a − u t )) 1 2 d B u t , (58a) dx t = F ( x t , u t ) dt + v ( t ) dt + Σ( x t , u t ) d B x t . (58b) When v ( t ) ≡ 0 n , this is nothing but ( 24 ) with ( 37 ) . When v ( t ) ≡ ˙ x ⋆ ( θ ( t )) and noise free (i.e., d B u t and d B x t are both identical to zero and u t ≡ θ ( t ) ), we hav e x t = x ⋆ ( θ ( t )) . Namely , x ⋆ ( θ ( t )) is a solution to ( 58 ). Similarly to the proof of Theorem 11 , we consider applying item 2 of Theorem 7 to the auxiliary SDE ( 58 ) . From the Feller condition (Assumption A7 ), u -dynamics has a weak solution staying in ( 0 m , a ) . Therefore, from the Lipschitz continuity and linear growth assumptions for the drift vector field F and the dispersion matrix Σ (Assumption 3 ), the auxiliary dynamics ( 58 ) satisfies the linear growth condition on ( 0 m , a ) , and thus a weak solution exists for any u 0 ∈ ( 0 m , a ) . In particular , u t ∈ ( 0 m , a ) for any t > 0 almost surely . Moreover , we can apply the Dynkin’ s formula (Proposition 6 ). Since u t ∈ (0 , a ) , we have trace(diag( u t ⊙ ( a − u t ))) ≤ ∥ a ∥ 2 2 4 . 17 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T Also, from the bounded dispersion assumption (Assumption A3 ), we hav e ( 59 ). trace  σ u diag( u t ⊙ ( a − u t )) 1 2 0 m × n 0 m × n Σ( x, u )  ⊤  ℓ c I m 0 m × n 0 n × m c ℓ P   σ u diag( u t ⊙ ( a − u t )) 1 2 0 m × n 0 m × n Σ( x, u )  ! = ℓ c σ 2 u trace(diag( u t ⊙ ( a − u t ))) + c ℓ trace(Σ( x, u ) ⊤ P Σ( x, u )) ≤ ℓ c ∥ a ∥ 2 2 4 σ 2 u + c ℓ σ 2 x . (59) Applying item 2 of Theorem 7 to the auxiliary SDE ( 58 ), we hav e for ( 24 ) with ( 37 ) and x ⋆ ( θ ( t )) , E       u t x t  −  θ ( t ) x ⋆ ( θ ( t ))      2 2 ,P 1 / 2 c ℓ  ≤ E       u 0 x 0  −  θ (0) x ⋆ ( θ (0))      2 2 ,P 1 / 2 c ℓ  e − cαt + 1 α  ℓ c ∥ a ∥ 2 2 4 σ 2 u + c ℓ σ 2 x  (1 − e − cαt ) + 1 c (1 − α ) Z t 0 e − cα ( t − τ ) ∥ ˙ x ⋆ ( θ ( τ )) ∥ 2 2 , ( c ℓ P ) 1 / 2 dτ . (60) Combining ( 52 ), ( 53 ), and ( 60 ) yields ( 42 ). Finally , ( 38 ) is obtained by taking the limit superior of ( 42 ). D Proof of Theor em 12 Pr oof. For an auxiliary dynamics ( 54 ) of the SDE ( 24 ) , in the Jacobi diffusion case, we ha ve ( 41 ) . W e consider applying item 2 of Theorem 7 to the auxiliary dynamics ( 54 ) . From the Feller condition (Assumption A7 ), u -dynamics has a weak solution staying in ( 0 m , a ) . Also, for continuity of ∂ 2 x ⋆ i ∂ u 2 , i = 1 , . . . , m (Assumption A5 ), v and Λ in ( 41 ) are bounded (when u ∈ ( 0 m , a ) ). Therefore, from the Lipschitz continuity and linear growth assumptions for the drift vector field F and the dispersion matrix Σ (Assumption 3 ), the auxiliary dynamics ( 54 ) satisfies the linear growth condition on ( 0 m , a ) , and thus a weak solution e xists for