Most Likely Trajectory: Optimality Under Uncertainty

• Most likely trajectory is the representative evolution path selected by an explicit optimality criterion, distinguishing it from deterministic or mean-path approaches.
• It encompasses varied methods such as relative entropy minimization, action minimization, and top-ranked proposal selection to handle uncertainty.
• Applications span Schrödinger bridges with killing, rare-event transition theory, and data-driven ranking in ocean drifters, highlighting its practical impact in research.

Searching arXiv for recent and directly relevant papers on “most likely trajectory” across domains. “Most likely trajectory” is a domain-dependent term for a trajectory, path, or evolution selected by an explicit optimality principle under uncertainty. In the research literature, the phrase does not denote a single universal construction. Instead, it refers to distinct but structurally related objects: a minimum-relative-entropy path-space law in Schrödinger bridge theory for diffusing and vanishing particles (Chen et al., 2021); a minimum-action transition path in small-noise large deviations for distribution-dependent stochastic systems (Wei et al., 2021); a maximum-probability path on a Markov transition graph for ocean drifters (O'Malley et al., 2020); a best-fit physically constrained trajectory under weighted least squares for meteoroid dynamics (Jansen-Sturgeon et al., 2019) and coronal ejecta tracking (Liewer et al., 2020); a deterministic nominal continuation inferred from online state estimation in interception and maneuver prediction (Bhise et al., 2020, Verma et al., 8 Jan 2025); a highest-ranked trajectory proposal in multimodal forecasting systems (Almeida et al., 2023, Jun et al., 19 May 2026); and, in monitored quantum dynamics, the dominant measurement record or noise-conditioned path obtained from a trajectory probability functional (Delmonte et al., 29 Sep 2025, Kokaew et al., 2022). Across these settings, the common idea is selection of one representative evolution from many admissible possibilities, but the selection criterion varies sharply with the underlying probabilistic, variational, or physical model.

1. Relative-entropy formulations on path space

In one major line of work, “most likely trajectory” is defined in the original Schrödinger sense: among all path-space laws consistent with observed endpoint marginals, choose the one that is closest to a prior stochastic law in relative entropy (Chen et al., 2021). For a classical Schrödinger bridge with diffusion prior

$dX_t = b(t,X_t)dt + \sigma(t,X_t)dW_t,$

with diffusion matrix

$a(t,x)=\sigma(t,x)\sigma(t,x)',$

the bridge solves

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$

Here the “most likely” evolution is the minimum-relative-entropy path law relative to the prior law $R$ (Chen et al., 2021).

The unbalanced extension for diffusing and vanishing particles generalizes this definition to priors with killing. The prior Fokker–Planck equation becomes

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$

or equivalently

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$

Because killing destroys mass, the endpoint marginals are unbalanced. The construction therefore augments the state space with a coffin state

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$

and works on path space

$\mathbf\Omega = D([0,1],\mathcal X).$

This converts a lossy transport problem into a probability-constrained path-space problem on an augmented space (Chen et al., 2021).

The resulting generalized Schrödinger bridge problem is

$$\mu^\star := {\rm arg}\min_{\mu\in \mathcal P(\mathbf \Omega)} \left\{H(\mu\mid R)~\mid~ \mu_0 = p_0, \mu_1 = p_1\right\},$$

with augmented marginals

$$p_0=(\rho_0(\cdot),0), \qquad p_1=\left(\rho_1(\cdot),\,1-\int_{\mathbb R^n}\rho_1(x)\,dx\right).$$

In this formulation, the most likely evolution describes not only transport of surviving particles but also when and where particles most likely vanish (Chen et al., 2021).

The optimizer has the multiplicative endpoint form

$a(t,x)=\sigma(t,x)\sigma(t,x)',$0

and is justified by a large-deviation principle: $a(t,x)=\sigma(t,x)\sigma(t,x)',$1 This gives a direct probabilistic meaning to “most likely”: among empirical path laws compatible with the observed marginals, the entropy minimizer is exponentially most probable (Chen et al., 2021).

2. Variational trajectories from large deviations and action minimization

A second major usage appears in Freidlin–Wentzell theory, where the most likely trajectory is the minimum-action path of a rare transition. For the McKean–Vlasov stochastic system

$a(t,x)=\sigma(t,x)\sigma(t,x)',$2

with $a(t,x)=\sigma(t,x)\sigma(t,x)',$3, the most likely transition path from $a(t,x)=\sigma(t,x)\sigma(t,x)',$4 to $a(t,x)=\sigma(t,x)\sigma(t,x)',$5 over fixed time $a(t,x)=\sigma(t,x)\sigma(t,x)',$6 is defined as the minimizer of the action

$a(t,x)=\sigma(t,x)\sigma(t,x)',$7

subject to

$a(t,x)=\sigma(t,x)\sigma(t,x)',$8

The associated quasipotential is

$a(t,x)=\sigma(t,x)\sigma(t,x)',$9

(Wei et al., 2021).

