arXiv subjects

Francesco Russo

Publications and source records attributed to Francesco Russo.

At least 19 recordsLinked to original sources

arXiv2026

Thermal One-point Functions and Asymptotic CFT Data: QFT in AdS

We investigate the thermal partition and one-point functions of the three-dimensional conformal field theory dual to a massive interacting scalar field in AdS$_4$. Using thermal inversion formulas, we determine the asymptotic behaviour of the spectral density and OPE coefficients involving heavy operators at fixed spin. We first analyse these CFT data for the generalised free field, corresponding to the non-interacting bulk theory. Then we compute the first-order perturbative corrections induced by the cubic and quartic bulk interactions. The thermal observables considered here probe a sector associated with operators of large dimension and, in the bulk description, a regime dominated by states with large particle number. This regime remains comparatively unexplored even in generalised free field theory. Remarkably, the asymptotic formulas obtained from thermal inversion remain quantitatively accurate far from the asymptotic regime, describing CFT data reliably already at intermediate conformal weights. Our results show that this feature survives the inclusion of bulk interactions and provide new analytic control over heavy-state data in conformal field theories.

hep-th↗

arXiv2026

A non-local singular non-linear Fokker-Planck PDE

The focus of this paper is a non-local singular non-linear Fokker-Planck partial differential equation (PDE). The peculiarity of this PDE feature is in its divergence coefficient, which presents a product between a Besov distribution and a non-linearity. The latter involves the convolution between an integrable kernel K and the solution of the PDE, which leads to a non-locality of the first order term in the PDE. We prove existence and uniqueness of a solution to the PDE as well as continuity results on its coefficients. Previous analytical results are then applied to the study of well-posedness in law for a non-local singular McKean stochastic differential equation. As byproduct of that probabilistic representation, we establish mass conservation and positivity preserving for the PDE.

math.PR↗

arXiv2026

On the tangent degree and the degree of the tangent variety of a projective variety

The tangent degree $\tau(X)$ of a projective variety $X^n\subset\mathbb P^N$ is the number of tangent spaces to $X$ at smooth points passing through a general point of the tangent variety $Tan(X)\subseteq\mathbb P^N$, if positive and finite; it is equal to zero if $\dim(Tan(X))<2n$. In this paper we focus on general properties of $\tau(X)$ and of $deg(Tan(X))$. For example $\tau(X)\neq 1$ if $N=2n$ and, as soon as $Tan(X)$ does not coincide with the secant variety, we prove a linear lower bound for the degree of $Tan(X)$ in terms of its codimension in the spirit of the paper Ciliberto.Russo.2006. Then we consider the cases in which the previous two invariants attain the lower bounds found here, either in small dimension/codimension and/or under the smoothness assumption. Finally for $N\geq 2n+1$ we consider varieties $X^n\subset\mathbb P^N$ having $\tau(X)>1$ and provide their classification in small dimension.

math.AG↗

arXiv2025

Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)

This paper provides the time-dependent $L^2$-martingale representation of the forward stochastic integral where the driving noise is the Riemann-Liouville fractional Brownian motion with parameter $\frac{1}{2} < H < 1$ and the integrand is a square-integrable adapted process. As a by-product, we obtain the exact $L^2$-isometry of the forward stochastic integrals based on suitable conditions on time-dependent martingale representations of adapted integrands combined with the Nelson's stochastic derivative of the underlying Gaussian driving noise.

math.PR↗

arXiv2025

McKean-Vlasov equations with singular coefficients - a review of recent results

This paper focuses on recent works on McKean-Vlasov stochastic differential equations (SDEs) involving singular coefficients. After recalling the classical framework, we review existing recent literature depending on the type of singularities of the coefficients: on the one hand they satisfy some integrability and measurability conditions only, while on the other hand the drift is allowed to be a generalised function. Different types of dependencies on the law of the unknown and different noises will also be considered. McKean-Vlasov SDEs are closely related to non-linear Fokker-Planck equations that are satisfied by the law (or its density) of the unknown. These connections are often established also in this singular setting and will be reviewed here. Important tools for dealing with singular coefficients are also included in the paper, such as Figalli-Trevisan superposition principle, Zvonkin transformation, Markov marginal uniqueness, and stochastic sewing lemma.