any u 0 ∈ ( 0 m , a ) . In particular , u t ∈ ( 0 m , a ) for any t > 0 almost surely . Moreover , we can apply the Dynkin’ s formula (Proposition 6 ). Next, from the bounded dispersion assumption (Assumption A3 ) and ( 41b ), we ha ve trace  Σ( x, u ) 0 n × m 0 n × r Λ( u )  ⊤  P − P − P P   Σ( x, u ) 0 n × m 0 n × r Λ( u )  = trace(Σ( x, u ) ⊤ P Σ( x, u )) + σ 2 u trace  diag( u t ⊙ ( a − u t )) 1 2 ∂ x ⋆ ∂ u ( u ) ⊤ P ∂ x ⋆ ∂ u ( u ) diag ( u t ⊙ ( a − u t )) 1 2  ≤ σ 2 x + 3 ∥ a ∥ 2 2 4 ℓ 2 c 2 σ 2 u ∥ P ∥ 2 = σ 2 x + 3 ∥ a ∥ 2 2 4 ℓ 2 c 2 σ 2 u . Repeating a procedure for deriving ( 56 ) in Theorem 10 , we ha ve E  ∥ x t − x ⋆ ( u t ) ∥ 2 2 ,P 1 / 2  ≤ E  ∥ x 0 − x ⋆ ( u 0 ) ∥ 2 2 ,P 1 / 2  e − 2 cαt + 1 α  σ 2 x 2 c + 3 ∥ a ∥ 2 2 4 ℓ 2 c 2 σ 2 u 2 c  (1 − e − 2 cαt ) + 1 1 − α 1 2 c Z t 0 e − 2 cα ( t − τ ) E  ∥ v ( θ ( τ ) , u τ ) ∥ 2 2 ,P 1 / 2  dτ (61) for each α ∈ (0 , 1) . It remains to compute E  ∥ v ( θ ( t ) , u t ) ∥ 2 2 ,P 1 / 2  . From ( 40 ) and ( 41a ), we hav e ∥ v ( θ ( t ) , u t ) ∥ 2 2 ,P 1 / 2 ≤ 2     ∂ x ⋆ ∂ θ ( u t )( − c ( u t − θ ( t )))     2 2 ,P 1 / 2 + 1 2 σ 4 u      m X i =1 u t,i ( a i − u t,i ) ∂ 2 x ⋆ ( u t ) ∂ u 2 t,i      2 2 ,P 1 / 2 ≤ 2 ℓ 2 ∥ u t − θ ( t ) ∥ 2 2 + h 2 JD 2 σ 4 u . 18 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T W e compute an upper bound on E [ ∥ u t − θ ( t ) ∥ 2 2 ] . From the Itô formula, we hav e L∥ u t − θ ( t ) ∥ 2 2 = 2( u t − θ ( t )) ⊤ ( du t − ˙ θ ( t ) dt ) + σ 2 u trace(diag( u t ⊙ ( a − u t ))) dt = − 2 c ∥ u t − θ ( t ) ∥ 2 2 dt − 2( u t − θ ( t )) ⊤ ˙ θ ( t ) dt + σ 2 u trace(diag( u t ⊙ ( a − u t ))) dt + 2 σ u ( u t − θ ( t )) ⊤ diag( u t ⊙ ( a − u t )) 1 2 d B u t ≤ − c ∥ u t − θ ( t ) ∥ 2 2 dt + 1 c ∥ ˙ θ ( t ) ∥ 2 2 dt + ∥ a ∥ 2 2 4 σ 2 u dt + 2 σ u ( u t − θ ( t )) ⊤ diag( u t ⊙ ( a − u t )) 1 2 d B u t where in the inequality , the Y oung’ s inequality and trace(diag( u t ⊙ ( a − u t ))) ≤ ∥ a ∥ 2 2 4 are used. Similarly to the proof of Theorem 7 , this leads to E  ∥ u t − θ ( t ) ∥ 2 2  ≤ e − ct E  ∥ u 0 − θ (0) ∥ 2 2  + 1 c Z t 0 e − c ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + ∥ a ∥ 2 2 4 σ 2 u c (1 − e − ct ) . In summary , we ha ve E  ∥ v ( θ ( t ) , u t ) ∥ 2 2 ,P 1 / 2  ≤ 2 ℓ 2 e − ct E  ∥ u 0 − θ (0) ∥ 2 2  + ℓ 2 ∥ a ∥ 2 2 2 σ 2 u c (1 − e − ct ) + 2 ℓ 2 c Z t 0 e − c ( t − τ ) ∥ ˙ θ ( τ ) ∥ 2 2 dτ + h 2 JD 2 σ 4 u . (62) Combining ( 61 ) and ( 62 ), we hav e ( 39 ). Finally , ( 39 ) is obtained by taking the limit superior of ( 43 ). E Minko wski’ s Inequality In this section ∥ · ∥ denotes an arbitrary norm on R n . Lemma 17 (Minkowski’ s inequality [ Billingsley , 1995 , Problems 5.10]) . Let Y and Z be random variables with values in R n on a pr obability space (Ω , F , P ) , and let E denote the expectation oper ator on R n . If E [ ∥ Y ∥ p ] 1 /p and E [ ∥ Z ∥ p ] 1 /p ar e finite for some p ∈ [1 , ∞ ] (wher e E [ ∥ · ∥ p ] 1 /p is r ead as ess sup ∥ · ∥ when p = ∞ ), then for such p , E [ ∥ Y + Z ∥ p ] 1 /p ≤ E [ ∥ Y ∥ p ] 1 /p + E [ ∥ Z ∥ p ] 1 /p . (63) Lemma 18 (Minko wski-type bound) . Let X and Y be random variables with values in R n on a pr obability space (Ω , F , P ) . Let b ∈ R and let p ∈ [1 , ∞ ] . Assume E [ ∥ Y ∥ p ] < ∞ and ∥ X ∥ ≤ ∥ Y ∥ + | b | a.s. (64) Then E [ ∥ X ∥ p ] < ∞ and E [ ∥ X ∥ p ] 1 /p ≤ E [ ∥ Y ∥ p ] 1 /p + | b | . Pr oof. The case p = ∞ is immediate from ( 64 ) , since taking essential suprema preserves the inequality . Now let p ∈ [1 , ∞ ) . From ( 64 ) and monotonicity of t 7→ t p on R ≥ 0 , ∥ X ∥ p ≤ ( ∥ Y ∥ + | b | ) p a.s. T aking expectations (which also sho ws E [ ∥ X ∥ p ] < ∞ ) and p th roots: E [ ∥ X ∥ p ] 1 /p ≤ E [( ∥ Y ∥ + | b | ) p ] 1 /p . Applying Minko wski’ s inequality to the random variables ∥ Y ∥ and the constant | b | giv es E [( ∥ Y ∥ + | b | ) p ] 1 /p ≤ E [ ∥ Y ∥ p ] 1 /p + | b | . This completes the proof. 19 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T References A. Davydov , V . Centorrino, A. Gokhale, G. Russo, and F . Bullo. T ime-varying con ve x optimization: A contrac- tion and equilibrium tracking approach. IEEE T ransactions on Automatic Control , 70(11):7446–7460, 2025. doi: 10.1109/T A C.2025.3576043 . W . Lohmiller and J.-J. E. Slotine. On contraction analysis for non-linear systems. Automatica , 34(6):683–696, 1998. doi: 10.1016/S0005-1098(98)00019-3 . F . Bullo. Contraction Theory for Dynamical Systems . Kindle Direct Publishing, 1.3 edition, 2026. ISBN 979- 8836646806. URL https://fbullo.github.io/ctds . R. I. Kachurovskii. Monotone operators and con ve x functionals. Uspekhi Matematicheskikh Nauk , 15(4):213–215, 1960. G. Qu and N. Li. On the exponential stability of primal-dual gradient dynamics. IEEE Contr ol Systems Letters , 3(1): 43–48, 2019. doi: 10.1109/LCSYS.2018.2851375 . A. Gokhale, A. Da vydov , and F . Bullo. Contractivity of distrib uted optimization and Nash seeking dynamics. IEEE Contr ol Systems Letters , 7:3896–3901, 2023. doi: 10.1109/LCSYS.2023.3341987 . A. Allibhoy and J. Cortés. Anytime solvers for variational inequalities: The (recursi ve) safe monotone flo ws. Automatica , 177:112287, 2025. doi: 10.1016/j.automatica.2025.112287 . J. W ang and N. Elia. A control perspectiv e for centralized and distributed con vex optimization. In IEEE Conf. on Decision and Contr ol and Eur opean Control Confer ence , pages 3800–3805, Orlando, USA, 2011. doi: 10.1109/CDC.2011.6161503 . A. Cherukuri, B. Gharesifard, and J. Cortes. Saddle-point dynamics: Conditions