Probabilistic interpretation for system of conservation law arising in adhesion particle dynamics

This Note presents a construction of a solution for the nonlinear stochastic differential equation X_t = X₀ + ∫₀^t $\text{E}$[u₀(X₀)|X_s]ds, t ≥ 0. The random variable X₀ with values in $\text{R}$ and the function u₀ are given. We denote by P_t the probability distribution of X_t and u(x, t) = $\text{E}$[u₀(X₀)|X_t = x]. We prove that (P_t, u(·, t), t ≥ 0) is a weak solution for system of conservation law arising in adhesion particle dynamics.     

Fractional part sums and divisor functions

Elementary methods are used to study sums of the form $\text{Σ}{}_{\mathrm{d\le x}}\text{{}\text{x}\text{d}\text{}}{}^{\mathrm{t}}$ for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form $\text{Σ}{}_{\mathrm{d\le x}}\text{d}{}^{\mathrm{p}}\text{P}{}_{\mathrm{t}}\text{(}\text{x}\text{d}\text{)}$ for integers p and t, t > 0, where P_t(x) = B_t({x}) and B_t(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σ_n≤xn^tσ_a(n) and Σ_n≤xE_t(n) for integers t and a, a ≥ 0, where σ_a(n) is the sum of the ath powers of the divisors of n and E_t(x) is the error term in the sum Σ_n≤xn^tσ_a(n).     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Some results about the cross-correlation function between two maximal linear sequences

Let {a₁} and {a_d1} be two maximal linear sequences of period pⁿ ? 1. The cross-correlation function is defined by C_d(t) =

for t = 0, t…pⁿ ? 2, where $\text{ζ = exp(2π}\text{1}\text{p}\text{)}$. We find some new general results about C_d(t). We also determine the values and the number of occurences of each value of C_d(t) for several new values of d.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) ∈ Rⁿ:t ∈ R^d} we describe the covariance functions of all scaling limits Y(t) = $\text{L}$lim_α↓0 B^?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions $\text{r(t, s) = EY(t) Y}{}^1\text{(s)}$ are found to be homogeneous in the sense that for some matrix L and each α > 0, $\text{(}{}^1\text{) r(αt, αs) = α}{}^{\mathrm{L1}}\text{r(t, s) α}{}^{\mathrm{L}}$. Processes with stationary increments satisfying (¹) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The functional differential equation x′(t) = x(x(t))

A classification of the solutions of the functional differential equation x′(t) = x(x(t)) is given and it is proved that every solution either vanishes identically or is strictly monotonie. For monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x(t₀) = x₀ where (t₀, x₀) is any given pair of reals in some specified subset of $\text{R}$². Every monotonically increasing solution is thus obtained. It is analytic and depends analytically on t₀ and x₀. Only for t₀ = x₀ = 1 is the question of analyticity still open.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Unitarization of symplectics and stability for causal differential equations in Hilbert space

Let Sp($\text{H}$) be the symplectic group for a complex Hibert space $\text{H}$. Its Lie algebra sp($\text{H}$) contains an open invariant convex cone C₀; each element of C₀ commutes with a unique sympletic complex structure. The Cayley transform $\text{C}$: X∈ sp($\text{H}$)→(I + X)¹∈ Sp($\text{H}$) is analyzed and compared with the exponential mapping. As an application we consider equations of the form $\text{(}\text{d}\text{dt}\text{) S = A(t)S,}\text{where}\text{t → A(t) ?}\text{C}\text{?}{}_0$ is strongly continuous, and show that if ∝_?∞^∞ ∥A(t)∥ dt < 2 and ∝_{? t8}^∞A(t) dt?C₀, the (scattering) operator

, where S_t′(t) is the solution such that S_t′(t′) = I, is in the range of $\text{B}$ restricted to C₀. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp($\text{H}$) to a unitary operator.     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.