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The aim of this work is to present a mathematical framework for the study of flickering inputs in visual processing tasks. When combined with geometric patterns, these inputs influence and induce interesting psychophysical phenomena, such as the MacKay and the Billock-Tsou effects, where the subjects perceive specific afterimages typically modulated by the flickering frequency. Due to the symmetry-breaking structure of the inputs, classical bifurcation theory and multi-scale analysis techniques are not very effective in our context. We thus take an approach based on the input-output framework of control theory for Amari-type neural fields. This allows us to prove that, when driven by periodic inputs, the dynamics converge to a periodic state. Moreover, we study under which assumptions these nonlinear dynamics can be effectively linearised, and in this case we present a precise approximation of the integral kernel for short-range excitatory and long-range inhibitory neuronal interactions. Finally, for inputs concentrated at the center of the visual field with a flickering background, we directly relate the width of the illusory contours appearing in the afterimage with both the flickering frequency and the strength of the inhibition., Comment: 21 pages, 8 figures

This paper investigates the online estimation of neural activity within the primary visual cortex (V1) in the framework of observability theory. We focus on a low-dimensional neural fields modeling hypercolumnar activity to describe activity in V1. We utilize the average cortical activity over V1 as measurement. Our contributions include detailing the model's observability singularities and developing a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance. The study emphasizes the intrinsic link between the model's non-linear nature and its observability. We also present numerical experiments highlighting the different properties of the observer., Comment: 23 pages, 4 figures

This paper focuses on the modeling of experiments conducted by Billock and Tsou [V. A. Billock and B. H. Tsou, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 8490--8495] using an Amari-type neural field that models the average membrane potential of neuronal activity in the primary visual cortex (V1). The study specifically focuses on a regular funnel pattern localized in the fovea or the peripheral visual field. It aims to comprehend and model the visual phenomena induced by this pattern, emphasizing their nonlinear nature. The research involves designing sensory inputs that mimic the visual stimuli from Billock and Tsou's experiments. The cortical outputs induced by these sensory inputs are then theoretically and numerically studied to assess their ability to model the experimentally observed visual effects at the V1 level. A crucial aspect of this study is the exploration of the effects induced by the nonlinear nature of neural responses. By highlighting the significance of excitatory and inhibitory neurons in the emergence of these visual phenomena, the research suggests that an interplay of both types of neuronal activities plays a crucial role in visual processes, challenging the assumption that the latter is primarily driven by excitatory activities alone.

This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849--850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay's psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay's funnel pattern "MacKay rays". From a control theory point of view, the Amari-type equation's exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.

Growing evidence indicates that only a sparse subset from a pool of sensory neurons is active for the encoding of visual stimuli at any instant in time. Traditionally, to replicate such biological sparsity, generative models have been using the $\ell_1$ norm as a penalty due to its convexity, which makes it amenable to fast and simple algorithmic solvers. In this work, we use biological vision as a test-bed and show that the soft thresholding operation associated to the use of the $\ell_1$ norm is highly suboptimal compared to other functions suited to approximating $\ell_q$ with $0 \leq q < 1 $ (including recently proposed Continuous Exact relaxations), both in terms of performance and in the production of features that are akin to signatures of the primary visual cortex. We show that $\ell_1$ sparsity produces a denser code or employs a pool with more neurons, i.e. has a higher degree of overcompleteness, in order to maintain the same reconstruction error as the other methods considered. For all the penalty functions tested, a subset of the neurons develop orientation selectivity similarly to V1 neurons. When their code is sparse enough, the methods also develop receptive fields with varying functionalities, another signature of V1. Compared to other methods, soft thresholding achieves this level of sparsity at the expense of much degraded reconstruction performance, that more likely than not is not acceptable in biological vision. Our results indicate that V1 uses a sparsity inducing regularization that is closer to the $\ell_0$ pseudo-norm rather than to the $\ell_1$ norm.

To study the interaction between retinal stimulation by redundant geometrical patterns and the cortical response in the primary visual cortex (V1), we focus on the MacKay effect (Nature, 1957) and Billock and Tsou's experiments (PNAS, 2007). We use a controllability approach to describe these phenomena starting from a classical biological model of neuronal field equations with a non-linear response function. The external input containing a localised control function is interpreted as a cortical representation of the static visual stimuli used in these experiments. We prove that while the MacKay effect is essentially a linear phenomenon (i.e., the nonlinear nature of the activation does not play any role in its reproduction), the phenomena reported by Billock and Tsou are wholly nonlinear and depend strongly on the shape of the nonlinearity used to model the response function.

Understanding sensory-induced cortical patterns in the primary visual cortex V1 is an important challenge both for physiological motivations and for improving our understanding of human perception and visual organisation. In this work, we focus on pattern formation in the visual cortex when the cortical activity is driven by a geometric visual hallucination-like stimulus. In particular, we present a theoretical framework for sensory-induced hallucinations which allows one to reproduce novel psychophysical results such as the MacKay effect (Nature, 1957) and the Billock and Tsou experiences (PNAS, 2007).

