stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with dirichlet boundary conditions we prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate markov diffusion process in particular our article focuses on the heston stochastic volatility process which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half plane the generator of this process with killing called the elliptic heston operator is a second order degenerate elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half plane in mathematical finance solutions to terminal boundary value or obstacle problems for the parabolic heston operator correspond to value functions for american style options on the underlying asset

degenerate stochastic differential equations with holder continuous coefficients and super markov chains we consider the operator formula math acting on functions in c 2 b r d we prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on ij and b i in contrast to previous work the b i need only be nonnegative on the boundary rather than strictly positive at the expense of the ij and b i being holder continuous applications to super markov chains are given the proof follows stroock and varadhan s perturbation argument but the underlying function space is now a weighted holder space and each component of the constant coefficient process being perturbed is the square of a bessel process

probability functionals for self consistent and invariant inference entropy and fisher information two existing methods of probabilistic inference are based on variational principles maximum entropy and minimum fisher information in each case a probability density function is inferred by setting the first variation of a functional to zero subject to information constraints this study considers whether other functionals could be used for this purpose and by starting with requirements for self consistency and invariance it is shown that the most general admissible functional is just a linear combination of entropy and fisher information with the proviso that the normal definition of fisher information is modified by the inclusion of a prior this amounts to an axiomatic derivation of entropy and fisher information the concern is with continuous random variables and both the single and multivariable cases are considered a number of examples are considered to compare inference based on entropy with that based on fisher information and to highlight the role of boundary conditions for inference based on fisher information

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

mimicking the one dimensional marginal distributions of processes having an ito differential let t be a stochastic process starting from 0 with ito differential d xi left t right delta left t omega right dw left t right beta left t omega right dt where left w left t right mathfrak f t right is a wiener process and are bounded mathfrak f processes such that t is uniformly positive definite then it is proved that there exists a stochastic differential equation dx left t right sigma left t x left t right right dw left t right b left t x left t right right dt with non random coefficients which admits a weak solutionx t having the same one dimensional probability distribution as t for everyt the coefficients andb have a simple interpretation sigma left t x right left e left delta delta t left t right xi left t right x right right tfrac 1 2 b left t x right e left beta left t right xi left t right x right

fisher based thermodynamics its legendre transform and concavity properties it is shown that the legendre transform structure of thermodynamics can be replicated without any change if one replaces the entropy s by fisher s information measure i also the important thermodynamic property of concavity is shown to be obeyed by i by this use of the fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one

fisher and jaynesian statistics compared in the description of classical fluids fisher statistics provides an information measure which like that of shannon is a functional of a distribution f the latter solves an euler equation to minimize the functional to make comparison with statistical derivations of extended thermodynamics calculations by frieden plastino and collaborators are specialized to the case where f is a phase space distribution approximating a solution of the liouville equation the euler equations are solved subject to conditions specifying values of internal energy and fluxes e g of heat and momentum the solution is compared with that of jaynes which with information theory provides an entropy model the fisher and jaynes entropies are identical only in equilibrium the distributions agree in linear approximation for a restricted set of variables this set is large enough to embrace all those needed in the classical non equilibrium description of simple fluids fisher euler solutions are not unique in non equilibrium

on the martingale problem for degenerate parabolic partial differential operators with unbounded coefficients and a mimicking theorem for ito processes using results from our companion article 26 on a schauder approach to existence of solutions to a degenerate parabolic partial differential equation we solve three intertwined problems motivated by probability theory and mathematical finance concerning degenerate diffusion processes we show that the martingale problem associated with a degenerate elliptic differential operator with unbounded locally holder continuous coefficients on a half space is wellposed in the sense of stroock and varadhan second we prove existence uniqueness and the strong markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable holder continuity properties third for an ito process with degenerate diffusion and unbounded but appropriately regular coefficients we prove existence of a strong markov process unique in the sense of probability law whose onedimensional marginal probability distributions match those of the given ito process

a schauder approach to degenerate parabolic partial differential equations with unbounded coefficients abstract motivated by applications to probability and mathematical finance we consider a parabolic partial differential equation on a half space whose coefficients are suitably holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half space we establish existence and uniqueness of solutions in weighted holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients in our companion article feehan and pop 12 we apply the main result of this article to show that the martingale problem associated with a degenerate elliptic partial differential operator is well posed in the sense of stroock and varadhan

fisher info and thermodynamics first law abstract we show starting from first principles that thermodynamics first law can be microscopically obtained for fisher s information measure without need of invoking the adiabatic theorem further it is proved that enforcing the fisher first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing fisher s information measure subject to appropriate constraints