math.PR↗

arXiv2025

Heavy-Heavy-Light Asymptotics from Thermal Correlators

We revisit the calculation of spectral densities and heavy-heavy-light (HHL) operator product expansion (OPE) coefficients in three-dimensional conformal field theories using thermal one-point functions on $S^1 \times S^2$. A central element of our analysis is a new inversion formula for one-point functions which is derived via Casimir differential equations. We develop systematic expansions of the spectral density and HHL OPE coefficients in the regime of large $\Delta_H$. We validate our analytic tools by comparing the results with the partial wave expansions of thermal one-point functions in free field theories. The algorithms developed for these expansions make full use of Casimir recursion relations, thereby extending their applicability into the heavy exchange regime. In the end, we observe excellent agreement with our analytic predictions and an improvement of up to three orders of magnitude compared to all previous leading order estimates of the CFT data even for moderate values of $\Delta_H$.

hep-th↗

arXiv2025

$C^{ 0,1}$ -It{\^o} chain rules and generalized solutions of parabolic PDEs

In this paper we first establish an It\^o formula for a finite quadratic variation process $X$ expanding $f(t,X_t),$ when $f$ is of class $C^2$ in space and is absolutely continuous in time. Second, via a Fukushima-Dirichlet decomposition we obtain an explicit chain rule for $f(t,X_t)$, when $X$ is a continuous semimartingale and $f$ is a ``quasi-strong solution'' (in the sense of approximation of classical solutions) of a parabolic PDE.

math.PR↗

arXiv2025

An entropy penalized approach for stochastic optimization with marginal law constraints. Complete version

This paper focuses on stochastic optimal control problems with constraints in law, which are rewritten as optimization (minimization) of probability measures problem on the canonical space. We introduce a penalized version of this type of problems by splitting the optimization variable and adding an entropic penalization term. We prove that this penalized version constitutes a good approximation of the original control problem and we provide an alternating procedure which converges, under a so called ''Stability Condition'', to an approximate solution of the original problem. We extend the approach introduced in a previous paperof the same authors including a jump dynamics, non-convex costs and constraints on the marginal laws of the controlled process. The interest of our approach is illustrated by numerical simulations related to demand-side management problems arising in power systems.

math.OC↗

arXiv2024

Thermal One-point Functions and Their Partial Wave Decomposition

In this work we address partial wave decompositions of thermal one-point functions in conformal field theories on $S^1 \times S^{d-1}$. With the help of Casimir differential equations we develop efficient algorithms to compute the relevant conformal blocks for an external field of arbitrary spin and with any spin exchange along the thermal circle, at least in three dimensions. This is achieved by identifying solutions to the Casimir equations with a special class of spherical functions in the harmonic analysis of the conformal group. The resulting blocks are then applied to study the decomposition of one-point functions of the scalar $\phi^2$ and the stress tensor $T$ for a three-dimensional free scalar field $\phi$. We are able to read off averaged OPE coefficients into exchanged fields of high weight and spin for a complete set of tensor structures. We also extract an asymptotic behaviour of conformal blocks and use it to analyse the density of heavy-heavy-light OPE coefficients for spinning operators, comparing it with semi-classical predictions, such as the dimensions of operators at large charge.

hep-th↗

arXiv2024

Characteristics and It{\^o}'s formula for weak Dirichlet processes: an equivalence result

The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C, $\nu$) and a It{\^o} formula type expansion of F (S), where F is a bounded function of class C2. This result connects weak solutions of path-dependent SDEs and related martingale problems. We extend this to the case when S is a weak Dirichlet process. A second aspect of the paper consists in discussing some untreated features of stochastic calculus for finite quadratic variation processes.

math.PR↗

arXiv2024

Exponential twist of probability measures: drift correction in term of a generalized gradient

In this paper we study the exponential twist, i.e. a path-integral exponential change of measure, of a Markovian reference probability measure $\P$. This type of transformation naturally appears in variational representation formulae originating from the theory of large deviations and can be interpreted in some cases, as the solution of a specific stochastic control problem. Under a very general Markovian assumption on $\P$, we fully characterize the exponential twist probability measure as the solution of a martingale problem and prove that it inherits the Markov property of the reference measure. The ''generator'' of the martingale problem shows a drift depending on a {\it generalized gradient} of some suitable {\it value function} $v$. The analysis focuses on the fact that any Markovian probability measure fulfills an {\it intrinsic martingale problem} for which no uniqueness is required.