In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov--Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland--Stein operator, that we prove in this paper and has an interest on its own., Comment: 44 pages

The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to refine the auditory cortex model introduced in [9], and inspired by the geometrical modelling of vision. The algorithm transforms the degraded sound in an 'image' in the time-frequency domain via a short-time Fourier transform. Such an image is then lifted in the Heisenberg group and it is reconstructed via a Wilson-Cowan differo-integral equation. Numerical experiments on a library of speech recordings are provided, showing the good reconstruction properties of the algorithm., Comment: arXiv admin note: substantial text overlap with arXiv:2004.02450

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide upper bounds for this rate of decay consistent with previously known lower bounds and analogous stability results for more general classes of persistently excited signals. The strategy of proof consists in relating the worst decay rate to optimal control questions and studying in details their solutions. As a byproduct of our analysis, we also obtain estimates for the worst $L_2$-gain of the time-varying linear control systems $\dot x=-cc^\top x+u$, where the signal $c$ is persistently excited, thus solving an open problem posed by A. Rantzer in 1999., Comment: 27 pages

The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to propose a mathematical model of sound reconstruction based on the functional architecture of the auditory cortex (A1). The model is inspired by the geometrical modelling of vision, which has undergone a great development in the last ten years. There are however fundamental dissimilarities, due to the different role played by the time and the different group of symmetries. The algorithm transforms the degraded sound in an 'image' in the time-frequency domain via a short-time Fourier transform. Such an image is then lifted in the Heisenberg group and it is reconstructed via a Wilson-Cowan differo-integral equation. Preliminary numerical experiments are provided, showing the good reconstruction properties of the algorithm on synthetic sounds concentrated around two frequencies.

We consider the evolution model proposed in [9, 6] to describe illusory contrast perception phenomena induced by surrounding orientations. Firstly, we highlight its analogies and differences with the widely used Wilson-Cowan equations [48], mainly in terms of efficient representation properties. Then, in order to explicitly encode local directional information, we exploit the model of the primary visual cortex (V1) proposed in [20] and largely used over the last years for several image processing problems [24,38,28]. The resulting model is thus defined in the space of positions and orientation and it is capable to describe assimilation and contrast visual bias at the same time. We report several numerical tests showing the ability of the model to reproduce, in particular, orientation-dependent phenomena such as grating induction and a modified version of the Poggendorff illusion. For this latter example, we empirically show the existence of a set of threshold parameters differentiating from inpainting to perception-type reconstructions and describing long-range connectivity between different hypercolumns in V1., Comment: This is the revised extended invited journal version of the SSVM 2019 conference proceeding arXiv:1812.07425

In this work we have aimed to reproduce supra-threshold perception phenomena, specifically visual illusions, with Wilson-Cowan-type models of neuronal dynamics. We have found that it is indeed possible to do so, but that the ability to replicate visual illusions is related to how well the neural activity equations comply with the efficient representation principle. Our first contribution is to show that the Wilson-Cowan equations can reproduce a number of brightness and orientation-dependent illusions, and that the latter type of illusions require that the neuronal dynamics equations consider explicitly the orientation, as expected. Then, we formally prove that there can't be an energy functional that the Wilson-Cowan equations are minimizing, but that a slight modification makes them variational and yields a model that is consistent with the efficient representation principle. Finally, we show that this new model provides a better reproduction of visual illusions than the original Wilson-Cowan formulation.

In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carath\'eodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma\subset \mathbb S^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function $\upsilon$ we construct a singular Riemannian structure whose spectrum is discrete and satisfies \[ N(\lambda) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} \upsilon(\lambda). \] Examples of slowly varying functions are $\log\lambda$, its iterations $\log_k \lambda = \log_{k-1}\log\lambda$, any rational function with positive coefficients of $\log_k \lambda$, and functions with non-lo\-ga\-rithmic growth such as $\exp\left((\log \lambda)^{\alpha_1} \dots (\log_k \lambda)^{\alpha_k} \right)$ for $\alpha_i \in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest., Comment: 39 pages. v2: added references and examples. v3: streamlined exponsition, additional comments and remarks. Final version accepted on Journal de Math\'ematiques Pures et Appliqu\'ees

In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $\Delta$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold $N$, whose associated Laplace-Beltrami operator is never essentially self-adjoint on $C^\infty_0(N\setminus\{q_0\})$, if $\dim N\le 3$. We then apply this result to the Schr\"odinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

We consider a differential model describing neuro-physiological contrast perception phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [10] largely used over the last years to describe neuron interactions in the primary visual cortex (V1) and applied to several image processing problems [12,19,13]. Our model connects to Wilson-Cowan-type equations [23] and it is analogous to the one used in [3,2,14] to describe assimilation and contrast phenomena, the main novelty being its explicit dependence on local image orientation. To confirm the validity of the model, we report some numerical tests showing its ability to explain orientation-dependent phenomena (such as grating induction) and geometric-optical illusions [21,16] classically explained only by filtering-based techniques [6,18].