For the class treated there, the action functional is

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$0

where the skeleton equation is

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$1

A key reduction occurs when $$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$2 and $$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$3 is an equilibrium of $$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$4. Then $$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$5, and the action reduces to that of an ordinary SDE with modified drift

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$6

In that case, the distribution-dependent problem can be computed through the non-distribution-dependent SDE

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$7

(Wei et al., 2021).

This notion of “most likely” is explicitly distinct from deterministic drift following. The deterministic reduced system

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$8

has zero action along its own orbits, but the most likely transition path is the path minimizing the action among rare trajectories that connect metastable states against the drift (Wei et al., 2021). The paper derives the corresponding Euler–Lagrange equation

$$P^\star := {\rm arg}\min_{P\in \mathcal P(\Omega)} \left\{H(P\mid R)\mid P_0=\rho_0,\;P_1=\rho_1\right\}.$$9

and computes solutions with the adaptive minimum action method (Wei et al., 2021).

A related but formally different variational use appears in unscented trajectory optimization, where the paper does not define a statistical mode path and instead optimizes what it calls the “most desirable tychastic trajectory” under uncertainty. The uncertain dynamics are

$R$0

and the optimization may penalize endpoint covariance

$R$1

or chance-constraint risk. This suggests a different but nearby interpretation: in some control settings, the representative trajectory is not the most probable one but the one that is most desirable under a chosen risk-sensitive criterion (Ross et al., 2024).

3. Dynamic laws, posterior corrections, and control interpretations

In the unbalanced Schrödinger bridge with killing, the optimal law remains Markovian and changes both drift and killing rate (Chen et al., 2021). The generalized Schrödinger system is

$R$2

$R$3

$R$4

$R$5

with boundary couplings

$R$6

$R$7

The one-time marginals are

$R$8

The optimal diffusion law is

$R$9

with posterior killing rate

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$0

Thus the most likely trajectory is not merely conditioned survival; it is a full posterior law over both surviving and vanished particles, with an inferred state-dependent killing mechanism (Chen et al., 2021).

The same object admits a fluid-dynamic and control-theoretic characterization: $$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$1 subject to

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$2

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$3

The stationarity conditions give

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$4

This suggests a general pattern: in several fields, “most likely trajectory” can be equivalently described either as a probabilistic posterior object or as a solution to a deterministic control problem with an action or entropy cost (Chen et al., 2021).

A different control-centered use appears in free-flight aircraft convergence. There the trajectory judged most appropriate is the minimum-time, bounded-turn, safe-separation convergence path toward a leader’s route. The Hamiltonian is affine in the turn-rate control, producing bang-bang structure and regular turn-straight-turn trajectories, later approximated online by a neural network (Shahzad et al., 2013). That work does not use a probabilistic definition, but it shares the same structural idea: a trajectory is selected by an explicit optimality principle under constraints.

4. Data-driven maximum-probability paths and best-fit trajectories

In data-driven ocean transport, the “most likely path” is defined directly as a maximum-probability path on a Markov chain inferred from drifter data (O'Malley et al., 2020). Continuous positions are discretized by

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$5

and an empirical transition matrix is estimated: $$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$6 For a path $$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$7,

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$8

The most likely path is

$$\partial_t R_t+ \nabla\cdot(b R_t) + V R_t = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2(a_{ij}R_t)}{\partial x_i\partial x_j},$$9

Taking logarithms gives a shortest-path problem with weights

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$0

so the maximum-probability path is exactly a shortest path in the transformed graph (O'Malley et al., 2020). This notion differs sharply from minimum-time or mean-path descriptions: the path can be circuitous in Euclidean space if dominant currents make that route more probable (O'Malley et al., 2020).

The same literature also estimates an expected travel time conditional on following the inferred path. For each step $$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$1,

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$2

and

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$3

This shows that “most likely path” and “fastest path” are not the same object (O'Malley et al., 2020).

A conceptually different data-driven maximum-selection procedure appears in vessel trajectory reconstruction from AIS data (Chen et al., 2020). There the next point of a vessel is chosen as the “best possible next point” using a local 3D feature space of time difference, scaled forward/backward prediction error, and space-time angle. For moving pairs, the score is

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$4

with angle screening by

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$5

Points are then linked by their selected next points to reconstruct complete trajectories (Chen et al., 2020). A plausible implication is that in reconstruction problems without explicit global probabilities, “most likely trajectory” may be operationalized through repeated local best-match decisions.