math.PR↗

arXiv2024

Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions

In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs' condition and the existence of a (what is termed) quasi-strong solution to the Bellman-Isaacs (BI) equations. If the diffusion coefficient of the state equation is non-degenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game: moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors.

math.OC↗

arXiv2024

About semilinear low dimension Bessel PDEs

We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator$L^\delta$ with low dimension $0 < \delta < 1$. $L^\delta$ is a local operator, whose drift is thederivative of $x \mapsto \log (\vert x\vert)$:in particular it is a Schwartz distribution, whichis not the derivative of a continuous function.The solutions are intended in a duality (''weak'') sensewith respect to state space$L^2(\mathbb{R}_+, d\mu),$ $\mu$ being an invariant measure for the Bessel semigroup.

math.PR↗

arXiv2024

Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces

The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak H{\"o}rmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.

math.PR↗

arXiv2023

On complete intersections of three quadrics in $\mathbb{P}^7$

We describe explicit birational maps from some rational complete intersections of three quadrics in $\mathbb{P}^7$ to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the cohomologically associated Castelnuovo surface of general type with $K^2=2$ and $\chi=4$ (the double cover of $\mathbb{P}^2$ ramified along the discriminant curve of the net of quadrics defining the complete intersection) as the minimal model of the non ruled irreducible component of the base locus of the inverse maps. In passing we also revisit and reformulate the results in [arXiv:1706.01371] about the existence of infinitely many loci of rational complete intersection of three quadrics in $\mathbb{P}^7$ to produce explicitly some of these loci of low codimension together with many other irreducible unirational components of the Noether--Lefschetz locus.

math.AG↗

arXiv2023

The $L^2$-norm of the forward stochastic integral w.r.t. Fractional Brownian motion $H > \frac{1}{2}$

In this article, we present the exact expression of the $L^2$-norm of the forward stochastic integral driven by the multi-dimensional fractional Brownian motion with parameter $\frac{1}{2} < H < 1$. The class of integrands only requires rather weak integrability conditions compatible w.r.t. a random finite measure whose density is expressed as a second-order polynomial of the underlying driving Gaussian noise. A simple consequence of our results is the exact expression of the $L^2$-norm for the pathwise Young integral.

math.PR↗

arXiv2023

The isometry of symmetric-Stratonovich integrals w.r.t. Fractional Brownian motion $H< \frac{1}{2}$

In this work, we present a detailed analysis on the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion $B$ with parameter $\frac{1}{4} < H < \frac{1}{2}$. Our main result is a complete description of a Hilbert space of integrand processes which realizes the $L^2$-isometry where none regularity condition in the sense of Malliavin calculus is imposed. The main idea is to exploit the regularity of the conditional expectation of the tensor product of the increments $B_{t-\delta,t+\delta}\otimes B_{s-\epsilon,s+\epsilon}$ onto the Gaussian space generated by $(B_s,B_t)$ as $(\delta,\epsilon)\downarrow 0$. The Hilbert space is characterized in terms of a random Radon $\sigma$-finite measure on $[0,T]^2$ off diagonal which can be characterized as a product of a non-Markovian version of the stochastic Nelson derivatives. As a by-product, we present the exact explicit expression of the $L^2$-norm of the pathwise rough integral in the sense of Gubinelli.

math.PR↗

arXiv2023

An entropy penalized approach for stochastic control problems. Complete version

In this paper, we propose an original approach to stochastic control problems. We consider a weak formulation that is written as an optimization (minimization) problem on the space of probability measures. We then introduce a penalized version of this problem obtained by splitting the minimization variables and penalizing the discrepancy between the two variables via an entropy term. We show that the penalized problem provides a good approximation of the original problem when the weight of the entropy penalization term is large enough. Moreover, the penalized problem has the advantage of giving rise to two optimization subproblems that are easy to solve in each of the two optimization variables when the other is fixed. We take advantage of this property to propose an alternating optimization procedure that converges to the infimum of the penalized problem with a rate $O(1/k)$, where $k$ is the number of iterations. The relevance of this approach is illustrated by solving a high-dimensional stochastic control problem aimed at controlling consumption in electrical systems.

math.OC↗