for asymptotic stability of saddle points. SIAM Journal on Contr ol and Optimization , 55(1):486–511, 2017. doi: 10.1137/15M1026924 . J. Cortés and S. K. Niederländer . Distributed coordination for nonsmooth con ve x optimization via saddle-point dynamics. Journal of Nonlinear Science , 29(4):1247–1272, 2019. doi: 10.1007/s00332-018-9516-4 . N. K. Dhingra, S. Z. Khong, and M. R. Jo vanovi ´ c. The proximal augmented Lagrangian method for nonsmooth compos- ite optimization. IEEE T ransactions on A utomatic Contr ol , 64(7):2861–2868, 2019. doi: 10.1109/T AC.2018.2867589 . Q. C. Pham, N. T abareau, and J.-J. E. Slotine. A contraction theory approach to stochastic incremental stability . IEEE T ransactions on A utomatic Control , 54(4):816–820, 2009. doi: 10.1109/tac.2008.2009619 . H. Tsukamoto, S.-J. Chung, and J.-J. E Slotine. Contraction theory for nonlinear stability analysis and learning-based control: A tutorial ov erview . Annual Reviews in Contr ol , 52:135–169, 2021. doi: 10.1016/j.arcontrol.2021.10.001 . N. T abareau, J.J. Slotine, and Q.-C. Pham. How synchronization protects from noise. PLoS Computational Biology , 6 (1):e1000637, 2010. doi: 10.1371/journal.pcbi.1000637 . N. M. Boffi and J.-J. E. Slotine. A continuous-time analysis of distributed stochastic gradient. Neural Computation , 32 (1):36–96, 2020. doi: 10.1162/neco_a_01248 . P . H. A. Ngoc. Contraction of stochastic differential equations. Communications in Nonlinear Science and Numerical Simulation , 95:105613, 2021. doi: 10.1016/j.cnsns.2020.105613 . P . H. A. Ngoc and L. T . Hieu. Contraction analysis of stochastic functional differential equations. International Journal of Systems Science , pages 1–10, 2026. doi: 10.1080/00207721.2026.2625788 . S. S. Ahmad and S. Raha. On estimation of transient stochastic stability of linear systems. Stochastics and Dynamics , 10(03):385–405, 2010. doi: 10.1142/s0219493710003017 . Z. Aminzare. Stochastic logarithmic Lipschitz constants: A tool to analyze contracti vity of stochastic dif ferential equations. IEEE Control Systems Letter s , 6:2311–2316, 2022. doi: 10.1109/LCSYS.2022.3148945 . Z. Aminzare and V . Sriv astav a. Stochastic synchronization in nonlinear network systems driven by intrinsic and coupling noise. Biological Cybernetics , 116(2):147–162, 2022. doi: 10.1007/s00422-022-00928-7 . C. V illani. Optimal T ransport: Old and New . Springer , 2009. doi: 10.1007/978-3-540-71050-9 . F . Bolley , I. Gentil, and A. Guillin. Conv ergence to equilibrium in W asserstein distance for Fokk er-Planck equations. Journal of Functional Analysis , 263(8):2430–2457, 2012. doi: 10.1016/j.jfa.2012.07.007 . J. Bouvrie and J.