Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N _X and N _Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L ^r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations

Let ${f:\Omega \rightarrow \mathbb{R}}$ be a smooth function on a domain   ${\Omega \subset \mathbb{C}^n}$ with its Hessian matrix ${\left( \frac{\partial^2 f}{\partial z^i \partial\bar{z}^j}\right)}$ positive Hermitian. In this paper, we investigate a class of partial differential equations $$\Delta \ln \det (f_{i\bar{j}}) = \beta \;\| \text{grad} \ln \det (f_{i\bar{j}}) \|^2, $$ where Δ and ${\| \cdot \|}$ are the Laplacian and tensor norm, respectively, with respect to the metric ${G = \sum f_{i\bar{j}} \,dz^i \otimes d\bar{z}^j}$ , and β > 1 is some real constant depending on the dimension n. We prove that the above PDEs have a Bernstein property when the metric G is complete, provided that ${\det (f_{i\bar{j}})}$ and the Ricci curvature are bounded.     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Levi Factors of the Special Fiber of a Parahoric Group Scheme and Tame Ramification

Let $\cal{A}$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p?>?0. Let G be a connected and reductive algebraic group over K, and let $\cal{P}$ be a parahoric group scheme over $\cal{A}$ with generic fiber ${\cal{P}}_{/K} = G$ . The special fiber ${\cal{P}}_{/k}$ is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the special fiber ${\cal{P}}_{/k}$ has a Levi factor, and that any two Levi factors of ${\cal{P}}_{/k}$ are geometrically conjugate. In the present paper, we extend a portion of this result. Following a suggestion of Gopal Prasad, we prove that if G splits over a tamely ramified extension of K, then the geometric special fiber ${\cal{P}}_{/k_{\rm{alg}}}$ has a Levi factor, where k _alg is an algebraic closure of k.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

Asymptotic analysis of a second-order singular perturbation model for phase transitions

We study the asymptotic behavior, as ${\varepsilon}$ tends to zero, of the functionals ${F^k_\varepsilon}$ introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i.e., $$F^k_\varepsilon(u):=\int\limits_{I} \left(\frac{W(u)}{\varepsilon}-k\,\varepsilon\,(u')^2+\varepsilon^3(u'')^2\right)\,dx,\quad u\in W^{2,2}(I),$$ where k?>?0 and ${W:\mathbb{R}\to[0,+\infty)}$ is a double-well potential with two potential wells of level zero at ${a,b\in\mathbb{R}}$ . By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k ₀ such that, for k?<?k ₀, and for a class of potentials W, ${(F^k_\varepsilon)}$ ??(L ¹)-converges to $$F^k(u):={\bf m}_k \, \#(S(u)),\quad u\in BV(I;\{a,b\}),$$ where m _k is a constant depending on W and k. Moreover, in the special case of the classical potential ${W(s)=\frac{(s^2-1)^2}{2}}$ , we provide an upper bound on the values of k such that the minimizers of ${F_\varepsilon^k}$ cannot develop oscillations on some fine scale and a lower bound on the values for which oscillations occur, the latter improving a previous estimate by Mizel, Peletier and Troy.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Warped product CR-submanifolds of cosymplectic manifolds

In the present paper, we study warped product CR-submanifolds of cosymplectic manifolds. It is shown that the warped product of the type ${N_\perp\times{_f}N_T}$ is trivial and obtain a characterization result for the warped product of the type ${N_T\times{_f}N_\perp}$ , where N _T and ${N_\perp}$ are invariant and anti-invariant submanifolds of a cosymplectic manifold ${\bar M}$ , respectively.     

On Rationality of W-algebras

We study the problem of classification of triples ( $ \mathfrak{g} $ ; f; k), where g is a simple Lie algebra, f its nilpotent element and k ∈ $ \mathbb{C} $ , for which the simple W-algebra W _k ( $ \mathfrak{g} $ ; f) is rational.     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.