In observational astronomy and heliophysics, the term often collapses into a best-fit trajectory under a physical model. For fireballs, the Dynamic Trajectory Fit treats the most plausible trajectory as the bounded weighted least-squares optimum of a 3D physical flight model fitted directly to multi-sensor lines of sight (Jansen-Sturgeon et al., 2019). The state includes position, velocity, ballistic coefficient, ablation coefficient, fragmentation parameters, timing offsets, and missing times, and the fit minimizes weighted along-track and cross-track angular residuals (Jansen-Sturgeon et al., 2019). For coronal ejecta observed by WISPR, the preferred 3D trajectory is the radial constant-speed parameter set

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$6

that best matches the measured angle tracks $$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$7 under weighted nonlinear least squares (Liewer et al., 2020). In both cases, “most likely” is effectively “best-fitting physically admissible trajectory,” not a path-space entropy or action minimizer.

5. Machine learning, ranking, and nominal extrapolation

In multimodal trajectory forecasting, “most likely trajectory” often denotes the highest-ranked candidate among generated proposals rather than a closed-form mode of a density. In the context-free clustering-based forecaster of (Almeida et al., 2023), the pipeline is: displacement-space transformation, clustering, cluster-conditioned proposal generation, and proposal ranking. One trajectory proposal is generated per cluster: $$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$8 Probabilities are assigned post hoc by

$$\partial_t R_t = -\nabla\cdot(bR_t) + \frac12 \sum_{i,j}\partial_{x_i x_j}^2(a_{ij}R_t) - VR_t.$$9

and the implied top-ranked trajectory is

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$0

The paper is explicit that this top-ranked generated proposal is its operational “most likely trajectory” (Almeida et al., 2023).

A recent egocentric prediction system uses a closely related top-1 selection idea, but with future camera motion as the latent intent variable (Jun et al., 19 May 2026). The trajectory representation is a “16-knot relative 6-DoF trajectory”

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$1

with each knot storing

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$2

At test time, the system retrieves a bank of plausible future trajectories, evaluates them with a gate-then-rank scorer, and either falls back to a no-trajectory predictor or uses the top-ranked candidate. In that setup, the most likely trajectory is the highest-ranked retrieved trajectory candidate conditioned on the current egocentric context (Jun et al., 19 May 2026).

By contrast, some forecasting papers use “most likely trajectory” only in a deterministic nominal sense. In high-speed interception, an extended Kalman filter estimates current position from visual observations, then a fitted local circular-motion model is rolled forward recursively. The resulting future path is best interpreted as the nominal continuation implied by the current filtered state and motion model, not as a trajectory distribution (Bhise et al., 2020). Similarly, AISE/FS trajectory prediction extrapolates a local Frenet–Serret geometry with estimated speed, curvature, and torsion: $\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$3

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$4

That work explicitly does not define a probabilistic “most likely trajectory”; it produces a single deterministic short-horizon continuation (Verma et al., 8 Jan 2025).

A probabilistic but still geometric variant appears in lateral aircraft ground-track prediction. There the future path is parameterized by piecewise-linear control points

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$5

with normalized times

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$6

and interpolation

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$7

The learned object is a context-conditioned distribution over control points $\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$8. The paper does not define a formal most-likely decoder, but a plausible reading is that it would correspond to the most probable or central control-point configuration under the predictive model (Pepper et al., 2023). This suggests a general caveat: in modern generative forecasting, “most likely trajectory” may be a downstream decoding choice, not an intrinsic object of the model.

6. Quantum and scientific uses of dominant trajectories

In monitored quantum dynamics, “most likely trajectory” acquires a literal path-probability meaning. For continuously monitored bosonic systems, the single-step readout probability under Gaussian weak measurement is

$\mathcal X = \mathbb R^n\cup\{\mathfrak c\},$9

Approximating $\mathbf\Omega = D([0,1],\mathcal X).$0 by a sharply peaked distribution yields

$\mathbf\Omega = D([0,1],\mathcal X).$1

and, in continuous time,

$\mathbf\Omega = D([0,1],\mathcal X).$2

Equivalently, in Wiener-process form, the most likely trajectory is the one with

$\mathbf\Omega = D([0,1],\mathcal X).$3

The resulting conditioned density matrix evolves by a deterministic nonlinear equation rather than a stochastic ensemble average (Delmonte et al., 29 Sep 2025).