power laws of complex systems from extreme physical information many complex systems obey allometric or power laws y y x a here y or 0 is the measured value of some system attribute a y or 0 is a constant and x is a stochastic variable remarkably for many living systems the exponent a is limited to values n 4 n 0 1 2 here x is the mass of a randomly selected creature in the population these quarter power laws hold for many attributes such as pulse rate n 1 allometry has in the past been theoretically justified on a case by case basis an ultimate goal is to find a common cause for allometry of all types and for both living and nonliving systems the principle i j extremum of extreme physical information is found to provide such a cause it describes the flow of fisher information j i from an attribute value a on the cell level to its exterior observation y data y are formed via a system channel function y identical to f x a with f x a to be found extremizing the difference i j through variation of f x a results in a general allometric law f x a identical to y y x a darwinian evolution is presumed to cause a second extremization of i j now with respect to the choice of a the solution is a n 4 n 0 1 2 defining the particular powers of biological allometry under special circumstances the model predicts that such biological systems are controlled by only two distinct intracellular information sources these sources are conjectured to be cellular dna and cellular transmembrane ion gradients

fisher transfer entropy quantifying the gain in transient sensitivity we introduce a novel measure fisher transfer entropy fte which quantifies a gain in sensitivity to a control parameter of a state transition in the context of another observable source the new measure captures both transient and contextual qualities of transfer entropy and the sensitivity characteristics of fisher information fte is exemplified for a ferromagnetic two dimensional lattice ising model with glauber dynamics and is shown to diverge at the critical point

the fisher information based uncertainty relation cramer rao inequality and kinetic energy for the d dimensional central problem the inequality with l being the grand orbital quantum number and its conjugate relation for r2 p 2 are shown to be fulfilled in the d dimensional central problem their use has allowed us to improve the fisher information based uncertainty relation i i const and the cramer rao inequalities r2i d2 p2i d2 in addition the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the fisher information in position and momentum spaces denoted by i and i respectively

c1 1 regularity for degenerate elliptic obstacle problems abstract the heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half plane the generator of this process with killing called the elliptic heston operator is a second order degenerate elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half plane in mathematical finance solutions to the obstacle problem for the elliptic heston operator correspond to value functions for perpetual american style options on the underlying asset with the aid of weighted sobolev spaces and weighted holder spaces we establish the optimal c 1 1 regularity up to the boundary of the half plane for solutions to obstacle problems for the elliptic heston operator when the obstacle functions are sufficiently smooth

special features of the relation between fisher information and schrodinger eigenvalue equation it is well known that a suggestive relation exists that links schrodinger s equation se to the information optimizing principle based on fisher s information measure the connection entails the existence of a legendre transform structure underlying the se here we show that appeal to this structure leads to a first order differential equation for the se s eigenvalues that in certain cases can be used to obtain the eigenvalues without explicitly solving the se complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue we show that the general solution is unique

c0 estimates and smoothness of solutions to the parabolic equation defined by kimura operators abstract kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics their infinitesimal generator is an elliptic differential operator whose second order coefficients matrix degenerates on the boundary of the domain in this article we consider the inhomogeneous initial value problem defined by generators of kimura diffusions and we establish c 0 estimates which allows us to prove that solutions to the inhomogeneous initial value problem are smooth up to the boundary of the domain where the operator degenerates even when the initial data is only assumed to be continuous

fisher information and nonlinear dynamics abstract the past decade has witnessed an impressive growth in the scope and variety of physical applications of fisher s information measure fim as a consequence fim is widely regarded today as providing one with a unifying principle for physics in the present paper we investigate its usefulness in connection with the description of important aspects of the behavior of dynamical systems and find that the unification referred to above can be extended to the field of these systems

existence conditions and spreading properties of extreme entropy d dimensional distributions abstract the extremization of the information theoretic measures fisher information shannon entropy tsallis entropy which complementary describe the spreading of the physical states of natural systems gives rise to fundamental equations of motion and or conservation laws at times the associated extreme entropy distributions are known for some given constraints usually moments or radial expectation values in this work first we give the existence conditions of the maxent probability distributions in a d dimensional scenario where two moments not necessarily of consecutive order are known then we find general relations which involve four elements the extremized entropy the other two information theoretic measures and the variance of the extremum density in scenarios with different dimensionalities and moment constraints

principle of maximum fisher information from hardy s axioms applied to statistical systems consider a finite sized multidimensional system in a parameter state a the system is in either a state of equilibrium or general non equilibrium and may obey either classical or quantum physics l hardy s mathematical axioms provide a basis for the physics obeyed by any such system one axiom is that the number n of distinguishable states a in the system obeys n max this assumes that n is known as deterministic prior knowledge however most observed systems suffer statistical fluctuations for which n is therefore only known approximately then what happens if the scope of the axiom n max is extended to include such observed systems it is found that the state a of the system must obey a principle of maximum fisher information i imax this is important because many physical laws have been derived assuming as a working hypothesis that i imax these derivations include uses of the principle of extreme physical information epi examples of such derivations were of the de broglie wave hypothesis quantum wave equations maxwell s equations new laws of biology e g of coulomb force directed cell development and of in situ cancer growth and new laws of economic fluctuation and investment that the principle i imax itself derives from suitably extended hardy axioms thereby eliminates its need to be assumed in these derivations thus uses of i imax and epi express physics at its most fundamental level its axiomatic basis in math