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$., Comment: 20 pages, v2 minor corrections, to appear on J. Math. Pures Appl

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to reconstruct only images that our visual system is still capable of recognize, we show that those of the second type completely transcend such limitation providing reconstructions at the state-of-the-art in image inpainting. This can be interpreted as a validation of the fact that our visual cortex actually encodes the first type of algorithm.

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points., Comment: 21 pages, 1 figure

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice., Comment: 123 pages, revised version

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on., Comment: 15 pages, 2 figures

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold $M$ equipped with a smooth measure $\omega$, possibly degenerate or singular near the metric boundary of $M$, and in presence of a real-valued potential $V\in L^2_{\mathrm{loc}}(M)$. The main merit of this paper is the identification of an intrinsic quantity, the effective potential $V_{\mathrm{eff}}$, which allows to formulate simple criteria for quantum confinement. Let $\delta$ be the distance from the possibly non-compact metric boundary of $M$. A simplified version of the main result guarantees quantum completeness if $V\ge -c\delta^2$ far from the metric boundary and \[ V_{\mathrm{eff}}+V\ge \frac3{4\delta^2}-\frac{\kappa}{\delta}, \qquad \text{close to the metric boundary}. \] These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of $M$; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [Boscain, Laurent - Ann. Inst. Fourier, 2013] ., Comment: 40 pages, 7 figures. (V2) corrected typos and updated references. (V3) corrected typos and updated references. Final version to appear on Journal of Spectral Theory

In this paper we prove a sub-Riemannian version of the classical Santal\'o formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g. CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a "reduction procedure" that allows to consider only a simple subset of sub-Riemannian geodesics. As an application, we derive isoperimetric-type and (p-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $\lambda_1(M)$ of the sub-Laplacian, \[ \lambda_1(M) \geq \frac{k \pi^2}{L^2}, \] in terms of the rank $k$ of the distribution and the length $L$ of the longest reduced sub-Riemannian geodesic contained in $M$. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations: \[ \mathbb{S}^1\hookrightarrow \mathbb{S}^{2d+1} \xrightarrow{p} \mathbb{CP}^d, \qquad \mathbb{S}^3\hookrightarrow \mathbb{S}^{4d+3} \xrightarrow{p} \mathbb{HP}^d, \qquad d \geq 1, \] where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = \pi$ and $k=2d$ or $4d$, respectively., Comment: 29 pages, 3 figures. Final version to appear on Journal of Differential Geometry

In this paper we propose a supervised object recognition method using new global features and inspired by the model of the human primary visual cortex V1 as the semidiscrete roto-translation group $SE(2,N) = \mathbb Z_N\rtimes \mathbb R^2$. The proposed technique is based on generalized Fourier descriptors on the latter group, which are invariant to natural geometric transformations (rotations, translations). These descriptors are then used to feed an SVM classifier. We have tested our method against the COIL-100 image database and the ORL face database, and compared it with other techniques based on traditional descriptors, global and local. The obtained results have shown that our approach looks extremely efficient and stable to noise, in presence of which it outperforms the other techniques analyzed in the paper., Comment: 16 pages, 5 figures -- Added acknowledgements

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art., Comment: 15 pages, 10 figures

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term $E\log E$. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator, Comment: Revised version. 18 pages, 6 figures

In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities., Comment: 28 pages, 5 figures

We study the evolution of the heat and of a free quantum particle (described by the Schr\"odinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{-2\alpha}d\theta^2$, where $x\in \mathbb R$, $\theta\in\mathbb T$ and the parameter $\alpha\in\mathbb R$. For $\alpha\le-1$ this metric describes cone-like manifolds (for $\alpha=-1$ it is a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha\ge 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $\Delta$ is essentially self-adjoint if and only if $\alpha\notin(-3,1)$. In this case the only self-adjoint extension is the Friedrichs extension $\Delta_F$, that does not allow communication through the singular set $\{x=0\}$ both for the heat and for a quantum particle. For $\alpha\in(-3,-1]$ we show that for the Schr\"odinger equation only the average on $\theta$ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is $\Delta_F$) cannot. For $\alpha\in(-1,1)$ we prove that there exists a canonical self-adjoint extension $\Delta_B$, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the $L^1$ norm for the heat equation) of the Markovian extensions $\Delta_F$ and $\Delta_B$, proving that $\Delta_F$ is stochastically complete at the singularity if and only if $\alpha\le -1$, while $\Delta_B$ is always stochastically complete at the singularity., Comment: 29 pages, 2 figures, accepted version

We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean distance of the value function associated with the $L^1$ cost of the control-affine system $\dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q)$, satisfying the strong H\"ormander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the $L^1$ cost of time-dependent systems of the form $\dot q = \sum_{j=1}^m u_j f_j^t(q)$., Comment: 25 pages, some corrections