The same paper defines the global trajectory probability functional

$\mathbf\Omega = D([0,1],\mathcal X).$4

and a fictitious action

$\mathbf\Omega = D([0,1],\mathcal X).$5

The most likely trajectory is then the saddle point satisfying

$\mathbf\Omega = D([0,1],\mathcal X).$6

For monitored free bosons, this construction is exact; for the interacting Sine-Gordon model, it is combined with a self-consistent time-dependent harmonic approximation and reveals an entanglement phase transition from area-law to logarithmic-law scaling (Delmonte et al., 29 Sep 2025). This is a case where “most likely trajectory” is neither an action-minimizing rare-event path nor a best-fit curve, but the dominant measurement record in a monitored quantum ensemble.

A related quantum-control use appears in state preparation under dephasing noise (Kokaew et al., 2022). There a stochastic path integral is built over noise histories, and the most-likely path is the state trajectory and associated noise realization that maximize path likelihood subject to fixed initial and final states. In the qubit proof of concept, the extremal equations imply a most-likely noise path $\mathbf\Omega = D([0,1],\mathcal X).$7, yielding analytic Rabi controls for arbitrary target states (Kokaew et al., 2022). This sharply contrasts with mean-path Lindblad control, which optimizes average fidelity over all noise realizations.

Outside mathematical trajectory inference, the phrase can also mean an inferred physical evolutionary path. In GW170817, the “most likely trajectory” refers to the post-merger evolution of the compact remnant, with the authors concluding that the system most likely formed a black hole well before day 107 rather than remaining a long-lived neutron star (Pooley et al., 2017). A plausible implication is that, in astrophysical discourse, the term may designate the most plausible physical scenario consistent with multi-epoch observations rather than a formal mathematical path.

7. Misconceptions, distinctions, and cross-domain synthesis

A recurring misconception is that “most likely trajectory” always means the deterministic trajectory of the noiseless system. The large-deviation formulation for McKean–Vlasov systems explicitly rejects this: the most likely transition path is the minimum-action rare path, not the zero-noise orbit (Wei et al., 2021). In Schrödinger bridge theory, the object is a posterior path law on path space, not a single sample trajectory (Chen et al., 2021). In monitored quantum systems, it is the dominant readout history, not the measurement-averaged state (Delmonte et al., 29 Sep 2025). In machine learning forecasting, it may be merely the top-ranked proposal among finitely many generated futures (Almeida et al., 2023, Jun et al., 19 May 2026).

Another common confusion is between “most likely,” “mean,” and “best fit.” The drifter path literature explicitly distinguishes the most likely path from the fastest path and from geometric shortest paths (O'Malley et al., 2020). Fireball and coronal ejecta studies identify a best-fit trajectory under measurement models, which is likelihood-based only through observational residuals and model assumptions (Jansen-Sturgeon et al., 2019, Liewer et al., 2020). Interception and local motion extrapolation papers output a single deterministic continuation without a full posterior over futures (Bhise et al., 2020, Verma et al., 8 Jan 2025).

The literature also shows that pathwise likelihood can live at different levels of abstraction:

Setting	Object selected as “most likely”	Selection principle
Schrödinger bridge with killing	Path-space law	Relative entropy minimization (Chen et al., 2021)
Small-noise transition theory	Continuous path	Action minimization (Wei et al., 2021)
Ocean drifters	Discrete state path	Maximum Markov path probability (O'Malley et al., 2020)
Proposal-based forecasting	Generated future proposal	Highest ranking score/probability (Almeida et al., 2023, Jun et al., 19 May 2026)
Monitored quantum dynamics	Measurement record / conditioned state path	Saddle point of trajectory probability functional (Delmonte et al., 29 Sep 2025)
Best-fit observational dynamics	Physical trajectory parameters	Weighted least squares under a physical model (Jansen-Sturgeon et al., 2019, Liewer et al., 2020)

This suggests that “most likely trajectory” is best treated as an umbrella term whose meaning is fixed by the measure on trajectories, the admissible class, and the optimization or inference rule. Where the underlying object is a path-space law, the phrase refers to a posterior distribution. Where the underlying object is a rare event, it refers to an action minimizer. Where the underlying object is a discrete candidate set, it refers to a highest-ranked proposal. Where the model is observational and deterministic, it may simply mean the trajectory that best matches data under a chosen physical parameterization.

The broad research trajectory of the concept therefore runs from classical large deviations and entropy projection (Chen et al., 2021, Wei et al., 2021), through data-derived graph and filtering formulations (O'Malley et al., 2020, Bhise et al., 2020), to modern proposal-ranking and latent-intent systems (Almeida et al., 2023, Jun et al., 19 May 2026), and to trajectory-probability methods in monitored quantum many-body dynamics (Delmonte et al., 29 Sep 2025). This suggests a unifying editor’s term, “selection principle,” for the rule that converts a set or distribution of possible evolutions into one designated trajectory. The literature does not support a single universal definition beyond that abstraction.