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

artinian rings with restricted primeness conditions abstract complete blocked triangular matrix rings over division rings are characterized as left artinian rings which satisfy suitably restricted primeness conditions a matrix representation is also provided for the class of nonsingular left artinian c prime rings

fisher information and the thermodynamics of scale invariant systems abstract we present a thermodynamic formulation for scale invariant systems based on the minimization with constraints of the fisher information measure in such a way a clear analogy between these systems thermal properties and those of gases and fluids is seen to emerge in a natural fashion we focus our attention on the non interacting scenario speaking thus of scale free ideal gases sfigs and present some empirical evidences regarding such disparate systems as electoral results city populations and total citations in physics journals that seem to indicate that sfigs do exist we also illustrate the way in which zipf s law can be understood in a thermodynamical context as the surface of a finite system finally we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature

legendre transform structure derived from quantum theorems abstract by recourse to i the hellmann feynman theorem and ii the virial one the information optimizing principle based on fisher s information measure uncovers a legendre transform structure associated with schrodinger s equation in close analogy with the structure that lies behind the standard thermodynamical formalism the present developments provide new evidence for the information theoretical links based on fisher s measure that exist between schrodinger s equation on the one hand and thermodynamics thermostatistics on the other hand

singular parabolic equations of second order on manifolds with singularities abstract the main aim of this article is to establish an l p theory for elliptic operators on manifolds with singularities the particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds such a theory is of importance for the study of elliptic and parabolic equations on non compact or even incomplete manifolds with or without boundary

weakly non local fluid mechanics the schrodinger equation a weakly non local extension of ideal fluid dynamics is derived from the second law of thermodynamics it is proved that in the reversible limit the additional pressure term can be derived from a potential the requirement of the additivity of the specific entropy function determines the quantum potential uniquely the relation to other known derivations of the schrodinger equation stochastic fisher information exact uncertainty is clarified

quantum trajectories emerging from classical phase space abstract starting with a classical hamiltonian function h we show that if we select among its phase space trajectories those that minimize fisher s information measure the resulting trajectories coincide with the stationary ones of their quantum counterparts h of h

integral conditions for uniqueness of solutions to degenerate parabolic equations abstract we investigate uniqueness of solutions to the initial value problem for degenerate parabolic equations posed in bounded domains where no boundary conditions are prescribed in order to obtain uniqueness we need that the solutions satisfy certain integral growth conditions which are crucially related to the degeneracy of the operator near the boundary in particular such solutions can be unbounded near the boundary our hypothesis on the behavior of the operator at the boundary is optimal in fact we show that if it fails then nonuniqueness of solutions prevails

schauder a priori estimates and regularity of solutions to boundary degenerate elliptic linear second order partial differential equations abstract we establish schauder a priori estimates and regularity for solutions to a class of boundary degenerate elliptic linear second order partial differential equations furthermore given a c smooth source function we prove c regularity of solutions up to the portion of the boundary where the operator is degenerate boundary degenerate elliptic operators of the kind described in our article appear in a diverse range of applications including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance 10 25 generators of diffusion processes arising in mathematical biology 3 11 and the study of porous media 7 8

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power

a classical perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate elliptic operators via holomorphic maps abstract we prove existence of solutions to boundary value problems and obstacle problems for degenerate elliptic linear second order partial differential operators with partial dirichlet boundary conditions using a new version of the perron method the elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by daskalopoulos and hamilton 9 in their study of the porous medium equation or the degeneracy of the heston operator 21 in mathematical finance existence of a solution to the partial dirichlet problem on a half ball where the operator becomes degenerate on the flat boundary and a dirichlet condition is only imposed on the spherical boundary provides the key additional ingredient required for our perron method surprisingly proving existence of a solution to this partial dirichlet problem with mixed boundary conditions on a half ball is more challenging than one might expect due to the difficulty in developing a global schauder estimate and due to compatibility conditions arising where the degenerate and non degenerate boundaries touch one cannot directly apply the continuity or approximate solution methods however in dimension two there is a holomorphic map from the half disk onto the infinite strip in the complex plane and one can extend this definition to higher dimensions to give a diffeomorphism from the half ball onto the infinite slab the solution to the partial dirichlet problem on the half ball can thus be converted to a partial dirichlet problem on the slab albeit for an operator which now has exponentially growing coefficients the required schauder regularity theory and existence of a solution to the partial dirichlet problem on the slab can nevertheless be obtained using previous work of the author and c pop 16 our perron method relies on weak and strong maximum principles for degenerate elliptic operators concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate elliptic operators and maximum and comparison principle estimates previously developed by the author 13