-J. Slotine. W asserstein contraction of stochastic nonlinear systems. arXiv preprint , 2019. L. Conger , F . Hoffmann, E. Mazumdar, and L. J. Ratlif f. Monotone multispecies flo ws. arXiv pr eprint , 2025. doi: 10.48550/arXiv .2506.22947 . 20 Incremental ISS and Equilibrium T racking for Stochastic Contracting Dynamics A P R E P R I N T A. Simonetto, E. Dall’Anese, S. Paternain, G. Leus, and G. B. Giannakis. T ime-varying con ve x optimiza- tion: T ime-structured algorithms and applications. Pr oceedings of the IEEE , 108(11):2032–2048, 2020. doi: 10.1109/JPR OC.2020.3003156 . E. Dall’Anese, A. Simonetto, S. Becker , and L. Madden. Optimization and learning with information streams: Time-v arying algorithms and applications. IEEE Signal Processing Ma gazine , 37(3):71–83, 2020. doi: 10.1109/MSP .2020.2968813 . Z. Marvi, F . Bullo, and A. G. Alleyne. Control barrier proximal dynamics: A contraction theoretic approach for safety verification. IEEE Contr ol Systems Letters , 8:880–885, 2024. doi: 10.1109/LCSYS.2024.3402188 . Z. Marvi, F . Bullo, and A. G. Alleyne. Discrete control barrier proximal dynamics: Quantized multi-actuator and sampled-data systems. Automatica , 2025a. Submitted. M. Fazlyab, S. Paternain, V . M. Preciado, and A. Ribeiro. Prediction-correction interior-point method for time-varying con vex optimization. IEEE T ransactions on Automatic Contr ol , 63(7):1973–1986, 2018. doi: 10.1109/T A C.2017.2760256 . Z. Marvi, F . Bullo, and A. G. Alleyne. Air cooled battery pack thermal management via control barrier proximal dynamics. IEEE T ransactions on Contr ol Systems T echnology , June 2025b. Conditionally accepted. I. Karatzas and S. Shrev e. Br ownian Motion and Stochastic Calculus . Springer , 2014. ISBN 978-1-4612-0949-2. B. Øksendal. Stochastic Differ ential Equations: an Intr oduction with Applications . Springer , 6 edition, 2013. ISBN 9783662036204. S. Coskun and R. K orn. Modeling the Intraday Electricity Demand in Germany , pages 3–23. Springer International Publishing, 2021. ISBN 9783030627324. doi: 10.1007/978-3-030-62732-4_1 . H. Brezis. Functional Analysis, Sobolev Spaces and P artial Differ ential Equations . Springer , 2011. ISBN 9780387709147. doi: 10.1007/978-0-387-70914-7 . H. K. Khalil. Nonlinear Systems . Prentice Hall, 3 edition, 2002. ISBN 0130673897. F . Bolley . Separability and completeness for the W asserstein distance , pages 371–377. Lecture Notes in Mathematics. 2008. ISBN 9783540779131. doi: 10.1007/978-3-540-77913-1_17 . C. R. Gi vens and R. M. Shortt. A class of W asserstein metrics for probability distrib utions. Michigan Mathematical Journal , 31(2), 1984. doi: 10.1307/mmj/1029003026 . P . Billingsley . Probability and Measur e . John Wile y & Sons, 3 edition, 1995. ISBN 0-471-00710-2. 21